User:Jon Awbrey/Differential Logic Archive 2004

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Differential Logic 2004

Differential Logic 2004 • Series A

Diff Log A 2004 • Note 1


One of the first things that you can do, once you
have a really decent calculus for boolean functions
or propositional logic, whatever you want to call it,
is to compute the differentials of these functions or
propositions.

Now there are many ways to dance around this idea,
and I feel like I have tried them all, before one
gets down to acting on it, and there many issues
of interpretation and justification that we will
have to clear up after the fact, that is, before
we can be sure that it all really makes any sense,
but I think this time I'll just jump in, and show
you the form in which this idea first came to me.

Start with a proposition of the form x & y, which
I graph as two labels attached to a root node, so:

o-------------------------------------------------o
|                                                 |
|                       x y                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                     x and y                     |
o-------------------------------------------------o

Written as a string, this is just the concatenation "x y".

The proposition xy may be taken as a boolean function f(x, y)
having the abstract type f : B x B -> B, where B = {0, 1} is
read in such a way that 0 means "false" and 1 means "true".

In this style of graphical representation,
the value "true" looks like a blank label
and the value "false" looks like an edge.

o-------------------------------------------------o
|                                                 |
|                                                 |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      true                       |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      false                      |
o-------------------------------------------------o

Back to the proposition xy.  Imagine yourself standing
in a fixed cell of the corresponding venn diagram, say,
the cell where the proposition xy is true, as pictured:

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /               o               \        |
|       /               /%\               \       |
|      /               /%%%\               \      |
|     o               o%%%%%o               o     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     |       x       |%%%%%|       y       |     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     o               o%%%%%o               o     |
|      \               \%%%/               /      |
|       \               \%/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o

Now ask yourself:  What is the value of the
proposition xy at a distance of dx and dy
from the cell xy where you are standing?

Don't think about it -- just compute:

o-------------------------------------------------o
|                                                 |
|                   dx o   o dy                   |
|                     / \ / \                     |
|                  x o---@---o y                  |
|                                                 |
o-------------------------------------------------o
|              (x + dx) and (y + dy)              |
o-------------------------------------------------o

To make future graphs easier to draw in Ascii land,
I will use devices like @=@=@ and o=o=o to identify
several nodes into one, as in this next redrawing:

o-------------------------------------------------o
|                                                 |
|                   x  dx y  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
|              (x + dx) and (y + dy)              |
o-------------------------------------------------o

However you draw it, these expressions follow because the
expression x + dx, where the plus sign indicates (mod 2)
addition in B, and thus corresponds to an exclusive-or
in logic, parses to a graph of the following form:

o-------------------------------------------------o
|                                                 |
|                     x    dx                     |
|                      o---o                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      x + dx                     |
o-------------------------------------------------o

Next question:  What is the difference between
the value of the proposition xy "over there" and
the value of the proposition xy where you are, all
expressed as general formula, of course?  Here 'tis:

o-------------------------------------------------o
|                                                 |
|             x  dx y  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         x y                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|           ((x + dx) & (y + dy)) - xy            |
o-------------------------------------------------o

Oh, I forgot to mention:  Computed over B,
plus and minus are the very same operation.
This will make the relationship between the
differential and the integral parts of the
resulting calculus slightly stranger than
usual, but never mind that now.

Last question, for now:  What is the value of this expression
from your current standpoint, that is, evaluated at the point
where xy is true?  Well, substituting 1 for x and 1 for y in
the graph amounts to the same thing as erasing those labels:

o-------------------------------------------------o
|                                                 |
|            dx    dy                             |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/                            |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|           ((1 + dx) & (1 + dy)) - 1&1           |
o-------------------------------------------------o

And this is equivalent to the following graph:

o-------------------------------------------------o
|                                                 |
|                     dx   dy                     |
|                      o   o                      |
|                       \ /                       |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                    dx or dy                     |
o-------------------------------------------------o

Enough for the moment.
Explanation to follow.

Diff Log A 2004 • Note 2


We have just met with the fact that
the differential of the "and" is
the "or" of the differentials.

x and y  --Diff-->  dx or dy.

o-------------------------------------------------o
|                                                 |
|                                    dx   dy      |
|                                     o   o       |
|                                      \ /        |
|                                       o         |
|        x y                            |         |
|         @          --Diff-->          @         |
|                                                 |
o-------------------------------------------------o
|        x y         --Diff-->     ((dx) (dy))    |
o-------------------------------------------------o

It will be necessary to develop a more refined analysis of
this statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of DeMorgan's rule,
it is no accident, as differentiation and negation turn out to be
closely related operations.  Indeed, one can find discussions of
logical difference calculus in the Boole-DeMorgan correspondence
and Peirce also made use of differential operators in a logical
context, but the exploration of these ideas has been hampered
by a number of factors, not the least of which being a syntax
adequate to handle the complexity of expressions that evolve.

For my part, it was definitely a case of the calculus being smarter
than the calculator thereof.  The graphical pictures were catalytic
in their power over my thinking process, leading me so quickly past
so many obstructions that I did not have time to think about all of
the difficulties that would otherwise have inhibited the derivation.
It did eventually became necessary to write all this up in a linear
script, and to deal with the various problems of interpretation and
justification that I could imagine, but that took another 120 pages,
and so, if you don't like this intuitive approach, then let that be
your sufficient notice.

Let us run through the initial example again, this time attempting
to interpret the formulas that develop at each stage along the way.

We begin with a proposition or a boolean function f(x, y) = xy.

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /               o               \        |
|       /               /`\               \       |
|      /               /```\               \      |
|     o               o`````o               o     |
|     |               |`````|               |     |
|     |               |`````|               |     |
|     |       x       |``f``|       y       |     |
|     |               |`````|               |     |
|     |               |`````|               |     |
|     o               o`````o               o     |
|      \               \```/               /      |
|       \               \`/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                       x y                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| f =                   x y                       |
o-------------------------------------------------o

A function like this has an abstract type and a concrete type.
The abstract type is what we invoke when we write things like
f : B x B -> B or f : B^2 -> B.  The concrete type takes into
account the qualitative dimensions or the "units" of the case,
which can be explained as follows.

1.  Let X be the set of values {(x), x} = {not x, x}.

2.  Let Y be the set of values {(y), y} = {not y, y}.

Then interpret the usual propositions about x, y
as functions of the concrete type f : X x Y -> B.

We are going to consider various "operators" on these functions.
Here, an operator F is a function that takes one function f into
another function Ff.

The first couple of operators that we need to consider are logical analogues
of those that occur in the classical "finite difference calculus", namely:

1.  The "difference" operator [capital Delta], written here as D.

2.  The "enlargement" operator [capital Epsilon], written here as E.

These days, E is more often called the "shift" operator.

In order to describe the universe in which these operators operate,
it will be necessary to enlarge our original universe of discourse.
We mount up from the space U = X x Y to its "differential extension",
EU = U x dU = X x Y x dX x dY, with dX = {(dx), dx} and dY = {(dy), dy}.
The interpretations of these new symbols can be diverse, but the easiest
for now is just to say that dx means "change x" and dy means "change y".
To draw the differential extension EU of our present universe U = X x Y
as a venn diagram, it would take us four logical dimensions X, Y, dX, dY,
but we can project a suggestion of what it's about on the universe X x Y
by drawing arrows that cross designated borders, labeling the arrows as
dx when crossing the border between x and (x) and as dy when crossing
the border between y and (y), in either direction, in either case.

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /       x       o       y       \        |
|       /               /`\               \       |
|      /               /```\               \      |
|     o               o`````o               o     |
|     |               |`````|               |     |
|     |         dy    |`````|    dx         |     |
|     |     <---------|--o--|--------->     |     |
|     |               |`````|               |     |
|     |               |`````|               |     |
|     o               o`````o               o     |
|      \               \```/               /      |
|       \               \`/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o

We can form propositions from these differential variables in the same way
that we would any other logical variables, for instance, interpreting the
proposition (dx (dy)) to say "dx => dy", in other words, however you wish
to take it, whether indicatively or injunctively, as saying something to
the effect that there is "no change in x without a change in y".

Given the proposition f(x, y) in U = X x Y,
the (first order) "enlargement" of f is the
proposition Ef in EU that is defined by the
formula Ef(x, y, dx, dy) = f(x + dx, y + dy).

In the example f(x, y) = xy, we obtain:

Ef(x, y, dx, dy)  =  (x + dx)(y + dy).

o-------------------------------------------------o
|                                                 |
|                   x  dx y  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef =            (x, dx) (y, dy)                 |
o-------------------------------------------------o

Given the proposition f(x, y) in U = X x Y,
the (first order) 'difference' of f is the
proposition Df in EU that is defined by the
formula Df = Ef - f, or, written out in full,
Df(x, y, dx, dy) = f(x + dx, y + dy) - f(x, y).

In the example f(x, y) = xy, the result is:

Df(x, y, dx, dy)  =  (x + dx)(y + dy) - xy.

o-------------------------------------------------o
|                                                 |
|             x  dx y  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         x y                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df =           ((x, dx)(y, dy), xy)             |
o-------------------------------------------------o

We did not yet go through the trouble to interpret this (first order)
"difference of conjunction" fully, but were happy simply to evaluate
it with respect to a single location in the universe of discourse,
namely, at the point picked out by the singular proposition xy,
in as much as if to say, at the place where x = 1 and y = 1.
This evaluation is written in the form Df|xy or Df|<1, 1>,
and we arrived at the locally applicable law that states
that f = xy = x & y  =>  Df|xy = ((dx)(dy)) = dx or dy.

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /       x       o       y       \        |
|       /               /`\               \       |
|      /               /```\               \      |
|     o               o`````o               o     |
|     |               |`````|               |     |
|     |      dy (dx)  |`````|  dx (dy)      |     |
|     |   o<----------|--o--|---------->o   |     |
|     |               |``|``|               |     |
|     |               |``|``|               |     |
|     o               o``|``o               o     |
|      \               \`|`/               /      |
|       \               \|/               /       |
|        \               |               /        |
|         \             /|\             /         |
|          o-----------o | o-----------o          |
|                        |                        |
|                      dx|dy                      |
|                        |                        |
|                        v                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                     dx   dy                     |
|                      o   o                      |
|                       \ /                       |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|xy =           ((dx) (dy))                   |
o-------------------------------------------------o

The picture illustrates the analysis of the inclusive disjunction ((dx)(dy))
into the exclusive disjunction:  dx(dy) + dy(dx) + dx dy, a proposition that
may be interpreted to say "change x or change y or both".  And this can be
recognized as just what you need to do if you happen to find yourself in
the center cell and desire a detailed description of ways to depart it.

Diff Log A 2004 • Note 3


Last time we computed what will variously be called
the "difference map", the "difference proposition",
or the "local proposition" Df_p for the proposition
f(x, y) = xy at the point p where x = 1 and y = 1.

In the universe U = X x Y, the four propositions
xy, x(y), (x)y, (x)(y) that indicate the "cells",
or the smallest regions of the venn diagram, are
called "singular propositions".  These serve as
an alternative notation for naming the points
<1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively.

Thus, we can write Df_p = Df|p = Df|<1, 1> = Df|xy,
so long as we know the frame of reference in force.

Sticking with the example f(x, y) = xy, let us compute the
value of the difference proposition Df at all of the points.

o-------------------------------------------------o
|                                                 |
|             x  dx y  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         x y                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df =        ((x, dx)(y, dy), xy)                |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                dx    dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/                            |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|xy =           ((dx) (dy))                   |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                   o                             |
|                dx |  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /         o                 |
|                \| |/          |                 |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|x(y) =          (dx) dy                      |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|             o                                   |
|             |  dx    dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /         o                 |
|                \| |/          |                 |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|(x)y =            dx (dy)                    |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|             o     o                             |
|             |  dx |  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /       o   o               |
|                \| |/         \ /                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|(x)(y) =          dx dy                      |
o-------------------------------------------------o

The easy way to visualize the values of these graphical
expressions is just to notice the following equivalents:

o-------------------------------------------------o
|                                                 |
|  x                                              |
|  o-o-o-...-o-o-o                                |
|   \           /                                 |
|    \         /                                  |
|     \       /                                   |
|      \     /                          x         |
|       \   /                           o         |
|        \ /                            |         |
|         @              =              @         |
|                                                 |
o-------------------------------------------------o
|  (x, , ... , , )       =             (x)        |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                o                                |
| x_1 x_2   x_k  |                                |
|  o---o-...-o---o                                |
|   \           /                                 |
|    \         /                                  |
|     \       /                                   |
|      \     /                                    |
|       \   /                                     |
|        \ /                       x_1 ... x_k    |
|         @              =              @         |
|                                                 |
o-------------------------------------------------o
|  (x_1, ..., x_k, ())   =         x_1 ... x_k    |
o-------------------------------------------------o

Laying out the arrows on the augmented venn diagram,
one gets a picture of a "differential vector field".

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                        |                        |
|                      dx|dy                      |
|                        |                        |
|          o-----------o | o-----------o          |
|         /             \|/             \         |
|        /       x       |       y       \        |
|       /               /|\               \       |
|      /               /`|`\               \      |
|     o               o``|``o               o     |
|     |      dy (dx)  |``v``|  dx (dy)      |     |
|     |   o-----------|->o<-|-----------o   |     |
|     |               |`````|               |     |
|     |   o<----------|--o--|---------->o   |     |
|     |      dy (dx)  |``|``|  dx (dy)      |     |
|     o               o``|``o               o     |
|      \               \`|`/               /      |
|       \               \|/               /       |
|        \               |               /        |
|         \             /|\             /         |
|          o-----------o | o-----------o          |
|                        |                        |
|                      dx|dy                      |
|                        |                        |
|                        v                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o

This really just constitutes a depiction of
the interpretations in EU = X x Y x dX x dY
that satisfy the difference proposition Df,
namely, these:

1.   x  y  dx  dy
2.   x  y  dx (dy)
3.   x  y (dx) dy
4.   x (y)(dx) dy
5.  (x) y  dx (dy)
6.  (x)(y) dx  dy

By inspection, it is fairly easy to understand Df
as telling you what you have to do from each point
of U in order to change the value borne by f(x, y).

Diff Log A 2004 • Note 4


We have been studying the action of the difference operator D,
also known as the "localization operator", on the proposition
f : X x Y -> B that is commonly known as the conjunction xy.
We described Df as a (first order) differential proposition,
that is, a proposition of the type Df : X x Y x dX x dY -> B.
Abstracting from the augmented venn diagram that illustrates
how the "models", or the "satisfying interpretations", of Df
distribute within the extended universe EU = X x Y x dX x dY,
we can depict Df in the form of a "digraph" or directed graph,
one whose points are labeled with the elements of  U =  X x Y
and whose arrows are labeled with the elements of dU = dX x dY.

o-------------------------------------------------o
|  f =                  x y                       |
o-------------------------------------------------o
|                                                 |
| Df =              x  y  ((dx)(dy))              |
|                                                 |
|           +       x (y)  (dx) dy                |
|                                                 |
|           +      (x) y    dx (dy)               |
|                                                 |
|           +      (x)(y)   dx  dy                |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                       x y                       |
|  x (y) o<------------->o<------------->o (x) y  |
|             (dx) dy    ^    dx (dy)             |
|                        |                        |
|                        |                        |
|                     dx | dy                     |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                     (x) (y)                     |
|                                                 |
o-------------------------------------------------o

Any proposition worth its salt has many equivalent ways to view it,
any one of which may reveal some unsuspected aspect of its meaning.
We will encounter more and more of these variant readings as we go.

Diff Log A 2004 • Note 5


The enlargement operator E, also known as the "shift operator",
has many interesting and very useful properties in its own right,
so let us not fail to observe a few of the more salient features
that play out on the surface of our simple example, f(x, y) = xy.

Introduce a suitably generic definition of the extended universe of discourse:

Let U = X_1 x ... x X_k and EU = U x dU = X_1 x ... x X_k x dX_1 x ... x dX_k.

For a proposition f : X_1 x ... x X_k -> B,
the (first order) "enlargement" of f is the
proposition Ef : EU -> B that is defined by:

   Ef(x_1, ..., x_k, dx_1, ..., dx_k)  =  f(x_1 + dx_1, ..., x_k + dx_k).

It should be noted that the so-called "differential variables" dx_j
are really just the same kind of boolean variables as the other x_j.
It is conventional to give the additional variables these brands of
inflected names, but whatever extra connotations we might choose to
attach to these syntactic conveniences are wholly external to their
purely algebraic meanings.

For the example f(x, y) = xy, we obtain:

   Ef(x, y, dx, dy)   =   (x + dx)(y + dy).

Given that this expression uses nothing more than the "boolean ring"
operations of addition (+) and multiplication (.), it is permissible
to "multiply things out" in the usual manner to arrive at the result:

   Ef(x, y, dx, dy)   =   x y  +  x dy  +  y dx  +  dx dy

To understand what this means in logical terms, for instance, as expressed
in a boolean expansion or a "disjunctive normal form" (DNF), it is perhaps
a little better to go back and analyze the expression the same way that we
did for Df.  Thus, let us compute the value of the enlarged proposition Ef
at each of the points in the universe of discourse U = X x Y.

o-------------------------------------------------o
|                                                 |
|                   x  dx y  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef =            (x, dx) (y, dy)                 |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                      dx    dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|xy =            (dx) (dy)                    |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                         o                       |
|                      dx |  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|x(y) =          (dx)  dy                     |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                   o                             |
|                   |  dx    dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|(x)y =           dx  (dy)                    |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                   o     o                       |
|                   |  dx |  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|(x)(y) =         dx   dy                     |
o-------------------------------------------------o

Given the sort of data that arises from this form of analysis,
we can now fold the disjoined ingredients back into a boolean
expansion or a DNF that is equivalent to the proposition Ef.

   Ef  =  xy Ef_xy  +  x(y) Ef_x(y)  +  (x)y Ef_(x)y  +  (x)(y) Ef_(x)(y)

Here is a summary of the result, illustrated by means of a digraph picture,
where the "no change" element (dx)(dy) is drawn as a loop at the point x y.

o-------------------------------------------------o
|  f =                  x y                       |
o-------------------------------------------------o
|                                                 |
| Ef =              x  y   (dx)(dy)               |
|                                                 |
|           +       x (y)  (dx) dy                |
|                                                 |
|           +      (x) y    dx (dy)               |
|                                                 |
|           +      (x)(y)   dx  dy                |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                    (dx) (dy)                    |
|                    .--->---.                    |
|                     \     /                     |
|                      \x y/                      |
|                       \ /                       |
|  x (y) o-------------->o<--------------o (x) y  |
|             (dx) dy    ^    dx (dy)             |
|                        |                        |
|                        |                        |
|                     dx | dy                     |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                     (x) (y)                     |
|                                                 |
o-------------------------------------------------o

We may understand the enlarged proposition Ef
as telling us all the different ways to reach
a model of f from any point of the universe U.

Diff Log A 2004 • Note 6


To broaden our experience with simple examples, let us now contemplate the
sixteen functions of concrete type X x Y -> B and abstract type B x B -> B.
For future reference, I will set here a few tables that detail the actions
of E and D and on each of these functions, allowing us to view the results
in several different ways.

By way of initial orientation, Table 1 lists equivalent expressions for the
sixteen functions in a number of different languages for zeroth order logic.

Table 1.  Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x : 1 1 0 0 |          |                  |          |
|         |       y : 1 0 1 0 |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
|         |         |         |          |                  |          |
| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
|         |         |         |          |                  |          |
| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
|         |         |         |          |                  |          |
| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
|         |         |         |          |                  |          |
| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
|         |         |         |          |                  |          |
| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
|         |         |         |          |                  |          |
| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
|         |         |         |          |                  |          |
| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
|         |         |         |          |                  |          |
| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
|         |         |         |          |                  |          |
| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
|         |         |         |          |                  |          |
| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
|         |         |         |          |                  |          |
| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
|         |         |         |          |                  |          |
| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
|         |         |         |          |                  |          |
| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
|         |         |         |          |                  |          |
| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
|         |         |         |          |                  |          |
| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o

The next four Tables expand the expressions of Ef and Df
in two different ways, for each of the sixteen functions.
Notice that the functions are given in a different order,
here being collected into a set of seven natural classes.

Table 2.  Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Ef | xy   | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   |  (dx)(dy)  |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   dx (dy)  |   dx  dy   |  (dx)(dy)  |  (dx) dy   |
|      |            |            |            |            |            |
| f_4  |    x (y)   |  (dx) dy   |  (dx)(dy)  |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |   dx       |   dx       |  (dx)      |  (dx)      |
|      |            |            |            |            |            |
| f_12 |    x       |  (dx)      |  (dx)      |   dx       |   dx       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       dy   |      (dy)  |       dy   |      (dy)  |
|      |            |            |            |            |            |
| f_10 |       y    |      (dy)  |       dy   |      (dy)  |       dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
|      |            |            |            |            |            |
| f_11 |   (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

Table 3.  Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Df | xy   | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
|      |            |            |            |            |            |
| f_4  |    x (y)   |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |   dx       |   dx       |   dx       |   dx       |
|      |            |            |            |            |            |
| f_12 |    x       |   dx       |   dx       |   dx       |   dx       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       dy   |       dy   |       dy   |       dy   |
|      |            |            |            |            |            |
| f_10 |       y    |       dy   |       dy   |       dy   |       dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

Table 4.  Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
|      |            |            |            |            |            |
|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
|      |            |            |            |            |            |
| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
|      |            |            |            |            |            |
| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
|      |            |            |            |            |            |
| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
|      |            |            |            |            |            |
| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|                   |            |            |            |            |
| Fixed Point Total |      4     |      4     |      4     |     16     |
|                   |            |            |            |            |
o-------------------o------------o------------o------------o------------o

Table 5.  Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |  ((x, y))  |    (y)     |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   (x, y)   |     y      |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x, y)   |    (y)     |     x      |     ()     |
|      |            |            |            |            |            |
| f_8  |    x  y    |  ((x, y))  |     y      |     x      |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
| f_12 |    x       |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
| f_10 |       y    |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x, y))  |     y      |     x      |     ()     |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |   (x, y)   |    (y)     |     x      |     ()     |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x, y)   |     y      |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |  ((x, y))  |    (y)     |    (x)     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

If the medium truly is the message,
the blank slate is the innate idea.

Diff Log A 2004 • Note 7


If you think that I linger in the realm of logical difference calculus
out of sheer vacillation about getting down to the differential proper,
it is probably out of a prior expectation that you derive from the art
or the long-engrained practice of real analysis.  But the fact is that
ordinary calculus only rushes on to the sundry orders of approximation
because the strain of comprehending the full import of E and D at once
whelm over its discrete and finite powers to grasp them.  But here, in
the fully serene idylls of ZOL, we find ourselves fit with the compass
of a wit that is all we'd ever wish to explore their effects with care.

So let us do just that.

I will first rationalize the novel grouping of propositional forms
in the last set of Tables, as that will extend a gentle invitation
to the mathematical subject of "group theory", and demonstrate its
relevance to differential logic in a strikingly apt and useful way.
The data for that account is contained in Table 4.

Table 4.  Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
|      |            |            |            |            |            |
|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
|      |            |            |            |            |            |
| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
|      |            |            |            |            |            |
| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
|      |            |            |            |            |            |
| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
|      |            |            |            |            |            |
| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|                   |            |            |            |            |
| Fixed Point Total |      4     |      4     |      4     |     16     |
|                   |            |            |            |            |
o-------------------o------------o------------o------------o------------o

The shift operator E can be understood as enacting a substitution operation
on the proposition that is given as its argument.  In our immediate example,
we have the following data and definition:

   E : (U -> B)  ->  (EU -> B),

   E :  f(x, y)  ->   Ef(x, y, dx, dy),

   Ef(x, y, dx, dy)  =  f(x + dx, y + dy).

Therefore, if we evaluate Ef at particular values of dx and dy,
for example, dx = i and dy = j, where i, j are in B, we obtain:

   E_ij : (U -> B)  ->  (U -> B),

   E_ij :    f      ->   E_ij f,

   E_ij f  =  Ef | <dx = i, dy = j>  =  f(x + i, y + j).

The notation is a little bit awkward, but the data of the Table should
make the sense clear.  The important thing to observe is that E_ij has
the effect of transforming each proposition f : U -> B into some other
proposition f' : U -> B.  As it happens, the action is one-to-one and
onto for each E_ij, so the gang of four operators {E_ij : i, j in B}
is an example of what is called a "transformation group" on the set
of sixteen propositions.  Bowing to a longstanding local and linear
tradition, I will therefore redub the four elements of this group
as T_00, T_01, T_10, T_11, to bear in mind their transformative
character, or nature, as the case may be.  Abstractly viewed,
this group of order four has the following operation table:

o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|    *     %   T_00   |   T_01   |   T_10   |   T_11   |
|          %          |          |          |          |
o==========o==========o==========o==========o==========o
|          %          |          |          |          |
|   T_00   %   T_00   |   T_01   |   T_10   |   T_11   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_01   %   T_01   |   T_00   |   T_11   |   T_10   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_10   %   T_10   |   T_11   |   T_00   |   T_01   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_11   %   T_11   |   T_10   |   T_01   |   T_00   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o

It happens that there are just two possible groups of 4 elements.
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
The other is Klein's four-group V_4 (German "Vier"), which it is.

More concretely viewed, the group as a whole pushes the set
of sixteen propositions around in such a way that they fall
into seven natural classes, called "orbits".  One says that
the orbits are preserved by the action of the group.  There
is an "Orbit Lemma" of immense utility to "those who count"
which, depending on your upbringing, you may associate with
the names of Burnside, Cauchy, Frobenius, or some subset or
superset of these three, vouching that the number of orbits
is equal to the mean number of fixed points, in other words,
the total number of points (in our case, propositions) that
are left unmoved by the separate operations, divided by the
order of the group.  In this instance, T_00 operates as the
group identity, fixing all 16 propositions, while the other
three group elements fix 4 propositions each, and so we get:
Number of orbits  =  (4 + 4 + 4 + 16) / 4  =  7.   Amazing!

Diff Log A 2004 • Note 8


We have been contemplating functions of the type f : U -> B,
studying the action of the operators E and D on this family.
These functions, that we may identify for our present aims
with propositions, inasmuch as they capture their abstract
forms, are logical analogues of "scalar potential fields".
These are the sorts of fields that are so picturesquely
presented in elementary calculus and physics textbooks
by images of snow-covered hills and parties of skiers
who trek down their slopes like least action heroes.
The analogous scene in propositional logic presents
us with forms more reminiscent of plateaunic idylls,
being all plains at one of two levels, the mesas of
verity and falsity, as it were, with nary a niche
to inhabit between them, restricting our options
for a sporting gradient of downhill dynamics to
just one of two, standing still on level ground
or falling off a bluff.

We are still working well within the logical analogue of the
classical finite difference calculus, taking in the novelties
that the logical transmutation of familiar elements is able to
bring to light.  Soon we will take up several different notions
of approximation relationships that may be seen to organize the
space of propositions, and these will allow us to define several
different forms of differential analysis applying to propositions.
In time we will find reason to consider more general types of maps,
having concrete types of the form X_1 x ... x X_k -> Y_1 x ... x Y_n
and abstract types B^k -> B^n.  We will think of these mappings as
transforming universes of discourse into themselves or into others,
in short, as "transformations of discourse".

Before we continue with this intinerary, however, I would like to highlight
another sort of "differential aspect" that concerns the "boundary operator"
or the "marked connective" that serves as one of the two basic connectives
in the cactus language for ZOL.

For example, consider the proposition f of concrete type f : X x Y x Z -> B
and abstract type f : B^3 -> B that is written "(x, y, z)" in cactus syntax.
Taken as an assertion in what Peirce called the "existential interpretation",
(x, y, z) says that just one of x, y, z is false.  It is useful to consider
this assertion in relation to the conjunction xyz of the features that are
engaged as its arguments.  A venn diagram of (x, y, z) looks like this:

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o            x            o                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
|         /          \%%%%%%/   \%%%%%%/          \         |
|        /            \%%%%/     \%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       o        y        o%%%%%%%o        z        o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o

In relation to the center cell indicated by the conjunction xyz,
the region indicated by (x, y, z) is comprised of the "adjacent"
or the "bordering" cells.  Thus they are the cells that are just
across the boundary of the center cell, as if reached by way of
Leibniz's "minimal changes" from the point of origin, here, xyz.

The same form of boundary relationship is exhibited for any cell
of origin that one might elect to indicate, say, by means of the
conjunction of positive and negative basis features u_1 ... u_k,
where u_j = x_j or u_j = (x_j), for j = 1 to k.  The proposition
(u_1, ..., u_k) indicates the disjunctive region consisting of
the cells that are "just next door" to the cell u_1 ... u_k.

Diff Log A 2004 • Note 9


| Consider what effects that might conceivably have
| practical bearings you conceive the objects of your
| conception to have.  Then, your conception of those
| effects is the whole of your conception of the object.
|
| Charles Sanders Peirce, "The Maxim of Pragmatism, CP 5.438.

One other subject that it would be opportune to mention at this point,
while we have an object example of a mathematical group fresh in mind,
is the relationship between the pragmatic maxim and what are commonly
known in mathematics as "representation principles".  As it turns out,
with regard to its formal characteristics, the pragmatic maxim unites
the aspects of a representation principle with the attributes of what
would ordinarily be known as a "closure principle".  We will consider
the form of closure that is invoked by the pragmatic maxim on another
occasion, focusing here and now on the topic of group representations.

Let us return to the example of the so-called "four-group" V_4.
We encountered this group in one of its concrete representations,
namely, as a "transformation group" that acts on a set of objects,
in this particular case a set of sixteen functions or propositions.
Forgetting about the set of objects that the group transforms among
themselves, we may take the abstract view of the group's operational
structure, say, in the form of the group operation table copied here:

o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    .    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o=========o=========o=========o=========o=========o
|         %         |         |         |         |
|    e    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    f    %    f    |    e    |    h    |    g    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    g    %    g    |    h    |    e    |    f    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    h    %    h    |    g    |    f    |    e    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o

This table is abstractly the same as, or isomorphic to, the versions with
the E_ij operators and the T_ij transformations that we discussed earlier.
That is to say, the story is the same -- only the names have been changed.
An abstract group can have a multitude of significantly and superficially
different representations.  Even after we have long forgotten the details
of the particular representation that we may have come in with, there are
species of concrete representations, called the "regular representations",
that are always readily available, as they can be generated from the mere
data of the abstract operation table itself.

For example, select a group element from the top margin of the Table,
and "consider its effects" on each of the group elements as they are
listed along the left margin.  We may record these effects as Peirce
usually did, as a logical "aggregate" of elementary dyadic relatives,
that is to say, a disjunction or a logical sum whose terms represent
the ordered pairs of <input : output> transactions that are produced
by each group element in turn.  This yields what is usually known as
one of the "regular representations" of the group, specifically, the
"first", the "post-", or the "right" regular representation.  It has
long been conventional to organize the terms in the form of a matrix:

Reading "+" as a logical disjunction:

   G  =  e  +  f  +  g  + h,

And so, by expanding effects, we get:

   G  =  e:e  +  f:f  +  g:g  +  h:h

      +  e:f  +  f:e  +  g:h  +  h:g

      +  e:g  +  f:h  +  g:e  +  h:f

      +  e:h  +  f:g  +  g:f  +  h:e

More on the pragmatic maxim as a representation principle later.

Diff Log A 2004 • Note 10


| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| objects of your 'conception' to have.  Then,
| your 'conception' of those effects is the
| whole of your 'conception' of the object.
|
| Peirce, "Maxim of Pragmaticism",
| 'Collected Papers', CP 5.438.

The genealogy of this conception of pragmatic representation is very intricate.
I will delineate some details that I presently fancy I remember clearly enough,
subject to later correction.  Without checking historical accounts, I will not
be able to pin down anything like a real chronology, but most of these notions
were standard furnishings of the 19th Century mathematical study, and only the
last few items date as late as the 1920's.

The idea about the regular representations of a group is universally known
as "Cayley's Theorem", usually in the form:  "Every group is isomorphic to
a subgroup of Aut(X), the group of automorphisms of an appropriate set X".
There is a considerable generalization of these regular representations to
a broad class of relational algebraic systems in Peirce's earliest papers.
The crux of the whole idea is this:

   Contemplate the effects of the symbol
   whose meaning you wish to investigate
   as they play out on all the stages of
   conduct on which you have the ability
   to imagine that symbol playing a role.

This idea of contextual definition is basically the same as Jeremy Bentham's
notion of "paraphrasis", a "method of accounting for fictions by explaining
various purported terms away" (Quine, in Van Heijenoort, page 216).  Today
we'd call these constructions "term models".  This, again, is the big idea
behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus,
and I reckon you know where that leads.

Diff Log A 2004 • Note 11


Let me return to Peirce's early papers on the algebra of relatives
to pick up the conventions that he used there, and then rewrite my
account of regular representations in a way that conforms to those.

Peirce expresses the action of an "elementary dual relative" like so:

   | [Let] A:B be taken to denote
   | the elementary relative which
   | multiplied into B gives A.
   |
   | Peirce, 'Collected Papers', CP 3.123.

And though he is well aware that it is not at all necessary to arrange
elementary relatives into arrays, matrices, or tables, when he does so
he tends to prefer organizing dyadic relations in the following manner:

   [  A:A   A:B   A:C  |
   |                   |
   |  B:A   B:B   B:C  |
   |                   |
   |  C:A   C:B   C:C  ]

That conforms to the way that the last school of thought
I matriculated into stipulated that we tabulate material:

   [  e_11  e_12  e_13  |
   |                    |
   |  e_21  e_22  e_23  |
   |                    |
   |  e_31  e_32  e_33  ]

So, for example, let us suppose that we have the small universe {A, B, C},
and the 2-adic relation m = "mover of" that is represented by this matrix:

   m  =

   [  m_AA (A:A)   m_AB (A:B)   m_AC (A:C)  |
   |                                        |
   |  m_BA (B:A)   m_BB (B:B)   m_BC (B:C)  |
   |                                        |
   |  m_CA (C:A)   m_CB (C:B)   m_CC (C:C)  ]

Also, let m be such that:

   A is a mover of A and B,
   B is a mover of B and C,
   C is a mover of C and A.

In sum:

   m  =

   [  1 * (A:A)   1 * (A:B)   0 * (A:C)  |
   |                                     |
   |  0 * (B:A)   1 * (B:B)   1 * (B:C)  |
   |                                     |
   |  1 * (C:A)   0 * (C:B)   1 * (C:C)  ]

For the sake of orientation and motivation,
compare with Peirce's notation in CP 3.329.

I think that will serve to fix notation
and set up the remainder of the account.

Diff Log A 2004 • Note 12


It is common in algebra to switch around
between different conventions of display,
as the momentary fancy happens to strike,
and I see that Peirce is no different in
this sort of shiftiness than anyone else.
A changeover appears to occur especially
whenever he shifts from logical contexts
to algebraic contexts of application.

In the paper "On the Relative Forms of Quaternions" (CP 3.323),
we observe Peirce providing the following sorts of explanation:

| If X, Y, Z denote the three rectangular components of a vector, and W denote
| numerical unity (or a fourth rectangular component, involving space of four
| dimensions), and (Y:Z) denote the operation of converting the Y component
| of a vector into its Z component, then
|
|     1  =  (W:W) + (X:X) + (Y:Y) + (Z:Z)
|
|     i  =  (X:W) - (W:X) - (Y:Z) + (Z:Y)
|
|     j  =  (Y:W) - (W:Y) - (Z:X) + (X:Z)
|
|     k  =  (Z:W) - (W:Z) - (X:Y) + (Y:X)
|
| In the language of logic (Y:Z) is a relative term whose relate is
| a Y component, and whose correlate is a Z component.  The law of
| multiplication is plainly (Y:Z)(Z:X) = (Y:X), (Y:Z)(X:W) = 0,
| and the application of these rules to the above values of
| 1, i, j, k gives the quaternion relations
|
|     i^2  =  j^2  =  k^2  =  -1,
|
|     ijk  =  -1,
|
|     etc.
|
| The symbol a(Y:Z) denotes the changing of Y to Z and the
| multiplication of the result by 'a'.  If the relatives be
| arranged in a block
|
|     W:W     W:X     W:Y     W:Z
|
|     X:W     X:X     X:Y     X:Z
|
|     Y:W     Y:X     Y:Y     Y:Z
|
|     Z:W     Z:X     Z:Y     Z:Z
|
| then the quaternion w + xi + yj + zk
| is represented by the matrix of numbers
|
|     w       -x      -y      -z
|
|     x        w      -z       y
|
|     y        z       w      -x
|
|     z       -y       x       w
|
| The multiplication of such matrices follows the same laws as the
| multiplication of quaternions.  The determinant of the matrix =
| the fourth power of the tensor of the quaternion.
|
| The imaginary x + y(-1)^(1/2) may likewise be represented by the matrix
|
|      x      y
|
|     -y      x
|
| and the determinant of the matrix = the square of the modulus.
|
| C.S. Peirce, 'Collected Papers', CP 3.323, (1882).
|'Johns Hopkins University Circulars', No. 13, p. 179.

This way of talking is the mark of a person who opts
to multiply his matrices "on the right", as they say.
Yet Peirce still continues to call the first element
of the ordered pair (i:j) its "relate" while calling
the second element of the pair (i:j) its "correlate".
That doesn't comport very well, so far as I can tell,
with his customary reading of relative terms, suited
more to the multiplication of matrices "on the left".

So I still have a few wrinkles to iron out before
I can give this story a smooth enough consistency.

Diff Log A 2004 • Note 13


Let us make up the model universe $1$ = A + B + C and the 2-adic relation
n = "noter of", as when "X is a data record that contains a pointer to Y".
That interpretation is not important, it's just for the sake of intuition.
In general terms, the 2-adic relation n can be represented by this matrix:

   n  =

   [  n_AA (A:A)   n_AB (A:B)   n_AC (A:C)  |
   |                                        |
   |  n_BA (B:A)   n_BB (B:B)   n_BC (B:C)  |
   |                                        |
   |  n_CA (C:A)   n_CB (C:B)   n_CC (C:C)  ]

Also, let n be such that:

   A is a noter of A and B,
   B is a noter of B and C,
   C is a noter of C and A.

Filling in the instantial values of the "coefficients" n_ij,
as the indices i and j range over the universe of discourse:

   n  =

   [  1 * (A:A)   1 * (A:B)   0 * (A:C)  |
   |                                     |
   |  0 * (B:A)   1 * (B:B)   1 * (B:C)  |
   |                                     |
   |  1 * (C:A)   0 * (C:B)   1 * (C:C)  ]

In Peirce's time, and even in some circles of mathematics today,
the information indicated by the elementary relatives (i:j), as
i, j range over the universe of discourse, would be referred to
as the "umbral elements" of the algebraic operation represented
by the matrix, though I seem to recall that Peirce preferred to
call these terms the "ingredients".  When this ordered basis is
understood well enough, one will tend to drop any mention of it
from the matrix itself, leaving us nothing but these bare bones:

   n  =

   [  1  1  0  |
   |           |
   |  0  1  1  |
   |           |
   |  1  0  1  ]

However the specification may come to be written, this
is all just convenient schematics for stipulating that:

   n  =  A:A  +  B:B  +  C:C  +  A:B  +  B:C  +  C:A

Recognizing !1! = A:A + B:B + C:C to be the identity transformation,
the 2-adic relation n = "noter of" may be represented by an element
!1! + A:B + B:C + C:A of the so-called "group ring", all of which
just makes this element a special sort of linear transformation.

Up to this point, we are still reading the elementary relatives of
the form i:j in the way that Peirce reads them in logical contexts:
i is the relate, j is the correlate, and in our current example we
read i:j, or more exactly, n_ij = 1, to say that i is a noter of j.
This is the mode of reading that we call "multiplying on the left".

In the algebraic, permutational, or transformational contexts of
application, however, Peirce converts to the alternative mode of
reading, although still calling i the relate and j the correlate,
the elementary relative i:j now means that i gets changed into j.
In this scheme of reading, the transformation A:B + B:C + C:A is
a permutation of the aggregate $1$ = A + B + C, or what we would
now call the set {A, B, C}, in particular, it is the permutation
that is otherwise notated as:

   ( A B C )
   <       >
   ( B C A )

This is consistent with the convention that Peirce uses in
the paper "On a Class of Multiple Algebras" (CP 3.324-327).

Diff Log A 2004 • Note 14


We have been contemplating the virtues and the utilities of
the pragmatic maxim as a standard heuristic in hermeneutics,
that is, as a principle of interpretation that guides us in
finding clarifying representations for a problematic corpus
of symbols by means of their actions on other symbols or in
terms of their effects on the syntactic contexts wherein we
discover them or where we might conceive to distribute them.

I began this excursion by taking off from the moving platform
of differential logic and passing by way of the corresponding
transformation groups, as they act on propositions, and on to
an exercise in applying the pragmatic maxim, by contemplating
the regular representations of groups as giving us one of the
simplest conceivable, relatively concrete applications of the
general principle of representation in question.

There are a few problems of implementation that have to be worked out
in practice, most of which are cleared up by keeping in mind which of
several possible conventions we have chosen to follow at a given time.

But there does appear to remain this rather more substantial question:
Are the effects we seek relates or correlates, or does it even matter?

I will have to leave that question as it is for now,
in hopes that a solution will evolve itself in time.

Diff Log A 2004 • Note 15


Obstacles to Applying the Pragmatic Maxim

No sooner do you get a good idea and try to apply it
than you find that a motley array of obstacles arise.

It would be good if we could in practice more consistently
apply the pragmatic maxim to the purpose for which it was
purportedly intended by its author.  That aim would be
the clarification of concepts, that is, intellectual
symbols or mental signs, to the point where their
inherent senses, or their lacks thereof, would
be rendered manifest to suitable interpreters.

There are big obstacles and little obstacles to applying the pragmatic maxim.
In good subgoaling fashion, I will merely mention a few of the bigger blocks,
as if in passing, but not really getting past them, and then I will get down
to the details of the problems that more immediately obstruct our advance.

Obstacle 1.  People do not always read the instructions very carefully.
There is a tendency in readers of particular prior persuasions to blow
the problem all out of proportion, to think that the maxim is meant to
reveal the absolutely positive and the totally unique meaning of every
preconception to which they might deign or elect to apply it.  Reading
the maxim with an even minimal attention, you can see that it promises
no such finality of unindexed sense, but ties what you conceive to you.
I have lately come to wonder at the tenacity of this misinterpretation.
Perhaps people reckon that nothing less would be worth their attention.
I am not sure.  I can only say the achievement of more modest goals is
the sort of thing on which our daily life depends, and there can be no
final end to inquiry nor any ultimate community without a continuation
of life, and that means life on a day to day basis.  All of which only
brings me back to the point of persisting with local meantime examples,
because if we can't apply the maxim there, we can't apply it anywhere.

Diff Log A 2004 • Note 16


Obstacles to Applying the Pragmatic Maxim (cont.)

Obstacle 2.  Applying the pragmatic maxim, even with a moderate aim, can be hard.
I think that my present example, deliberately impoverished as it is, affords us
with an embarassing richness of evidence of just how complex the simple can be.

All the better reason for me to see if I can finish it up before moving on.

Expressed most simply, the idea is to replace the question of "what it is",
which modest people know is far too difficult for them to answer right off,
with the question of "what it does", which most of us know a modicum about.

In the case of regular representations of groups we found
a non-plussing surplus of answers to sort our way through.
So let us track back one more time to see if we can learn
any lessons that might carry over to more realistic cases.

Here is is the operation table of V_4 once again:

Table 1.  Klein Four-Group V_4
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    .    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o=========o=========o=========o=========o=========o
|         %         |         |         |         |
|    e    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    f    %    f    |    e    |    h    |    g    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    g    %    g    |    h    |    e    |    f    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    h    %    h    |    g    |    f    |    e    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o

A group operation table is really just a device for
recording a certain 3-adic relation, to be specific,
the set of triples of the form <x, y, z> satisfying
the equation x.y = z, where "." signifies the group
operation, usually omitted as understood in context.

In the case of V_4 = (G, .), where G is the "underlying set"
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
whose triples are listed below:

   <e, e, e>
   <e, f, f>
   <e, g, g>
   <e, h, h>

   <f, e, f>
   <f, f, e>
   <f, g, h>
   <f, h, g>

   <g, e, g>
   <g, f, h>
   <g, g, e>
   <g, h, f>

   <h, e, h>
   <h, f, g>
   <h, g, f>
   <h, h, e>

It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.
Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:

1.  Sub(x, <_, y>) puts any specified x into
    the empty slot of the rheme <_, y>, with
    the effect of producing the saturated
    rheme <x, y> that evaluates to xy.

2.  Sub(x, <y, _>) puts any specified x into
    the empty slot of the rheme <y, _>, with
    the effect of producing the saturated
    rheme <y, x> that evaluates to yx.

In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {(y : xy) : y in G}.
The pairs (y : xy) can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run across the top margin.  This aspect of
pragmatic definition we recognize as the regular
ante-representation:

   e  =  e:e  +  f:f  +  g:g  +  h:h

   f  =  e:f  +  f:e  +  g:h  +  h:g

   g  =  e:g  +  f:h  +  g:e  +  h:f

   h  =  e:h  +  f:g  +  g:f  +  h:e

In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {(y : yx) : y in G}.
The pairs (y : yx) can be found by picking an x
from the top margin of the group operation table
and considering its effects on each y in turn as
these run down the left margin.  This aspect of
pragmatic definition we recognize as the regular
post-representation:

   e  =  e:e  +  f:f  +  g:g  +  h:h

   f  =  e:f  +  f:e  +  g:h  +  h:g

   g  =  e:g  +  f:h  +  g:e  +  h:f

   h  =  e:h  +  f:g  +  g:f  +  h:e

If the ante-rep looks the same as the post-rep,
now that I'm writing them in the same dialect,
that is because V_4 is abelian (commutative),
and so the two representations have the very
same effects on each point of their bearing.

Diff Log A 2004 • Note 17


So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
a non-abelian (non-commutative) group.  This is a group of six
elements, say, G = {e, f, g, h, i, j}, with no relation to any
other employment of these six symbols being implied, of course,
and it can be most easily represented as the permutation group
on a set of three letters, say, X = {A, B, C}, usually notated
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
Here are the permutation (= substitution) operations in Sym(X):

Table 1.  Permutations or Substitutions in Sym_{A, B, C}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Here is the operation table for S_3, given in abstract fashion:

Table 2.  Symmetric Group S_3

|                        ^
|                     e / \ e
|                      /   \
|                     /  e  \
|                  f / \   / \ f
|                   /   \ /   \
|                  /  f  \  f  \
|               g / \   / \   / \ g
|                /   \ /   \ /   \
|               /  g  \  g  \  g  \
|            h / \   / \   / \   / \ h
|             /   \ /   \ /   \ /   \
|            /  h  \  e  \  e  \  h  \
|         i / \   / \   / \   / \   / \ i
|          /   \ /   \ /   \ /   \ /   \
|         /  i  \  i  \  f  \  j  \  i  \
|      j / \   / \   / \   / \   / \   / \ j
|       /   \ /   \ /   \ /   \ /   \ /   \
|      (  j  \  j  \  j  \  i  \  h  \  j  )
|       \   / \   / \   / \   / \   / \   /
|        \ /   \ /   \ /   \ /   \ /   \ /
|         \  h  \  h  \  e  \  j  \  i  /
|          \   / \   / \   / \   / \   /
|           \ /   \ /   \ /   \ /   \ /
|            \  i  \  g  \  f  \  h  /
|             \   / \   / \   / \   /
|              \ /   \ /   \ /   \ /
|               \  f  \  e  \  g  /
|                \   / \   / \   /
|                 \ /   \ /   \ /
|                  \  g  \  f  /
|                   \   / \   /
|                    \ /   \ /
|                     \  e  /
|                      \   /
|                       \ /
|                        v

By the way, we will meet with the symmetric group S_3 again
when we return to take up the study of Peirce's early paper
"On a Class of Multiple Algebras" (CP 3.324-327), and also
his late unpublished work "The Simplest Mathematics" (1902)
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).

Diff Log A 2004 • Note 18


By way of collecting a short-term pay-off for all the work that we
did on the regular representations of the Klein 4-group V_4, let us
write out as quickly as possible in "relative form" a minimal budget
of representations for the symmetric group on three letters, Sym(3).
After doing the usual bit of compare and contrast among the various
representations, we will have enough concrete material beneath our
abstract belts to tackle a few of the presently obscur'd details
of Peirce's early "Algebra + Logic" papers.

Table 1.  Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Writing this table in relative form generates
the following "natural representation" of S_3.

   e  =  A:A + B:B + C:C

   f  =  A:C + B:A + C:B

   g  =  A:B + B:C + C:A

   h  =  A:A + B:C + C:B

   i  =  A:C + B:B + C:A

   j  =  A:B + B:A + C:C

I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as X:Y
constitutes the turning of X into Y.  It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.

Diff Log A 2004 • Note 19


To construct the regular representations of S_3,
we pick up from the data of its operation table:

Table 1.  Symmetric Group S_3

|                        ^
|                     e / \ e
|                      /   \
|                     /  e  \
|                  f / \   / \ f
|                   /   \ /   \
|                  /  f  \  f  \
|               g / \   / \   / \ g
|                /   \ /   \ /   \
|               /  g  \  g  \  g  \
|            h / \   / \   / \   / \ h
|             /   \ /   \ /   \ /   \
|            /  h  \  e  \  e  \  h  \
|         i / \   / \   / \   / \   / \ i
|          /   \ /   \ /   \ /   \ /   \
|         /  i  \  i  \  f  \  j  \  i  \
|      j / \   / \   / \   / \   / \   / \ j
|       /   \ /   \ /   \ /   \ /   \ /   \
|      (  j  \  j  \  j  \  i  \  h  \  j  )
|       \   / \   / \   / \   / \   / \   /
|        \ /   \ /   \ /   \ /   \ /   \ /
|         \  h  \  h  \  e  \  j  \  i  /
|          \   / \   / \   / \   / \   /
|           \ /   \ /   \ /   \ /   \ /
|            \  i  \  g  \  f  \  h  /
|             \   / \   / \   / \   /
|              \ /   \ /   \ /   \ /
|               \  f  \  e  \  g  /
|                \   / \   / \   /
|                 \ /   \ /   \ /
|                  \  g  \  f  /
|                   \   / \   /
|                    \ /   \ /
|                     \  e  /
|                      \   /
|                       \ /
|                        v

Just by way of staying clear about what we are doing,
let's return to the recipe that we worked out before:

It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.

Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:

1.  Sub(x, <_, y>) puts any specified x into
    the empty slot of the rheme <_, y>, with
    the effect of producing the saturated
    rheme <x, y> that evaluates to xy.

2.  Sub(x, <y, _>) puts any specified x into
    the empty slot of the rheme <y, _>, with
    the effect of producing the saturated
    rheme <y, x> that evaluates to yx.

In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {(y : xy) : y in G}.
The pairs (y : xy) can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run along the right margin.  This produces
the regular ante-representation of S_3, like so:

   e   =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j

   f   =   e:f  +  f:g  +  g:e  +  h:j  +  i:h  +  j:i

   g   =   e:g  +  f:e  +  g:f  +  h:i  +  i:j  +  j:h

   h   =   e:h  +  f:i  +  g:j  +  h:e  +  i:f  +  j:g

   i   =   e:i  +  f:j  +  g:h  +  h:g  +  i:e  +  j:f

   j   =   e:j  +  f:h  +  g:i  +  h:f  +  i:g  +  j:e

In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {(y : yx) : y in G}.
The pairs (y : yx) can be found by picking an x
on the right margin of the group operation table
and considering its effects on each y in turn as
these run along the left margin.  This generates
the regular post-representation of S_3, like so:

   e   =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j

   f   =   e:f  +  f:g  +  g:e  +  h:i  +  i:j  +  j:h

   g   =   e:g  +  f:e  +  g:f  +  h:j  +  i:h  +  j:i

   h   =   e:h  +  f:j  +  g:i  +  h:e  +  i:g  +  j:f

   i   =   e:i  +  f:h  +  g:j  +  h:f  +  i:e  +  j:g

   j   =   e:j  +  f:i  +  g:h  +  h:g  +  i:f  +  j:e

If the ante-rep looks different from the post-rep,
it is just as it should be, as S_3 is non-abelian
(non-commutative), and so the two representations
differ in the details of their practical effects,
though, of course, being representations of the
same abstract group, they must be isomorphic.

Diff Log A 2004 • Note 20


| the way of heaven and earth
| is to be long continued
| in their operation
| without stopping
|
| i ching, hexagram 32

You may be wondering what happened to the announced subject of "Differential Logic".
If you think that we have been taking a slight excursion my reply to the charge of
a scenic rout would be both "yes and no".  What happened was this.  We chanced to
make the observation that the shift operators E_ij form a transformation group
that acts on the set of propositions of the form f : B^2 -> B.  Group theory
is a very attractive subject, but it did not have the effect of drawing us
so far off our initial course as one might at first think.  For one thing,
groups, in particular, the special family of groups that have come to be
named after the Norwegian mathematician Marius Sophus Lie, turn out to
be of critical importance in the solution of differential equations.
For another thing, group operations afford us examples of 3-adic
relations that have been extremely well-studied over the years,
and thus they supply us with no small bit of guidance in the
study of sign relations, another class of 3-adic relations
that have significance for logical studies, in our brief
acquaintance with which we have scarcely even begun to
break the ice.  Finally, I could not resist taking up
the connection between group representations, which
constitute a very generic class of logical models,
and the all-important pragmatic maxim.

Biographical Data for Marius Sophus Lie (1842-1899):
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html

Diff Log A 2004 • Note 21


We've seen a couple of groups, V_4 and S_3, represented in various ways, and
we've seen their representations presented in a variety of different manners.
Let us look at one other stylistic variant for presenting a representation
that is frequently seen, the so-called "matrix representation" of a group.

Recalling the manner of our acquaintance with the symmetric group S_3,
we began with the "bigraph" (bipartite graph) picture of its natural
representation as the set of all permutations or substitutions on
the set X = {A, B, C}.

Table 1.  Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Then we rewrote these permutations -- since they are
functions f : X -> X they can also be recognized as
2-adic relations f c X x X -- in "relative form",
in effect, in the manner to which Peirce would
have made us accustomed had he been given
a relative half-a-chance:

   e  =  A:A + B:B + C:C

   f  =  A:C + B:A + C:B

   g  =  A:B + B:C + C:A

   h  =  A:A + B:C + C:B

   i  =  A:C + B:B + C:A

   j  =  A:B + B:A + C:C

These days one is much more likely to encounter the natural representation
of S_3 in the form of a "linear representation", that is, as a family of
linear transformations that map the elements of a suitable vector space
into each other, all of which would in turn usually be represented by
a set of matrices like these:

Table 2.  Matrix Representations of the Permutations in Sym(3)
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

The key to the mysteries of these matrices is revealed by noting that their
coefficient entries are arrayed and overlayed on a place mat marked like so:

   [ A:A  A:B  A:C |
   | B:A  B:B  B:C |
   | C:A  C:B  C:C ]

Of course, the place-settings of convenience at different symposia may vary.

Differential Logic 2004 • Series B

Diff Log B 2004 • Note 1


| The most fundamental concept in cybernetics is that of "difference",
| either that two things are recognisably different or that one thing
| has changed with time.
|
| William Ross Ashby, 
|'An Introduction to Cybernetics',
| Chapman & Hall, London, UK, 1956,
| Methuen & Company, London, UK, 1964.

Linear Topics.  The Differential Theory of Qualitative Equations

This chapter is titled "Linear Topics" because that is the heading
under which the derivatives and the differentials of any functions
usually come up in mathematics, namely, in relation to the problem
of computing "locally linear approximations" to the more arbitrary,
unrestricted brands of functions that one finds in a given setting.

To denote lists of propositions and to detail their components,
we use notations like:

   !a!  =  <a, b, c>,     !p!  =  <p, q, r>,     !x!  =  <x, y, z>,

or, in more complicated situations:

    x = <x_1, x_2, x_3>,   y = <y_1, y_2, y_3>,   z = <z_1, z_2, z_3>.

In a universe where some region is ruled by a proposition,
it is natural to ask whether we can change the value of that
proposition by changing the features of our current state.

Given a venn diagram with a shaded region and starting from
any cell in that universe, what sequences of feature changes,
what traverses of cell walls, will take us from shaded to
unshaded areas, or the reverse?

In order to discuss questions of this type, it is useful
to define several "operators" on functions.  An operator
is nothing more than a function between sets that happen
to have functions as members.

A typical operator F takes us from thinking about a given function f
to thinking about another function g.  To express the fact that g can
be obtained by applying the operator F to f, we write g = Ff.
 
The first operator, E, associates with a function f : X -> Y
another function Ef, where Ef : X x X -> Y is defined by the
following equation:

   Ef(x, y)  =  f(x + y).

E is called a "shift operator" because it takes us from contemplating the
value of f at a place x to considering the value of f at a shift of y away.
Thus, E tells us the absolute effect on f that is obtained by changing its
argument from x by an amount that is equal to y.

Historical Note.  The protean "shift operator" E was originally called
the "enlargement operator", hence the initial "E" of the usual notation.

The next operator, D, associates with a function f : X -> Y
another function Df, where Df : X x X -> Y is defined by the
following equation:

   Df(x, y)  =  Ef(x, y) - f(x),

or, equivalently,

   Df(x, y)  =  f(x + y) - f(x).

D is called a "difference operator" because it tells us about the
relative change in the value of f along the shift from x to x + y.
 
In practice, one of the variables, x or y, is often
considered to be "less variable" than the other one,
being fixed in the context of a concrete discussion.
Thus, we might find any one of the following idioms:

1.  Df : X x X -> Y,

    Df(c, x)  =  f(c + x) - f(c).

Here, c is held constant and Df(c, x) is regarded
mainly as a function of the second variable x,
giving the relative change in f at various
distances x from the center c.

2.  Df : X x X -> Y,

    Df(x, h)  =  f(x + h) - f(x).

Here, h is either a constant (usually 1), in discrete contexts,
or a variably "small" amount (near to 0) over which a limit is
being taken, as in continuous contexts.  Df(x, h) is regarded
mainly as a function of the first variable x, in effect, giving
the differences in the value of f between x and a neighbor that
is a distance of h away, all the while that x itself ranges over
its various possible locations.

3.  Df : X x X -> Y,

    Df(x, dx)  =  f(x + dx) - f(x).

This is yet another variant of the previous form,
with dx denoting small changes contemplated in x.
 
That's the basic idea.  The next order of business is to develop
the logical side of the analogy a bit more fully, and to take up
the elaboration of some moderately simple applications of these
ideas to a selection of relatively concrete examples.

Diff Log B 2004 • Note 2


Example 1.  A Polymorphous Concept

I start with an example that is simple enough that it will allow us to compare
the representations of propositions by venn diagrams, truth tables, and my own
favorite version of the syntax for propositional calculus all in a relatively
short space.  To enliven the exercise, I borrow an example from a book with
several independent dimensions of interest, 'Topobiology' by Gerald Edelman.
One finds discussed there the notion of a "polymorphous set".  Such a set
is defined in a universe of discourse whose elements can be described in
terms of a fixed number k of logical features.  A "polymorphous set" is
one that can be defined in terms of sets whose elements have a fixed
number j of the k features.

As a rule in the following discussion, I will use upper case letters as names
for concepts and sets, lower case letters as names for features and functions.

The example that Edelman gives (1988, Fig. 10.5, p. 194) involves sets of
stimulus patterns that can be described in terms of the three features
"round" 'u', "doubly outlined" 'v', and "centrally dark" 'w'.  We may
regard these simple features as logical propositions u, v, w : X -> B.
The target concept Q is one whose extension is a polymorphous set Q,
the subset Q of the universe X where the complex feature q : X -> B
holds true.  The Q in question is defined by the requirement:
"Having at least 2 of the 3 features in the set {u, v, w}".

Taking the symbols u = "round", v = "doubly outlined", w = "centrally dark",
and using the corresponding capitals to label the circles of a venn diagram,
we get a picture of the target set Q as the shaded region in Figure 1.  Using
these symbols as "sentence letters" in a truth table, let the truth function q
mean the very same thing as the expression "{u and v} or {u and w} or {v and w}".

o-----------------------------------------------------------o
| X                                                         |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |            U            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/%\%%%%%%%%/        \          |
|         /          \%%%%%%/%%%\%%%%%%/          \         |
|        /            \%%%%/%%%%%\%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |        V        |%%%%%%%|        W        |       |
|       |                 |%%%%%%%|                 |       |
|       o                 o%%%%%%%o                 o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 1.  Polymorphous Set Q

In other words, the proposition q is a truth-function of the 3 logical variables u, v, w,
and it may be evaluated according to the "truth table" scheme that is shown in Table 2.
In this representation the polymorphous set Q appears in the guise of what some people
call the "pre-image" or the "fiber of truth" under the function q.  More precisely,
the 3-tuples for which q evaluates to true are in an obvious correspondence with
the shaded cells of the venn diagram.  No matter how we get down to the level
of actual information, it's all pretty much the same stuff.

Table 2.  Polymorphous Function q
o---------------o-----------o-----------o-----------o-------o
|   u   v   w   |   u & v   |   u & w   |   v & w   |   q   |
o---------------o-----------o-----------o-----------o-------o
|               |           |           |           |       |
|   0   0   0   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   0   0   1   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   0   1   0   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   0   1   1   |     0     |     0     |     1     |   1   |
|               |           |           |           |       |
|   1   0   0   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   1   0   1   |     0     |     1     |     0     |   1   |
|               |           |           |           |       |
|   1   1   0   |     1     |     0     |     0     |   1   |
|               |           |           |           |       |
|   1   1   1   |     1     |     1     |     1     |   1   |
|               |           |           |           |       |
o---------------o-----------o-----------o-----------o-------o

With the pictures of the venn diagram and the truth table before us,
we have come to the verge of seeing how the word "model" is used in
logic, namely, to distinguish whatever things satisfy a description.

In the venn diagram presentation, to be a model of some conceptual
description !F! is to be a point x in the corresponding region F
of the universe of discourse X.

In the truth table representation, to be a model of a logical
proposition f is to be a data-vector !x! (a row of the table)
on which a function f evaluates to true.

This manner of speaking makes sense to those who consider the ultimate meaning of
a sentence to be not the logical proposition that it denotes but its truth value
instead.  From the point of view, one says that any data-vector of this type
(k-tuples of truth values) may be regarded as an "interpretation" of the
proposition with k variables.  An interpretation that yields a value
of true is then called a "model".

For the most threadbare kind of logical system that we find residing
in propositional calculus, this notion of model is almost too simple
to deserve the name, yet it can be of service to fashion some form
of continuity between the simple and the complex.

| Edelman, Gerald M.,
|'Topobiology:  An Introduction to Molecular Embryology',
| Basic Books, New York, NY, 1988.

Diff Log B 2004 • Note 3


| The present is big with the future.
|
| ~~ Leibniz

Here I now delve into subject matters
that are more specifically logical in
the character of their interpretation.

Imagine that we are sitting in one of the cells of a venn diagram,
contemplating the walls.  There are k of them, one for each positive
feature x_1, ..., x_k in our universe of discourse.  Our particular cell
is described by a concatenation of k signed assertions, positive or negative,
regarding each of these features, and this description of our position amounts
to what is called an "interpretation" of whatever proposition may rule the space,
or reign on the universe of discourse.  But are we locked into this interpretation?

With respect to each edge x of the cell we consider a test proposition dx
that determines our decision whether or not we will make a difference in
how we stand regarding to x.  If dx is true then it marks our decision,
intention, or plan to cross over the edge x at some point within the
purview of the contemplated plan.

To reckon the effect of several such decisions on our current interpretation,
or the value of the reigning proposition, we transform that position or that
proposition by making the following array of substitutions everywhere in its
expression:

   1.  Substitute "( x_1 , dx_1 )"  for  "x_1"
   2.  Substitute "( x_2 , dx_2 )"  for  "x_2"
   3.  Substitute "( x_3 , dx_3 )"  for  "x_3"
   ...
   k.  Substitute "( x_k , dx_k )"  for  "x_k"

For concreteness, consider the polymorphous set Q of Example 1
and focus on the central cell, specifically, the cell described
by the conjunction of logical features in the expression "u v w".

o-----------------------------------------------------------o
| X                                                         |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |            U            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/%\%%%%%%%%/        \          |
|         /          \%%%%%%/%%%\%%%%%%/          \         |
|        /            \%%%%/%%%%%\%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |        V        |%%%%%%%|        W        |       |
|       |                 |%%%%%%%|                 |       |
|       o                 o%%%%%%%o                 o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 1.  Polymorphous Set Q

The proposition or the truth-function q that describes Q is:

   (( u v )( u w )( v w ))

Conjoining the query that specifies the center cell gives:

   (( u v )( u w )( v w )) u v w

And we know the value of the interpretation by
whether this last expression issues in a model.

Applying the enlargement operator E
to the initial proposition q yields:

   ((  ( u , du )( v , dv )
   )(  ( u , du )( w , dw )
   )(  ( v , dv )( w , dw )
   ))

Conjoining a query on the center cell yields:

   ((  ( u , du )( v , dv )
   )(  ( u , du )( w , dw )
   )(  ( v , dv )( w , dw )
   ))

   u v w

The models of this last expression tell us which combinations of
feature changes among the set {du, dv, dw} will take us from our
present interpretation, the center cell expressed by "u v w", to
a true value under the target proposition (( u v )( u w )( v w )).

The result of applying the difference operator D
to the initial proposition q, conjoined with
a query on the center cell, yields:

   (
      ((  ( u , du )( v , dv )
      )(  ( u , du )( w , dw )
      )(  ( v , dv )( w , dw )
      ))
   ,
      ((  u v
      )(  u w
      )(  v w
      ))
   )

   u v w

The models of this last proposition are:

   1.  u v w  du  dv  dw
   2.  u v w  du  dv (dw)
   3.  u v w  du (dv) dw
   4.  u v w (du) dv  dw

This tells us that changing any two or more of the
features u, v, w will take us from the center cell
to a cell outside the shaded region for the set Q.

Diff Log B 2004 • Note 4


| It is one of the rules of my system of general harmony,
| 'that the present is big with the future', and that he
| who sees all sees in that which is that which shall be.
|
| Leibniz, 'Theodicy'
|
| Gottfried Wilhelm, Freiherr von Leibniz,
|'Theodicy:  Essays on the Goodness of God,
| The Freedom of Man, & The Origin of Evil',
| Edited with an Introduction by Austin Farrer,
| Translated by E.M. Huggard from C.J. Gerhardt's
| Edition of the 'Collected Philosophical Works',
| 1875-90;  Routledge & Kegan Paul, London, UK, 1951;
| Open Court, La Salle, IL, 1985.  Paragraph 360, Page 341.

To round out the presentation of the "Polymorphous" Example 1,
I will go through what has gone before and lay in the graphic
forms of all of the propositional expressions.  These graphs,
whose official botanical designation makes them out to be
a species of "painted and rooted cacti" (PARC's), are not
too far from the actual graph-theoretic data-structures
that result from parsing the Cactus string expressions,
the "painted and rooted cactus expressions" (PARCE's).
Finally, I will add a couple of venn diagrams that
will serve to illustrate the "difference opus" Dq.
If you apply an operator to an operand you must
arrive at either an opus or an opera, no?

Consider the polymorphous set Q of Example 1 and focus on the central cell,
described by the conjunction of logical features in the expression "u v w".

o-------------------------------------------------o
| X                                               |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o           U           o            |
|            |                       |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \%%%%%%%%%\ /%%%%%%%%%/     \       |
|      /       \%%%%%%%%%o%%%%%%%%%/       \      |
|     /         \%%%%%%%/%\%%%%%%%/         \     |
|    /           \%%%%%/%%%\%%%%%/           \    |
|   o             o---o-----o---o             o   |
|   |                 |%%%%%|                 |   |
|   |        V        |%%%%%|        W        |   |
|   |                 |%%%%%|                 |   |
|   o                 o%%%%%o                 o   |
|    \                 \%%%/                 /    |
|     \                 \%/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 1.  Polymorphous Set Q

The proposition or truth-function q : X -> B that
describes Q is represented by the following graph
and text expressions:

o-------------------------------------------------o
| q                                               |
o-------------------------------------------------o
|                                                 |
|                 u v   u w   v w                 |
|                    o   o   o                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|             (( u v )( u w )( v w ))             |
o-------------------------------------------------o

Conjoining the query that specifies the center cell gives:

o-------------------------------------------------o
| q.uvw                                           |
o-------------------------------------------------o
|                                                 |
|                 u v   u w   v w                 |
|                    o   o   o                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        @ u v w                  |
|                                                 |
o-------------------------------------------------o
|             (( u v )( u w )( v w )) u v w       |
o-------------------------------------------------o

And we know the value of the interpretation by
whether this last expression issues in a model.

Applying the enlargement operator E
to the initial proposition q yields:

o-------------------------------------------------o
| Eq                                              |
o-------------------------------------------------o
|                                                 |
|      u  du v  dv  u  du w  dw  v  dv w  dw      |
|      o---o o---o  o---o o---o  o---o o---o      |
|       \  | |  /    \  | |  /    \  | |  /       |
|        \ | | /      \ | | /      \ | | /        |
|         \| |/        \| |/        \| |/         |
|          o=o          o=o          o=o          |
|             \          |          /             |
|              \         |         /              |
|               \        |        /               |
|                \       |       /                |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                                                 |
|          ((  ( u , du ) ( v , dv )              |
|          )(  ( u , du ) ( w , dw )              |
|          )(  ( v , dv ) ( w , dw )              |
|          ))                                     |
|                                                 |
o-------------------------------------------------o

Conjoining a query on the center cell yields:

o-------------------------------------------------o
| Eq.uvw                                          |
o-------------------------------------------------o
|                                                 |
|      u  du v  dv  u  du w  dw  v  dv w  dw      |
|      o---o o---o  o---o o---o  o---o o---o      |
|       \  | |  /    \  | |  /    \  | |  /       |
|        \ | | /      \ | | /      \ | | /        |
|         \| |/        \| |/        \| |/         |
|          o=o          o=o          o=o          |
|             \          |          /             |
|              \         |         /              |
|               \        |        /               |
|                \       |       /                |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        @ u v w                  |
|                                                 |
o-------------------------------------------------o
|                                                 |
|          ((  ( u , du ) ( v , dv )              |
|          )(  ( u , du ) ( w , dw )              |
|          )(  ( v , dv ) ( w , dw )              |
|          ))                                     |
|                                                 |
|          u v w                                  |
|                                                 |
o-------------------------------------------------o

The models of this last expression tell us which combinations of
feature changes among the set {du, dv, dw} will take us from our
present interpretation, the center cell expressed by "u v w", to
a true value under the target proposition (( u v )( u w )( v w )).

The result of applying the difference operator D
to the initial proposition q, conjoined with
a query on the center cell, yields:

o-------------------------------------------------o
| Dq.uvw                                          |
o-------------------------------------------------o
|                                                 |
|    u  du v  dv  u  du w  dw  v  dv w  dw        |
|    o---o o---o  o---o o---o  o---o o---o        |
|     \  | |  /    \  | |  /    \  | |  /         |
|      \ | | /      \ | | /      \ | | /          |
|       \| |/        \| |/        \| |/           |
|        o=o          o=o          o=o            |
|           \          |          /               |
|            \         |         /                |
|             \        |        /                 |
|              \       |       /                  |
|               \      |      /                   |
|                \     |     /                    |
|                 \    |    /    u v  u w  v w    |
|                  \   |   /       o   o   o      |
|                   \  |  /         \  |  /       |
|                    \ | /           \ | /        |
|                     \|/             \|/         |
|                      o               o          |
|                      |               |          |
|                      |               |          |
|                      |               |          |
|                      o---------------o          |
|                       \             /           |
|                        \           /            |
|                         \         /             |
|                          \       /              |
|                           \     /               |
|                            \   /                |
|                             \ /                 |
|                              @ u v w            |
|                                                 |
o-------------------------------------------------o
|                                                 |
|       (                                         |
|          ((  ( u , du ) ( v , dv )              |
|          )(  ( u , du ) ( w , dw )              |
|          )(  ( v , dv ) ( w , dw )              |
|          ))                                     |
|       ,                                         |
|          ((  u v                                |
|          )(  u w                                |
|          )(  v w                                |
|          ))                                     |
|       )                                         |
|                                                 |
|       u v w                                     |
|                                                 |
o-------------------------------------------------o

The models of this last proposition are:

   1.  u v w  du  dv  dw
   2.  u v w  du  dv (dw)
   3.  u v w  du (dv) dw
   4.  u v w (du) dv  dw

This tells us that changing any two or more of the
features u, v, w will take us from the center cell,
as described by the conjunctive expression "u v w",
to a cell outside the shaded region for the set Q.

o-------------------------------------------------o
| X                                               |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /        U        \               |
|              /                   \              |
|             /                     \             |
|            o                  @    o            |
|            |                  ^    |            |
|            |                  |dw  |            |
|            |                  |    |         @  |
|        o---o---------o   o----|----o---o    ^   |
|       /     \`````````\ /`````|```/     \  /dw  |
|      /    du \`````dw``o``dv``|``/       \/     |
|     /  @<-----\-o<----/+\---->o`/        /\     |
|    /           \`````/`|`\`````/        /  \    |
|   o             o---o--|--o---o        /    o   |
|   |                 |``|``|           /     |   |
|   |  V              |`du``|          /   W  |   |
|   |                 |` |``|         /       |   |
|   o                 o``v``o   dv   /        o   |
|    \                 \`o-/------->@        /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 3.  Effect of the Difference Operator D
           Acting on a Polymorphous Function q

Figure 3 shows one way to picture this kind of a situation,
by superimposing the paths of indicated feature changes on
the venn diagram of the underlying proposition.  Here, the
models, or the satisfying interpretations, of the relevant
"difference proposition" Dq are marked with "@" signs, and
the boundary crossings along each path are marked with the
corresponding "differential features" among the collection
{du, dv, dw}.  In sum, starting from the cell uvw, we have
the following four paths:

   1.   du  dv  dw   =>  Change u, v, w.
   2.   du  dv (dw)  =>  Change u and v.
   3.   du (dv) dw   =>  Change u and w.
   4.  (du) dv  dw   =>  Change v and w.

Next I will discuss several applications of logical differentials,
developing along the way their logical and practical implications.

Diff Log B 2004 • Note 5


We have come to the point of making a connection,
at a very primitive level, between propositional
logic and the classes of mathematical structures
that are employed in mathematical systems theory
to model dynamical systems of very general sorts.

Here is a flash montage of what has gone before,
retrospectively touching on just the highpoints,
and highlighting mostly just Figures and Tables,
all directed toward the aim of ending up with a
novel style of pictorial diagram, one that will
serve us well in the future, as I have found it
readily adaptable and steadily more trustworthy
in my previous investigations, whenever we have
to illustrate these very basic sorts of dynamic
scenarios to ourselves, to others, to computers.

We typically start out with a proposition of interest,
for example, the proposition q : X -> B depicted here:

o-------------------------------------------------o
| q                                               |
o-------------------------------------------------o
|                                                 |
|                 u v   u w   v w                 |
|                    o   o   o                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|             (( u v )( u w )( v w ))             |
o-------------------------------------------------o

The proposition q is properly considered as an "abstract object",
in some acceptation of those very bedevilled and egging-on terms,
but it enjoys an interpretation as a function of a suitable type,
and all we have to do in order to enjoy the utility of this type
of representation is to observe a decent respect for what befits.

I will skip over the details of how to do this for right now.
I started to write them out in full, and it all became even
more tedious than my usual standard, and besides, I think
that everyone more or less knows how to do this already.

Once we have survived the big leap of re-interpreting these
abstract names as the names of relatively concrete dimensions
of variation, we can begin to lay out all of the familiar sorts
of mathematical models and pictorial diagrams that go with these
modest dimensions, the functions that can be formed on them, and
the transformations that can be entertained among this whole crew.

Here is the venn diagram for the proposition q.

o-----------------------------------------------------------o
| X                                                         |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |            U            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/%\%%%%%%%%/        \          |
|         /          \%%%%%%/%%%\%%%%%%/          \         |
|        /            \%%%%/%%%%%\%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |        V        |%%%%%%%|        W        |       |
|       |                 |%%%%%%%|                 |       |
|       o                 o%%%%%%%o                 o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 1.  Venn Diagram for the Proposition q

By way of excuse, if not yet a full justification, I probably ought to give
an account of the reasons why I continue to hang onto these primitive styles
of depiction, even though I can hardly recommend that anybody actually try to
draw them, at least, not once the number of variables climbs much higher than
three or four or five at the utmost.  One of the reasons would have to be this:
that in the relationship between their continuous aspect and their discrete aspect,
venn diagrams constitute a form of "iconic" reminder of a very important fact about
all "finite information depictions" (FID's) of the larger world of reality, and that
is the hard fact that we deceive ourselves to a degree if we imagine that the lines
and the distinctions that we draw in our imagination are all there is to reality,
and thus, that as we practice to categorize, we also manage to discretize, and
thus, to distort, to reduce, and to truncate the richness of what there is to
the poverty of what we can sieve and sift through our senses, or what we can
draw in the tangled webs of our own very tenuous and tinctured distinctions.
  
Another common scheme for description and evaluation of a proposition
is the so-called "truth table" or the "semantic tableau", for example:

Table 2.  Truth Table for the Proposition q
o---------------o-----------o-----------o-----------o-------o
|   u   v   w   |   u & v   |   u & w   |   v & w   |   q   |
o---------------o-----------o-----------o-----------o-------o
|               |           |           |           |       |
|   0   0   0   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   0   0   1   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   0   1   0   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   0   1   1   |     0     |     0     |     1     |   1   |
|               |           |           |           |       |
|   1   0   0   |     0     |     0     |     0     |   0   |
|               |           |           |           |       |
|   1   0   1   |     0     |     1     |     0     |   1   |
|               |           |           |           |       |
|   1   1   0   |     1     |     0     |     0     |   1   |
|               |           |           |           |       |
|   1   1   1   |     1     |     1     |     1     |   1   |
|               |           |           |           |       |
o---------------o-----------o-----------o-----------o-------o

Reading off the shaded cells of the venn diagram or the
rows of the truth table that have a "1" in the q column,
we see that the "models", or satisfying interpretations,
of the proposition q are the four that can be expressed,
in either the "additive" or the "multiplicative" manner,
as follows:

1.  The points of the space X that are assigned the coordinates:
    <u, v, w> = <0, 1, 1> or <1, 0, 1> or <1, 1, 0> or <1, 1, 1>.

2.  The points of the space X that have the conjunctive descriptions:
    "(u) v w", "u (v) w", "u v (w)", "u v w", where "(x)" is "not x".

The next thing that one typically does is to consider the effects
of various "operators" on the proposition of interest, which may
be called the "operand" or the "source" proposition, leaving the
corresponding terms "opus" or "target" as names for the result.

In our initial consideration of the proposition q, we naturally
interpret it as a function of the three variables that it wears
on its sleeve, as it were, namely, those that we find contained
in the basis {u, v, w}.  As we begin to regard this proposition
from the standpoint of a differential analysis, however, we may
need to regard it as "tacitly embedded" in any number of higher
dimensional spaces.  Just by way of starting out, our immediate
interest is with the "first order differential analysis" (FODA),
and this requires us to regard all of the propositions in sight
as functions of the variables in the first order extended basis,
specifically, those in the set {u, v, w, du, dv, dw}.  Now this
does not change the expression of any proposition, like q, that
does not mention the extra variables, only changing how it gets
interpreted as a function.  A level of interpretive flexibility
of this order is very useful, and it is quite common throughout
mathematics.  In this discussion, I will invoke its application
under the name of the "tacit extension" of a proposition to any
universe of discourse based on a superset of its original basis.

Diff Log B 2004 • Note 6


I think that we finally have enough of the preliminary
set-ups and warm-ups out of the way that we can begin
to tackle the differential analysis proper of our
sample proposition q = (( u v )( u w )( v w )).

When X is the type of space that is generated by {u, v, w},
let dX be the type of space that is generated by (du, dv, dw},
and let X x dX be the type of space that is generated by the
extended set of boolean basis elements {u, v, w, du, dv, dw}.
For convenience, define a notation "EX" so that EX = X x dX.
Even though the differential variables are in some abstract
sense no different than other boolean variables, it usually
helps to mark their distinctive roles and their differential
interpretation by means of the distinguishing domain name "dB".
Using these designations of logical spaces, the propositions
over them can be assigned both abstract and concrete types.

For instance, consider the proposition q<u, v, w>, as before,
and then consider its tacit extension q<u, v, w, du, dv, dw>,
the latter of which may be indicated more explicitly as "eq".

1.  Proposition q is abstractly typed as q : B^3 -> B.
    Proposition q is concretely typed as q :  X  -> B.

2.  Proposition eq is abstractly typed as eq : B^3 x dB^3 -> B.
    Proposition eq is concretely typed as eq :  X  x  dX  -> B.
    Succinctly, eq : EX -> B.

We now return to our consideration of the effects
of various differential operators on propositions.
This time around we have enough exact terminology
that we shall be able to explain what is actually
going on here in a rather more articulate fashion.

The first transformation of the source proposition q that we may
wish to stop and examine, though it is not unusual to skip right
over this stage of analysis, frequently regarding it as a purely
intermediary stage, holding scarcely even so much as the passing
interest, is the work of the "enlargement" or "shift" operator E.

Applying the operator E to the operand proposition q yields:

o-------------------------------------------------o
| Eq                                              |
o-------------------------------------------------o
|                                                 |
|      u  du v  dv  u  du w  dw  v  dv w  dw      |
|      o---o o---o  o---o o---o  o---o o---o      |
|       \  | |  /    \  | |  /    \  | |  /       |
|        \ | | /      \ | | /      \ | | /        |
|         \| |/        \| |/        \| |/         |
|          o=o          o=o          o=o          |
|             \          |          /             |
|              \         |         /              |
|               \        |        /               |
|                \       |       /                |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                                                 |
|          ((  ( u , du ) ( v , dv )              |
|          )(  ( u , du ) ( w , dw )              |
|          )(  ( v , dv ) ( w , dw )              |
|          ))                                     |
|                                                 |
o-------------------------------------------------o

The enlarged proposition Eq is a minimally interpretable as
as a function on the six variables of {u, v, w, du, dv, dw}.
In other words, Eq : EX -> B, or Eq : X x dX -> B.

Conjoining a query on the center cell, c = uvw, yields:

o-------------------------------------------------o
| Eq.c                                            |
o-------------------------------------------------o
|                                                 |
|      u  du v  dv  u  du w  dw  v  dv w  dw      |
|      o---o o---o  o---o o---o  o---o o---o      |
|       \  | |  /    \  | |  /    \  | |  /       |
|        \ | | /      \ | | /      \ | | /        |
|         \| |/        \| |/        \| |/         |
|          o=o          o=o          o=o          |
|             \          |          /             |
|              \         |         /              |
|               \        |        /               |
|                \       |       /                |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        @ u v w                  |
|                                                 |
o-------------------------------------------------o
|                                                 |
|          ((  ( u , du ) ( v , dv )              |
|          )(  ( u , du ) ( w , dw )              |
|          )(  ( v , dv ) ( w , dw )              |
|          ))                                     |
|                                                 |
|          u v w                                  |
|                                                 |
o-------------------------------------------------o

The models of this last expression tell us which combinations of
feature changes among the set {du, dv, dw} will take us from our
present interpretation, the center cell expressed by "u v w", to
a true value under the given proposition (( u v )( u w )( v w )).

The models of Eq.c can be described in the usual ways as follows:

1.  The points of the space EX that have
    the following coordinate descriptions:
    
    <u, v, w, du, dv, dw> =

    <1, 1, 1,  0,  0,  0>,
    <1, 1, 1,  0,  0,  1>,
    <1, 1, 1,  0,  1,  0>,
    <1, 1, 1,  1,  0,  0>.

2.  The points of the space EX that have
    the following conjunctive expressions:

    u v w (du)(dv)(dw),
    u v w (du)(dv) dw ,
    u v w (du) dv (dw),
    u v w  du (dv)(dw).

In summary, Eq.c informs us that we can get from c to a model of q by
making the following changes in our position with respect to u, v, w,
to wit, "change none or just one among {u, v, w}".

I think that it would be worth our time to diagram the models
of the "enlarged" or "shifted" proposition, Eq, at least, the
selection of them that we find issuing from the center cell c.

Figure 4 is an extended venn diagram for the proposition Eq.c,
where the shaded area gives the models of q and the "@" signs
mark the terminal points of the requisite feature alterations.

o-------------------------------------------------o
| X                                               |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o           U           o            |
|            |                       |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /`````````/     \       |
|      /       \`````dw``o``dv`````/       \      |
|     /         \`@<----/@\---->@`/         \     |
|    /           \`````/`|`\`````/           \    |
|   o             o---o--|--o---o             o   |
|   |                 |``|``|                 |   |
|   |        V        |`du``|        W        |   |
|   |                 |` |``|                 |   |
|   o                 o``v``o                 o   |
|    \                 \`@`/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 4.  Effect of the Enlargement Operator E
           On the Proposition q, Evaluated at c

Diff Log B 2004 • Note 7


One more piece of notation will save us a few bytes
in the length of many of our schematic formulations.

Let !X!  =  {x_1, ..., x_k} be a finite class of variables --
whose names I list, according to the usual custom, without
what seems to my semiotic consciousness like the necessary
quotation marks around their particular characters, though
not without not a little trepidation, or without a worried
cognizance that I may be obligated to reinsert them all to
their rightful places at a subsequent stage of development --
with regard to which we may now define the following items:

1.  The "(first order) differential alphabet",

    d!X!  =  {dx_1, ..., dx_k}.

2.  The "(first order) extended alphabet",

    E!X!  =  !X! |_| d!X!,

    E!X!  =  {x_1, ..., x_k,  dx_1, ..., dx_k}.

Before we continue with the differential analysis
of the source proposition q, we need to pause and
take another look at just how it shapes up in the
light of the extended universe EX, in other words,
to examine in utter detail its tacit extension eq.

The models of eq in EX can be comprehended as follows:

1.  Working in the "summary coefficient" form of representation,
    if the coordinate list x is a model of q in X, then one can
    construct a coordinate list ex as a model for eq in EX just
    by appending any combination of values for the differential
    variables in d!X!.

    For example, to focus once again on the center cell c,
    which happens to be a model of the proposition q in X,
    one can extend c in eight different ways into EX, and
    thus get eight models of the tacit extension eq in EX.
    Though it may seem an utter triviality to write these
    out, I will do it for the sake of seeing the patterns.

    The models of eq in EX that are tacit extensions of c:

    <u, v, w, du, dv, dw> =

    <1, 1, 1,  0,  0,  0>,
    <1, 1, 1,  0,  0,  1>,
    <1, 1, 1,  0,  1,  0>,
    <1, 1, 1,  0,  1,  1>,
    <1, 1, 1,  1,  0,  0>,
    <1, 1, 1,  1,  0,  1>,
    <1, 1, 1,  1,  1,  0>,
    <1, 1, 1,  1,  1,  1>.

2.  Working in the "conjunctive product" form of representation,
    if the conjunct symbol x is a model of q in X, then one can
    construct a conjunct symbol ex as a model for eq in EX just
    by appending any combination of values for the differential
    variables in d!X!.

    The models of eq in EX that are tacit extensions of c:

    u v w (du)(dv)(dw),
    u v w (du)(dv) dw ,
    u v w (du) dv (dw),
    u v w (du) dv  dw ,
    u v w  du (dv)(dw),
    u v w  du (dv) dw ,
    u v w  du  dv (dw),
    u v w  du  dv  dw .

In short, eq.c just enumerates all of the possible changes in EX
that "derive from", "issue from", or "stem from" the cell c in X.

Okay, that was pretty tedious, and I know that it all appears
to be totally trivial, which is precisely why we usually just
leave it "tacit" in the first place, but hard experience, and
a real acquaintance with the confusion that can beset us when
we do not render these implicit grounds explicit, have taught
me that it will ultimately be necessary to get clear about it,
and by this "clear" to say "marked", not merely "transparent".

Diff Log B 2004 • Note 8


Before going on -- in order to keep alive the will to go on! --
it would probably be a good idea to remind ourselves of just
why we are going through with this exercise.  It is to unify
the world of change, for which aspect or regime of the world
I occasionally evoke the eponymous figures of Prometheus and
Heraclitus, and the world of logic, for which facet or realm
of the world I periodically recur to the prototypical shades
of Epimetheus and Parmenides, at least, that is, to state it
more carefully, to encompass the antics and the escapades of
these all too manifestly strife-born twins within the scopes
of our thoughts and within the charts of our theories, as it
is most likely the only places where ever they will, for the
moment and as long as it lasts, be seen or be heard together.

With that intermezzo, with all of its echoes of the opening overture,
over and done, let us now return to that droller drama, already fast
in progress, the differential disentanglements, hopefully toward the
end of a grandly enlightening denouement, of the ever-polymorphous Q.

The next transformation of the source proposition q, that we are
typically aiming to contemplate in the process of carrying out a
"differential analysis" of its "dynamic" effects or implications,
is the yield of the so-called "difference" or "delta" operator D.
The resultant "difference proposition" Dq is defined in terms of
the source proposition q and the "shifted proposition" Eq thusly:

   | Dq  =  Eq - q  =  Eq - eq.
   |
   | Since "+" and "-" signify the same operation over B, we have:
   |
   | Dq  =  Eq + q  =  Eq + eq.
   |
   | Since "+" = "exclusive-or", RefLog syntax expresses this as:
   |
   |          Eq   q         Eq   eq
   |           o---o          o---o
   |            \ /            \ /
   | Dq  =       @      =       @
   |
   | Dq  =  ( Eq , q )  =  ( Eq , eq ).
   |
   | Recall that a k-place bracket "(x_1, x_2, ..., x_k)"
   | is interpreted (in the "existential interpretation")
   | to mean "Exactly one of the x_j is false", thus the
   | two-place bracket is equivalent to the exclusive-or.

The result of applying the difference operator D to the source
proposition q, conjoined with a query on the center cell c, is:

o-------------------------------------------------o
| Dq.uvw                                          |
o-------------------------------------------------o
|                                                 |
|    u  du v  dv  u  du w  dw  v  dv w  dw        |
|    o---o o---o  o---o o---o  o---o o---o        |
|     \  | |  /    \  | |  /    \  | |  /         |
|      \ | | /      \ | | /      \ | | /          |
|       \| |/        \| |/        \| |/           |
|        o=o          o=o          o=o            |
|           \          |          /               |
|            \         |         /                |
|             \        |        /                 |
|              \       |       /                  |
|               \      |      /                   |
|                \     |     /                    |
|                 \    |    /    u v  u w  v w    |
|                  \   |   /       o   o   o      |
|                   \  |  /         \  |  /       |
|                    \ | /           \ | /        |
|                     \|/             \|/         |
|                      o               o          |
|                      |               |          |
|                      |               |          |
|                      |               |          |
|                      o---------------o          |
|                       \             /           |
|                        \           /            |
|                         \         /             |
|                          \       /              |
|                           \     /               |
|                            \   /                |
|                             \ /                 |
|                              @ u v w            |
|                                                 |
o-------------------------------------------------o
|                                                 |
|       (                                         |
|          ((  ( u , du ) ( v , dv )              |
|          )(  ( u , du ) ( w , dw )              |
|          )(  ( v , dv ) ( w , dw )              |
|          ))                                     |
|       ,                                         |
|          ((  u v                                |
|          )(  u w                                |
|          )(  v w                                |
|          ))                                     |
|       )                                         |
|                                                 |
|       u v w                                     |
|                                                 |
o-------------------------------------------------o

The models of the difference proposition Dq.uvw are:

   1.  u v w  du  dv  dw

   2.  u v w  du  dv (dw)

   3.  u v w  du (dv) dw

   4.  u v w (du) dv  dw

This tells us that changing any two or more of the
features u, v, w will take us from the center cell
that is marked by the conjunctive expression "uvw",
to a cell outside the shaded region for the area Q.

o-------------------------------------------------o
| X                                               |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /        U        \               |
|              /                   \              |
|             /                     \             |
|            o                  @    o            |
|            |                  ^    |            |
|            |                  |dw  |            |
|            |                  |    |         @  |
|        o---o---------o   o----|----o---o    ^   |
|       /     \`````````\ /`````|```/     \  /dw  |
|      /    du \`````dw``o``dv``|``/       \/     |
|     /  @<-----\-o<----/+\---->o`/        /\     |
|    /           \`````/`|`\`````/        /  \    |
|   o             o---o--|--o---o        /    o   |
|   |                 |``|``|           /     |   |
|   |  V              |`du``|          /   W  |   |
|   |                 |` |``|         /       |   |
|   o                 o``v``o   dv   /        o   |
|    \                 \`o-/------->@        /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 3.  Effect of the Difference Operator D
           Acting on a Polymorphous Function q

Figure 3 shows one way to picture this kind of a situation,
by superimposing the paths of indicated feature changes on
the venn diagram of the underlying proposition.  Here, the
models, or the satisfying interpretations, of the relevant
"difference proposition" Dq are marked with "@" signs, and
the boundary crossings along each path are marked with the
corresponding "differential features" among the collection
{du, dv, dw}.  In sum, starting from the cell uvw, we have
the following four paths:

   1.   du  dv  dw   =  Change u, v, w.

   2.   du  dv (dw)  =  Change u and v.

   3.   du (dv) dw   =  Change u and w.

   4.  (du) dv  dw   =  Change v and w.

That sums up, but rather more carefully, the material that
I ran through just a bit too quickly the first time around.
Next time, I will begin to develop an alternative style of
diagram for depicting these types of differential settings.

Diff Log B 2004 • Note 9


Another way of looking at this situation is by letting the (first order)
differential features du, dv, dw be viewed as the features of another
universe of discourse, called the "tangent universe to X with respect
to the interpretation c" and represented as dX.c.  In this setting,
Dq.c, the "difference proposition of q at the interpretation c",
where c = uvw, is marked by the shaded region in Figure 4.

o-----------------------------------------------------------o
| dX.c                                                      |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |           dU            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \``````````\ /``````````/    \            |
|           /      \````2`````o`````3````/      \           |
|          /        \````````/`\````````/        \          |
|         /          \``````/```\``````/          \         |
|        /            \````/``1``\````/            \        |
|       o              o--o-------o--o              o       |
|       |                 |```````|                 |       |
|       |                 |```````|                 |       |
|       |                 |```````|                 |       |
|       |       dV        |```4```|       dW        |       |
|       |                 |```````|                 |       |
|       o                 o```````o                 o       |
|        \                 \`````/                 /        |
|         \                 \```/                 /         |
|          \                 \`/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 4.  Tangent Venn Diagram for Dq.c

Taken in the context of the tangent universe to X at c = uvw,
written dX.c or dX.uvw, the shaded area of Figure 4 indicates
the models of the difference proposition Dq.uvw, specifically:

   1.  u v w  du  dv  dw

   2.  u v w  du  dv (dw)

   3.  u v w  du (dv) dw

   4.  u v w (du) dv  dw

Diff Log B 2004 • Note 10


Sub*Title.  There's Gonna Be A Rumble Tonight!

From:  "Theme One:  A Program of Inquiry",
Jon Awbrey & Susan Awbrey, August 9, 1989.

Example 5.  Jets and Sharks

The propositional calculus that is based on the boundary operator
can be interpreted in a way that resembles the logic of activation
states and competition constraints in certain neural network models.
One way to do this is by interpreting the blank or unmarked state as
the resting state of a neural pool, the bound or marked state as its
activated state, and by representing a mutually inhibitory pool of
neurons A, B, C in the expression "(A, B, C)".  To illustrate this
possibility, we transcribe a well-known example from the parallel
distributed processing literature (McClelland & Rumelhart, 1988)
and work through two of the associated exercises as portrayed
in Existential Graph format.

File "jas.log".  Jets and Sharks Example
o-----------------------------------------------------------o
|                                                           |
|   (( art    ),( al   ),( sam  ),( clyde ),( mike  ),      |
|    ( jim    ),( greg ),( john ),( doug  ),( lance ),      |
|    ( george ),( pete ),( fred ),( gene  ),( ralph ),      |
|    ( phil   ),( ike  ),( nick ),( don   ),( ned   ),      |
|    ( karl   ),( ken  ),( earl ),( rick  ),( ol    ),      |
|    ( neal   ),( dave ))                                   |
|                                                           |
|   ( jets , sharks )                                       |
|                                                           |
|   ( jets ,                                                |
|    ( art    ),( al   ),( sam  ),( clyde ),( mike  ),      |
|    ( jim    ),( greg ),( john ),( doug  ),( lance ),      |
|    ( george ),( pete ),( fred ),( gene  ),( ralph ))      |
|                                                           |
|   ( sharks ,                                              |
|    ( phil ),( ike  ),( nick ),( don ),( ned  ),( karl ),  |
|    ( ken  ),( earl ),( rick ),( ol  ),( neal ),( dave ))  |
|                                                           |
|   (( 20's ),( 30's ),( 40's ))                            |
|                                                           |
|   ( 20's ,                                                |
|    ( sam    ),( jim  ),( greg ),( john ),( lance ),       |
|    ( george ),( pete ),( fred ),( gene ),( ken   ))       |
|                                                           |
|   ( 30's ,                                                |
|    ( al  ),( mike ),( doug ),( ralph ),( phil ),          |
|    ( ike ),( nick ),( don  ),( ned   ),( rick ),          |
|    ( ol  ),( neal ),( dave ))                             |
|                                                           |
|   ( 40's ,                                                |
|    ( art ),( clyde ),( karl ),( earl ))                   |
|                                                           |
|   (( junior_high ),( high_school ),( college ))           |
|                                                           |
|   ( junior_high ,                                         |
|    ( art  ),( al    ),( clyde  ),( mike  ),( jim ),       |
|    ( john ),( lance ),( george ),( ralph ),( ike ))       |
|                                                           |
|   ( high_school ,                                         |
|    ( greg ),( doug ),( pete ),( fred ),                   |
|    ( nick ),( karl ),( ken  ),( earl ),                   |
|    ( rick ),( neal ),( dave ))                            |
|                                                           |
|   ( college ,                                             |
|    ( sam ),( gene ),( phil ),( don ),( ned ),( ol ))      |
|                                                           |
|   (( single ),( married ),( divorced ))                   |
|                                                           |
|   ( single ,                                              |
|    ( art  ),( sam  ),( clyde ),( mike  ),( doug ),        |
|    ( pete ),( fred ),( gene  ),( ralph ),( ike  ),        |
|    ( nick ),( ken  ),( neal  ))                           |
|                                                           |
|   ( married ,                                             |
|    ( al  ),( greg ),( john ),( lance ),( phil ),          |
|    ( don ),( ned  ),( karl ),( earl  ),( ol   ))          |
|                                                           |
|   ( divorced ,                                            |
|    ( jim ),( george ),( rick ),( dave ))                  |
|                                                           |
|   (( bookie ),( burglar ),( pusher ))                     |
|                                                           |
|   ( bookie ,                                              |
|    ( sam  ),( clyde ),( mike ),( doug ),                  |
|    ( pete ),( ike   ),( ned  ),( karl ),( neal ))         |
|                                                           |
|   ( burglar ,                                             |
|    ( al     ),( jim ),( john ),( lance ),                 |
|    ( george ),( don ),( ken  ),( earl  ),( rick ))        |
|                                                           |
|   ( pusher ,                                              |
|    ( art   ),( greg ),( fred ),( gene ),                  |
|    ( ralph ),( phil ),( nick ),( ol   ),( dave ))         |
|                                                           |
o-----------------------------------------------------------o

We now apply 'Study' to the proposition
defining the Jets and Sharks data base.

With a query on the name "ken" we obtain the following
output, giving all the features associated with Ken:

File "ken.sen".  Output of Query on "ken"
o-----------------------------------------------------------o
|                                                           |
|   ken                                                     |
|    sharks                                                 |
|     20's                                                  |
|      high_school                                          |
|       single                                              |
|        burglar                                            |
|                                                           |
o-----------------------------------------------------------o

With a query on the two features "college" and "sharks" we obtain
the following outline of all features satisfying these constraints:

File "cos.sen".  Output of Query on "college" and "sharks"
o-----------------------------------------------------------o
|                                                           |
|   college                                                 |
|    sharks                                                 |
|     30's                                                  |
|      married                                              |
|       bookie                                              |
|        ned                                                |
|       burglar                                             |
|        don                                                |
|       pusher                                              |
|        phil                                               |
|        ol                                                 |
|                                                           |
o-----------------------------------------------------------o

From this we discover that all college Sharks are 30-something and married.
Further, we have a complete listing of their names broken down by occupation,
as no doubt all of them will be, eventually.

Reference.

| McClelland, James L. & Rumelhart, David E.,
|'Explorations in Parallel Distributed Processing:
| A Handbook of Models, Programs, and Exercises',
| MIT Press, Cambridge, MA, 1988.

Those who already know the tune, 
Be at liberty to sing out of it.

Diff Log B 2004 • Note 11


| "The burden of genius is undeliverable"
|  From a poster, as I once misread it,
|  Marlboro, Vermont, c. 1976

How does Cosmo, and by this I mean my pet personification
of cosmic order in the universe, not to be too tautologous
about it, preserve a memory like that, a goodly fraction of
a century later, whether localized to this body that's kept
going by this heart, and whether by common assumption still
more localized to the spongey fibres of this brain, or not?

It strikes me, as it has struck others, that it's terribly
unlikely to be stored in persistent patterns of activation,
for "activation" and "persistent" are nigh a contradiction
in terms, as even the author, Cosmo, of the 'I Ching' knew.

But that was then, this is now, so let me try to say it planar.

Diff Log B 2004 • Note 12


I happened on the graphical syntax for propositional calculus that
I now call the "cactus language" while exploring the confluence of
three streams of thought.  There was C.S. Peirce's use of operator
variables in logical forms and the operational representations of
logical concepts, there was George Spencer Brown's explanation of
a variable as the contemplated presence or absence of a constant,
and then there was the graph theory and group theory that I had
been picking up, bit by bit, since I first encountered them in
tandem in Frank Harary's foundations of math course, c. 1970.

More on that later, as the memories unthaw, but for the moment
I want very much to take care of some long-unfinished business,
and give a more detailed explanation of how I used this syntax
to represent a popular exercise from the PDP literature of the
late 1980's, McClelland's and Rumelhart's "Jets and Sharks".

The knowledge base of the case can be expressed as a single proposition.
The following display presents it in the corresponding text file format.

File "jas.log".  Jets and Sharks Example
o-----------------------------------------------------------o
|                                                           |
|   (( art    ),( al   ),( sam  ),( clyde ),( mike  ),      |
|    ( jim    ),( greg ),( john ),( doug  ),( lance ),      |
|    ( george ),( pete ),( fred ),( gene  ),( ralph ),      |
|    ( phil   ),( ike  ),( nick ),( don   ),( ned   ),      |
|    ( karl   ),( ken  ),( earl ),( rick  ),( ol    ),      |
|    ( neal   ),( dave ))                                   |
|                                                           |
|   ( jets , sharks )                                       |
|                                                           |
|   ( jets ,                                                |
|    ( art    ),( al   ),( sam  ),( clyde ),( mike  ),      |
|    ( jim    ),( greg ),( john ),( doug  ),( lance ),      |
|    ( george ),( pete ),( fred ),( gene  ),( ralph ))      |
|                                                           |
|   ( sharks ,                                              |
|    ( phil ),( ike  ),( nick ),( don ),( ned  ),( karl ),  |
|    ( ken  ),( earl ),( rick ),( ol  ),( neal ),( dave ))  |
|                                                           |
|   (( 20's ),( 30's ),( 40's ))                            |
|                                                           |
|   ( 20's ,                                                |
|    ( sam    ),( jim  ),( greg ),( john ),( lance ),       |
|    ( george ),( pete ),( fred ),( gene ),( ken   ))       |
|                                                           |
|   ( 30's ,                                                |
|    ( al  ),( mike ),( doug ),( ralph ),( phil ),          |
|    ( ike ),( nick ),( don  ),( ned   ),( rick ),          |
|    ( ol  ),( neal ),( dave ))                             |
|                                                           |
|   ( 40's ,                                                |
|    ( art ),( clyde ),( karl ),( earl ))                   |
|                                                           |
|   (( junior_high ),( high_school ),( college ))           |
|                                                           |
|   ( junior_high ,                                         |
|    ( art  ),( al    ),( clyde  ),( mike  ),( jim ),       |
|    ( john ),( lance ),( george ),( ralph ),( ike ))       |
|                                                           |
|   ( high_school ,                                         |
|    ( greg ),( doug ),( pete ),( fred ),                   |
|    ( nick ),( karl ),( ken  ),( earl ),                   |
|    ( rick ),( neal ),( dave ))                            |
|                                                           |
|   ( college ,                                             |
|    ( sam ),( gene ),( phil ),( don ),( ned ),( ol ))      |
|                                                           |
|   (( single ),( married ),( divorced ))                   |
|                                                           |
|   ( single ,                                              |
|    ( art  ),( sam  ),( clyde ),( mike  ),( doug ),        |
|    ( pete ),( fred ),( gene  ),( ralph ),( ike  ),        |
|    ( nick ),( ken  ),( neal  ))                           |
|                                                           |
|   ( married ,                                             |
|    ( al  ),( greg ),( john ),( lance ),( phil ),          |
|    ( don ),( ned  ),( karl ),( earl  ),( ol   ))          |
|                                                           |
|   ( divorced ,                                            |
|    ( jim ),( george ),( rick ),( dave ))                  |
|                                                           |
|   (( bookie ),( burglar ),( pusher ))                     |
|                                                           |
|   ( bookie ,                                              |
|    ( sam  ),( clyde ),( mike ),( doug ),                  |
|    ( pete ),( ike   ),( ned  ),( karl ),( neal ))         |
|                                                           |
|   ( burglar ,                                             |
|    ( al     ),( jim ),( john ),( lance ),                 |
|    ( george ),( don ),( ken  ),( earl  ),( rick ))        |
|                                                           |
|   ( pusher ,                                              |
|    ( art   ),( greg ),( fred ),( gene ),                  |
|    ( ralph ),( phil ),( nick ),( ol   ),( dave ))         |
|                                                           |
o-----------------------------------------------------------o

Let's start with the simplest clause of the conjoint proposition:

    ( jets , sharks )

Drawn as the corresponding cactus graph, we have:

      jets   sharks
        o-----o
         \   /
          \ /
           @

According to my earlier, if somewhat sketchy interpretive suggestions,
we are supposed to picture a quasi-neural pool that contains a couple
of quasi-neural agents or "units", that between the two of them stand
for the logical variables "jets" and "sharks", respectively.  Further,
we imagine these agents to be mutually inhibitory, so that settlement
of the dynamic between them achieves equilibrium when just one of the
two is "active" or "changing" and the other is "stable" or "enduring".

Diff Log B 2004 • Note 13


We were focussing on a particular figure of syntax,
presented here in both graph and string renditions:

o-------------------------------------------------o
|                                                 |
|                     x     y                     |
|                     o-----o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                    ( x , y )                    |
o-------------------------------------------------o

In traversing the cactus graph, in this case a cactus
of one rooted lobe, one starts at the root, reads off
a left parenthesis "(" on the ascent up the left side
of the lobe, reads off the variable "x", counts off a
comma "," as one transits the interior expanse of the
lobe, reads off the variable "y", and then sounds out
a right parenthesiss ")" on the descent down the last
slope that closes out the clause of this cactus lobe.

According to the current story about how the abstract logical situation
is embodied in the concrete physical situation, the whole pool of units
that corresponds to this expression comes to its resting condition when
just one of the two units in {x, y} is resting and the other is charged.
We may think of the state of the whole pool as associated with the root
node of the cactus, here distinguished by an "amphora" or "at" sign "@",
but the root of the cactus is not represented by an individual agent of
the system, at least, not yet.  We may summarize these facts in tabular
form, as shown in Table 5.  Simply by way of a common term, let's count
a single unit as a "pool of one".

Table 5.  Dynamics of (x , y)
o---------o---------o---------o
|    x    |    y    | (x , y) |
o=========o=========o=========o
| charged | charged | charged |
o---------o---------o---------o
| charged | resting | resting |
o---------o---------o---------o
| resting | charged | resting |
o---------o---------o---------o
| resting | resting | charged |
o---------o---------o---------o

I'm going to let that settle a while.

Diff Log B 2004 • Note 14


Table 5 sums up the facts of the physical situation at equilibrium.
If we let B = {note, rest} = {moving, steady} = {charged, resting},
or whatever candidates you pick for the 2-membered set in question,
the Table shows a function f : B x B -> B, where f[x, y] = (x , y).

Table 5.  Dynamics of (x , y)
o---------o---------o---------o
|    x    |    y    | (x , y) |
o=========o=========o=========o
| charged | charged | charged |
o---------o---------o---------o
| charged | resting | resting |
o---------o---------o---------o
| resting | charged | resting |
o---------o---------o---------o
| resting | resting | charged |
o---------o---------o---------o

There are two ways that this physical function
might be taken to represent a logical function:

1.  If we make the identifications:
    
    charged  =  true   (= indicated),
    
    resting  =  false  (= otherwise),
    
    then the physical function f : B x B -> B
    is tantamount to the logical function that
    is commonly known as "logical equivalence",
    or just plain "equality":

    Table 6.  Equality Function
    o---------o---------o---------o
    | x       | y       | (x , y) |
    o=========o=========o=========o
    | true    | true    | true    |
    o---------o---------o---------o
    | true    | false   | false   |
    o---------o---------o---------o
    | false   | true    | false   |
    o---------o---------o---------o
    | false   | false   | true    |
    o---------o---------o---------o

2.  If we make the identifications:
    
    resting  =  true   (= indicated),
    
    charged  =  false  (= otherwise),
    
    then the physical function f : B x B -> B
    is tantamount to the logical function that
    is commonly known as "logical difference",
    or "exclusive disjunction":

    Table 7.  Difference Function
    o---------o---------o---------o
    | x       | y       | (x , y) |
    o=========o=========o=========o
    | false   | false   | false   |
    o---------o---------o---------o
    | false   | true    | true    |
    o---------o---------o---------o
    | true    | false   | true    |
    o---------o---------o---------o
    | true    | true    | false   |
    o---------o---------o---------o

Although the syntax of the cactus language modifies the
syntax of Peirce's graphical formalisms to some extent,
the first interpretation corresponds to what he called
the "entitative graphs" and the second interpretation
corresponds to what he called the "existential graphs".
In working through the present example, I have chosen
the existential interpretation of cactus expressions,
and so the form "(jets , sharks)" is interpreted as
saying that everything in the universe of discourse
is either a Jet or a Shark, but never both at once.

Diff Log B 2004 • Note 15


Before we tangle with the rest of the Jets and Sharks example,
let's look at a cactus expression that's next in the series
we just considered, this time a lobe with three variables.
For instance, let's analyze the cactus form whose graph
and string expressions are shown in the next display.

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                     o--o--o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                    (x, y, z)                    |
o-------------------------------------------------o

As always in this competitive paradigm, we assume that
the units x, y, z are mutually inhibitory, so that the
only states that are possible at equilibrium are those
with exactly one unit charged and all the rest at rest.
Table 8 gives the lobal dynamics of the form (x, y, z).

Table 8.  Lobal Dynamics of the Form (x, y, z)
o-----------o-----------o-----------o-----------o
|     x     |     y     |     z     | (x, y, z) |
o-----------o-----------o-----------o-----------o
|           |           |           |           |
|  charged  |  charged  |  charged  |  charged  |
|           |           |           |           |
|  charged  |  charged  |  resting  |  charged  |
|           |           |           |           |
|  charged  |  resting  |  charged  |  charged  |
|           |           |           |           |
|  charged  |  resting  |  resting  |  resting  |
|           |           |           |           |
|  resting  |  charged  |  charged  |  charged  |
|           |           |           |           |
|  resting  |  charged  |  resting  |  resting  |
|           |           |           |           |
|  resting  |  resting  |  charged  |  resting  |
|           |           |           |           |
|  resting  |  resting  |  resting  |  charged  |
|           |           |           |           |
o-----------o-----------o-----------o-----------o

Given B = {charged, resting} the Table presents the appearance
of a function f : B x B x B -> B, where f[x, y, z] = (x, y, z).

If we make the identifications, charged = false, resting = true,
in accord with the so-called "existential" interpretation, then
the physical function f : B^3 -> B is tantamount to the logical
function that is suggested by the phrase "just 1 of 3 is false".
Table 9 is the truth table for the logical function that we get,
this time using 0 for false and 1 for true in the customary way.

Table 9.  Existential Interpretation of (x, y, z)
o-----------o-----------o-----------o-----------o
|     x     |     y     |     z     | (x, y, z) |
o-----------o-----------o-----------o-----------o
|                                   |           |
|     0           0           0     |     0     |
|                                   |           |
|     0           0           1     |     0     |
|                                   |           |
|     0           1           0     |     0     |
|                                   |           |
|     0           1           1     |     1     |
|                                   |           |
|     1           0           0     |     0     |
|                                   |           |
|     1           0           1     |     1     |
|                                   |           |
|     1           1           0     |     1     |
|                                   |           |
|     1           1           1     |     0     |
|                                   |           |
o-----------------------------------o-----------o

Diff Log B 2004 • Note 16


I sometimes refer to the cactus lobe operators in the series
(), (x_1), (x_1, x_2), (x_1, x_2, x_3), ..., (x_1, ..., x_k)
as "boundary operators" and one of the reasons for this can
be seen most easily in the venn diagram for the k-argument
boundary operator (x_1, ..., x_k).  Figure 10 shows the
venn diagram for the 3-fold boundary form (x, y, z).

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |            X            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
|         /          \%%%%%%/   \%%%%%%/          \         |
|        /            \%%%%/     \%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |        Y        |%%%%%%%|        Z        |       |
|       |                 |%%%%%%%|                 |       |
|       o                 o%%%%%%%o                 o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 10.  Venn Diagram for (x, y, z)

In this picture, the "oval" (actually, octangular) regions that
are customarily said to be "indicated" by the basic propositions
x, y, z : B^3 -> B, that is, where the simple arguments x, y, z,
respectively, evaluate to true, are marked with the corresponding
capital letters X, Y, Z, respectively.  The proposition (x, y, z)
comes out true in the region that is shaded with per cent signs.
Invoking various idioms of general usage, one may refer to this
region as the indicated region, truth set, or fibre of truth
of the proposition in question.

It is useful to consider the truth set of the proposition (x, y, z)
in relation to the logical conjunction xyz of its arguments x, y, z.

In relation to the central cell indicated by the conjunction xyz,
the region indicated by "(x, y, z)" is composed of the "adjacent"
or the "bordering" cells.  Thus they are the cells that are just
across the boundary of the center cell, arrived at by taking all
of Leibniz's "minimal changes" from the given point of departure.

Diff Log B 2004 • Note 17


Any cell in a venn diagram has a well-defined set of nearest neighbors,
and so we can apply a boundary operator of the appropriate rank to the
list of signed features that conjoined would indicate the cell in view.

For example, having computed the "boundary", or what is more properly
called the "point omitted neighborhood" (PON) of the center cell in a
3-dimensional universe of discourse, what is the PON of the cell that
is furthest from it, namely, the "origin cell" indicated as (x)(y)(z)?

The region bordering the origin cell, (x)(y)(z), can be computed by placing
its three signed conjuncts in a 3-place bracket like (__, __, __), arriving
at the cactus expression that is shown below in both graph and string forms.

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                     o  o  o                     |
|                     |  |  |                     |
|                     o--o--o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                  ((x),(y),(z))                  |
o-------------------------------------------------o

Figure 11 shows the venn diagram of this expression,
whose meaning is adequately suggested by the phrase
"just 1 of 3 is true".

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o`````````````````````````o                |
|                |``````````` X ```````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o```o----------o--o             |
|            /````\          \`/          /````\            |
|           /``````\          o          /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    /     \    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |``````` Y ```````|       |`````` Z ````````|       |
|       |`````````````````|       |`````````````````|       |
|       o`````````````````o       o`````````````````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 11.  Venn Diagram for ((x),(y),(z))

Diff Log B 2004 • Note 18


Given the foregoing explanation of the k-fold boundary operator,
along with its use to express such forms of logical constraints
as "just 1 of k is false" and "just 1 of k is true", there will
be no trouble interpreting an expression of the following shape
from the Jets and Sharks example:

   (( art    ),( al   ),( sam  ),( clyde ),( mike  ),
    ( jim    ),( greg ),( john ),( doug  ),( lance ),
    ( george ),( pete ),( fred ),( gene  ),( ralph ),
    ( phil   ),( ike  ),( nick ),( don   ),( ned   ),
    ( karl   ),( ken  ),( earl ),( rick  ),( ol    ),
    ( neal   ),( dave ))

This expression says that everything in the universe of discourse
is either Art, or Al, or ..., or Neal, or Dave, but never any two
of them at once.  In effect, I've exploited the circumstance that
the universe contains but finitely many ostensible individuals to
dedicate its own predicate to each one of them, imposing only the
requirement that these predicates must be disjoint and exhaustive.

Likewise, each of the following clauses has the effect of
partitioning the universe of discourse among the factions
or features that are enumerated in the clause in question.

   ( jets , sharks )

   (( 20's ),( 30's ),( 40's ))

   (( junior_high ),( high_school ),( college ))

   (( single ),( married ),( divorced ))

   (( bookie ),( burglar ),( pusher ))

We may note in passing that ( x , y ) = ((x),(y)),
but a rule of this form holds only in the case of
the 2-fold boundary operator.

Diff Log B 2004 • Note 19


Let's collect the various ways of representing the structure
of a universe of discourse that is described by the following
cactus expressions, verbalized as "just 1 of x, y, z is true".

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                     o  o  o                     |
|                     |  |  |                     |
|                     o--o--o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                  ((x),(y),(z))                  |
o-------------------------------------------------o

Table 12 shows the truth table for the existential
interpretation of the cactus formula ((x),(y),(z)).

Table 12.  Existential Interpretation of ((x),(y),(z))
o-----------o-----------o-----------o-------------o
|     x     |     y     |     z     |  (x, y, z)  |
o-----------o-----------o-----------o-------------o
|                                   |             |
|     0           0           0     |      0      |
|                                   |             |
|     0           0           1     |      1      |
|                                   |             |
|     0           1           0     |      1      |
|                                   |             |
|     0           1           1     |      0      |
|                                   |             |
|     1           0           0     |      1      |
|                                   |             |
|     1           0           1     |      0      |
|                                   |             |
|     1           1           0     |      0      |
|                                   |             |
|     1           1           1     |      0      |
|                                   |             |
o-----------------------------------o-------------o

Figure 13 shows the same data as a 2-colored 3-cube,
coloring a node with a hollow dot (o) for "false"
or a star (*) for "true".

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                        o                        |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /   x (y) z   \                 |
|       x  y (z) o       o       o (x) y  z       |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|       x (y)(z) *       *       * (x)(y) z       |
|                 \  (x) y (z)  /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                    (x)(y)(z)                    |
|                                                 |
o-------------------------------------------------o

Figure 14 repeats the venn diagram that we've already seen.

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o`````````````````````````o                |
|                |``````````` X ```````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o```o----------o--o             |
|            /````\          \`/          /````\            |
|           /``````\          o          /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    /     \    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |``````` Y ```````|       |`````` Z ````````|       |
|       |`````````````````|       |`````````````````|       |
|       o`````````````````o       o`````````````````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 14.  Venn Diagram for ((x),(y),(z))

Figure 15 shows an alternate form of venn diagram for the same
proposition, where we collapse to a nullity all of the regions
on which the proposition in question evaluates to false.  This
leaves a structure that partitions the universe into precisely
three parts.  In mathematics, operations that identify diverse
elements are called "quotient operations".  In this case, many
regions of the universe are being identified with the null set,
leaving only this 3-fold partition as the "quotient structure".

o-----------------------------------------------------------o
| \                                                       / |
|   \                                                   /   |
|     \                                               /     |
|       \                                           /       |
|         \                                       /         |
|           \                 X                 /           |
|             \                               /             |
|               \                           /               |
|                 \                       /                 |
|                   \                   /                   |
|                     \               /                     |
|                       \           /                       |
|                         \       /                         |
|                           \   /                           |
|                             o                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|              Y              |              Z              |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
o-----------------------------o-----------------------------o
Figure 15.  Quotient Structure Venn Diagram for ((x),(y),(z))

Diff Log B 2004 • Note 20


Let's now look at the last type of clause that we find in my
transcription of the Jets and Sharks data base, for instance,
as exemplified by the following couple of lobal expressions:

   ( jets ,
    ( art    ),( al   ),( sam  ),( clyde ),( mike  ),
    ( jim    ),( greg ),( john ),( doug  ),( lance ),
    ( george ),( pete ),( fred ),( gene  ),( ralph ))

   ( sharks ,
    ( phil ),( ike  ),( nick ),( don ),( ned  ),( karl ),
    ( ken  ),( earl ),( rick ),( ol  ),( neal ),( dave ))

Each of these clauses exhibits a generic pattern whose logical properties
may be studied well enough in the form of the following schematic example.

o-------------------------------------------------o
|                                                 |
|                        y  z                     |
|                        o  o                     |
|                     x  |  |                     |
|                     o--o--o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                  ( x ,(y),(z))                  |
o-------------------------------------------------o

The proposition (u, v, w) evaluates to true
if and only if just one of u, v, w is false.
In the same way, the proposition (x,(y),(z))
evaluates to true if and only if exactly one
of x, (y), (z) is false.  Taking it by cases,
let us first suppose that x is true.  Then it
has to be that just one of (y) or (z) is false,
which is tantamount to the proposition ((y),(z)),
which is equivalent to the proposition ( y , z ).
On the other hand, let us suppose that x is the
false one.  Then both (y) and (z) must be true,
which is to say that y is false and z is false.

What we have just said here is that the region
where x is true is partitioned into the regions
where y and z are true, respectively, while the
region where x is false has both y and z false.
In other words, we have a "pie-chart" structure,
where the genus X is divided into the disjoint
and X-haustive couple of species Y and Z.

The same analysis applies to the generic form
(x, (x_1), ..., (x_k)), specifying a pie-chart
with a genus X and the k species X_1, ..., X_k.

Differential Logic 2004 • Series C

Diff Log C 2004 • Note 1


It would be good to summarize, in rough but intuitive terms,
the outlook on differential logic that we have reached so far.

We have been considering a class of operators on universes of discourse,
each of which takes us from considering one universe of discourse, X,
to considering a larger universe of discourse, EX.

Each of these operators, in general terms having the form F : X -> EX,
acts on each proposition p : X -> B of the source universe X to produce
a proposition Fp : EX -> B of the target universe EX.

The two main operators that we have worked with up to this point are the
enlargement operator E : X -> EX and the difference operator D : X -> EX.

E and D take a proposition in X, that is, a proposition p : X -> B
that is said to be "about" the subject matter of X, and produce the
extended propositions Ep, Dp : EX -> B, which may be interpreted as
being about specified collections of changes that might occur in X.

Here we have need of visual representations,
some array of concrete pictures to anchor our
more earthy intuitions and to help us keep our
wits about us before we try to climb any higher
into the ever more rarefied air of abstractions.

One good picture comes to us by way of the "field" concept.
Given a space X, a "field" of a specified type T over X is
formed by assigning to each point of X an object of type T.
If that sounds like the same thing as a function from X to
the space of things of type T, it is, but it does seems to
help to vary the mental pictures and the figures of speech
that naturally spring to mind within these fertile fields.

In the field picture, a proposition p : X -> B becomes a "scalar" field,
that is, a field of values in B, or a "field of true-false indications".

Let us take a moment to view an old proposition
in this new light, for example, the conjunction
uv : X -> B that is depicted in Figure 1.

o-------------------------------------------------o
| X                                               |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /                 o                 \      |
|     /                 /`\                 \     |
|    /                 /```\                 \    |
|   o                 o`````o                 o   |
|   |                 |`````|                 |   |
|   |        U        |`````|        V        |   |
|   |                 |`````|                 |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
|  f =                  u v                       |
o-------------------------------------------------o
Figure 1.  Conjunction uv : X -> B

Each of the operators E, D : X -> EX takes us from considering
propositions p : X -> B, here viewed as "scalar fields" over X,
to considering the corresponding "differential fields" over X,
analogous to what are usually called "vector fields" over X.

The structure of these differential fields can be described this way.
To each point of X there is attached an object of the following type,
a proposition about changes in X, that is, a proposition g : dX -> B.
In this setting, if X is the universe that is generated by the set of
coordinate propositions {u, v}, then dX is the differential universe
that is generated by the set of differential propositions {du, dv}.
These differential propositions may be interpreted as indicating
"change in u" and "change in v", respectively.

A differential operator F, of the first order sort that we have
been considering, takes a proposition p : X -> B and gives back
a differential proposition Fp : EX -> B.

In the field view, we see the proposition p : X -> B as a scalar field
and we see the differential proposition Fp : EX -> B as a vector field,
specifically, a field of propositions about contemplated changes in X.

The field of changes produced by E on uv is shown in Figure 2.

o-------------------------------------------------o
| X                                               |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        U        o        V        \      |
|     /                 /`\                 \     |
|    /                 /```\                 \    |
|   o                 o.->-.o                 o   |
|   |    u(v)(du)dv   |`\`/`|  (u)v du(dv)    |   |
|   | o---------------|->o<-|---------------o |   |
|   |                 |``^``|                 |   |
|   o                 o``|``o                 o   |
|    \                 \`|`/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                  (u)(v) du dv                   |
|                                                 |
o-------------------------------------------------o
|  f =                  u v                       |
o-------------------------------------------------o
|                                                 |
| Ef =              u  v   (du)(dv)               |
|                                                 |
|           +       u (v)  (du) dv                |
|                                                 |
|           +      (u) v    du (dv)               |
|                                                 |
|           +      (u)(v)   du  dv                |
|                                                 |
o-------------------------------------------------o
Figure 2.  Enlargement E[uv] : EX -> B

The differential field E[uv] specifies the changes
that need to be made from each point of X in order
to reach one of the models of the proposition uv,
that is, in order to satisfy the proposition uv.

The field of changes produced by D on uv is shown in Figure 3.

o-------------------------------------------------o
| X                                               |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        U        o        V        \      |
|     /                 /`\                 \     |
|    /                 /```\                 \    |
|   o                 o`````o                 o   |
|   |       (du)dv    |`````|    du(dv)       |   |
|   | o<--------------|->o<-|-------------->o |   |
|   |                 |``^``|                 |   |
|   o                 o``|``o                 o   |
|    \                 \`|`/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                      du dv                      |
|                                                 |
o-------------------------------------------------o
|  f =                  u v                       |
o-------------------------------------------------o
|                                                 |
| Df =              u  v  ((du)(dv))              |
|                                                 |
|           +       u (v)  (du) dv                |
|                                                 |
|           +      (u) v    du (dv)               |
|                                                 |
|           +      (u)(v)   du  dv                |
|                                                 |
o-------------------------------------------------o
Figure 3.  Difference D[uv] : EX -> B

The differential field D[uv] specifies the changes
that need to be made from each point of X in order
to change the value of the proposition uv.

Differential Logic 2004 • Series D


Differential Logic and Dynamic Systems

Author:   Jon Awbrey
Created:  16 Dec 1993
Relayed:  31 Oct 1994
Revised:  03 Jun 2003

0.  Purpose

1.  Review and Transition

2.  A Functional Conception of Propositional Calculus
2.1.  Qualitative Logic and Quantitative Analogy
2.2.  Philosophy of Notation:  Formal Terms and Flexible Types
2.3.  Special Classes of Propositions
2.4.  Basis Relativity and Type Ambiguity
2.5.  The Analogy Between Real and Boolean Types
2.6.  Theory of Control and Control of Theory
2.7.  Propositions as Types and Higher Order Types
2.8.  Reality at the Threshold of Logic
2.9.  Tables of Propositional Forms

3.  A Differential Extension of Propositional Calculus
3.1.  Differential Propositions:  The Qualitative Analogue of Differential Equations
3.2.  An Interlude on the Path
3.3.  The Extended Universe of Discourse
3.4.  Intentional Propositions
3.5.  Life on Easy Street

4.  Back to the Beginning:  Some Exemplary Universes
4.1.  A One-Dimensional Universe
4.2.  Example 1.  A Square Rigging
4.3.  Back to the Feature
4.4.  Tacit Extensions
4.5.  Example 2.  Drives and Their Vicissitudes

5.  Transformations of Discourse

5.1.  Foreshadowing Transformations:  Extensions and Projections of Discourse
5.1.1.  Extension from 1 to 2 Dimensions
5.1.2.  Extension from 2 to 4 Dimensions

5.2.  Thematization of Functions:  And a Declaration of Independence for Variables
5.2.1.  Thematization:  Venn Diagrams
5.2.2.  Thematization:  Truth Tables

5.3.  Propositional Transformations
5.3.1.  Alias and Alibi Transformations
5.3.2.  Transformations of General Type

5.4.  Analytic Expansions:  Operators and Functors
5.4.1.  Operators on Propositions and Transformations
5.4.2.  Differential Analysis of Propositions and Transformations
5.4.2.1.  The Secant Operator:  $E$
5.4.2.2.  The Radius Operator:  $e$
5.4.2.3.  The Phantom of the Operators:  !h!
5.4.2.4.  The Chord Operator:  $D$
5.4.2.5.  The Tangent Operator:  $T$

5.5.  Transformations of Type B^2 -> B^1
5.5.1.  Analytic Expansion of Conjunction
5.5.1.1.  Tacit Extension of Conjunction
5.5.1.2.  Enlargement Map of Conjunction
5.5.1.3.  Digression:  Reflection on Use and Mention
5.5.1.4.  Difference Map of Conjunction
5.5.1.5.  Differential of Conjunction
5.5.1.6.  Remainder of Conjunction
5.5.1.7.  Summary of Conjunction
5.5.2.  Analytic Series:  Coordinate Method
5.5.3.  Analytic Series:  Recap
5.5.4.  Terminological Interlude
5.5.5.  End of Perfunctory Chatter:  Time to Roll the Clip!

5.6.  Taking Aim at Higher Dimensional Targets

5.7.  Transformations of Type B^2 -> B^2

Epilogue, Enchoiry, Exodus

Diff Log D 2004 • Note 1


Differential Logic and Dynamic Systems

| Stand and unfold yourself.
|
| 'Hamlet', 1.1.2

Purpose

This series of reports develops a differential extension of
propositional calculus and applies it to a context of problems
arising in dynamic systems.  The work pursued here is coordinated
with a parallel application that focuses on neural network systems,
but the dependencies are arranged to make the present series the
main and the more self-contained work, to serve as a conceptual
frame and a technical background for the network project.

Review and Transition

This note continues a previous discussion on the problem of dealing
with change and diversity in logic-based intelligent systems.  For
ease of reference, I begin by summarizing essential material from
previous reports.

Table 1 outlines the notation that I use for propositional calculus.
Explained as briefly as possible, I am using only two basic kinds
of truth-functional connectives, both of variable k-ary scope.

1.  For the first, I use concatenation as a connective
    to indicate the logical conjunction of k arguments.

2.  For the other, I use a bracket of the form ( , , , )
    as a connective which says that exactly one of its k
    arguments is false.

All other truth-functional connectives can be obtained in a very
efficient style of representation through combinations of these
two forms.

This treatment of propositional logic is derived from the work of
C.S. Peirce [P1, P2], who gave this approach an extensive development
in his graphical systems of predicate, relational, and modal logic [Rob].
More recently, these ideas were revived and supplemented in an alternative
interpretation by G. Spencer-Brown [SpB].  Both of these authors used other
forms of enclosure where I use parentheses, but the structural topologies of
expression and the functional varieties of interpretation are fundamentally
the same.

While working with expressions solely in propositional calculus, the use
of plain parentheses to represent logical connectives is simplest to write
and easiest to read for both human and machine parsers.  In the present text
I preserve this form of expression in tables and set-off displays, but in
contexts where parentheses are needed for functional notation I will use
a distinctive font for logical operators.  [Not available in Ascii.]

The briefest expression for logical truth is the empty word, usually denoted by
epsilon or lambda in formal languages, where it forms the identity element for
concatenation.  To make it visible in this text, I denote it by the equivalent
expression "(())", or, especially if operating in an algebraic context, by a
simple "1".  Also when working in an algebraic mode, I use the plus sign "+"
for exclusive disjunction.  Thus, we may express the following paraphrases
of algebraic forms:

A + B      =  (A, B)

A + B + C  =  ((A, B), C)  =  (A, (B, C))

One should be careful to observe that these last two
expressions are not equivalent to the form (A, B, C).

Table 1.  Syntax & Semantics of a Calculus for Propositional Logic
o-------------------o-------------------o-------------------o
|    Expression     |  Interpretation   |  Other Notations  |
o-------------------o-------------------o-------------------o
|  " "              | True.             |  1                |
o-------------------o-------------------o-------------------o
|  ()               | False.            |  0                |
o-------------------o-------------------o-------------------o
|  A                | A.                |  A                |
o-------------------o-------------------o-------------------o
|  (A)              | Not A.            |  A'               |
|                   |                   |  ~A               |
o-------------------o-------------------o-------------------o
|  A B C            | A and B and C.    |  A & B & C        |
o-------------------o-------------------o-------------------o
|  ((A)(B)(C))      | A or B or C.      |  A v B v C        |
o-------------------o-------------------o-------------------o
|  (A (B))          | A implies B.      |  A => B           |
|                   | If A then B.      |                   |
o-------------------o-------------------o-------------------o
|  (A, B)           | A not equal to B. |  A =/= B          |
|                   | A exclusive-or B. |  A  +  B          |
o-------------------o-------------------o-------------------o
|  ((A, B))         | A is equal to B.  |  A  =  B          |
|                   | A if & only if B. |  A <=> B          |
o-------------------o-------------------o-------------------o
|  (A, B, C)        | Just one of       |  A'B C  v         |
|                   | A, B, C           |  A B'C  v         |
|                   | is false.         |  A B C'           |
o-------------------o-------------------o-------------------o
|  ((A),(B),(C))    | Just one of       |  A B'C' v         |
|                   | A, B, C           |  A'B C' v         |
|                   | is true.          |  A'B'C            |
|                   |                   |                   |
|                   | Partition all     |                   |
|                   | into A, B, C.     |                   |
o-------------------o-------------------o-------------------o
|  ((A, B), C)      | Oddly many of     |  A + B + C        |
|  (A, (B, C))      | A, B, C           |                   |
|                   | are true.         |  A B C  v         |
|                   |                   |  A B'C' v         |
|                   |                   |  A'B C' v         |
|                   |                   |  A'B'C            |
o-------------------o-------------------o-------------------o
|  (Q, (A),(B),(C)) | Partition  Q      |  Q'A'B'C' v       |
|                   | into A, B, C.     |  Q A B'C' v       |
|                   |                   |  Q A'B C' v       |
|                   | Genus Q comprises |  Q A'B'C          |
|                   | species A, B, C.  |                   |
o-------------------o-------------------o-------------------o

NB. The usage that one often sees, of a plus sign "+"
to represent inclusive disjunction, and the reference
to this operation as "boolean addition", is a misnomer
on at least two counts.  Boole used the plus sign to
represent exclusive disjunction (at any rate, an
operation of aggregation restricted in its logical
interpretation to cases where the represented sets
are disjoint [Boo, 32]), as any mathematician with
a sensitivity to the ring and field properties of
algebra would do:

| The expression x + y seems indeed uninterpretable,
| unless it be assumed that the things represented
| by x and the things represented by y are entirely
| separate;  that they embrace no individuals in
| common.  [Boo, 66].

It was only later that Peirce and Jevons treated inclusive
disjunction as a fundamental operation, but these authors,
with a respect for the algebraic properties that were already
associated with the plus sign, used a variety of other symbols
for inclusive disjunction [Sty, 177, 189].  It seems to have
been Schroeder who later reassigned the plus sign to inclusive
disjunction [Sty, 208].  Additional information, discussion,
and references can be found in [Boo] and [Sty, 177-263].
Aside from these historical points, which never really
count against a current practice that has gained a life
of its own, this usage does have a further disadvantage
of cutting or confounding the lines of communication
between algebra and logic.  For this reason, I am
forced to avoid it here.

Diff Log D 2004 • Note 2


| Out of the dimness opposite equals advance . . . .
|    Always substance and increase,
| Always a knit of identity . . . . always distinction . . . .
|    always a breed of life.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 28]

A Functional Conception of Propositional Calculus

In the general case, we start with a set of logical features {a_1, ..., a_n}
that represent properties of objects or propositions about the world.  In
concrete examples the features {a_i} commonly appear as capital letters
from an "alphabet" like {A, B, C, ...} or as meaningful words from a
linguistic "vocabulary" of codes.  This language can be drawn from
any sources, whether natural, technical, or artificial in character
and interpretation.  In the application to dynamic systems we tend
to use the letters {x_1, ... , x_n} as our coordinate propositions,
and to interpret them as denoting properties of a system's "state",
that is, as propositions about its location in configuration space.
Because I have to consider non-deterministic systems from the outset,
I often use the word "state" in a loose sense, to denote the position
or configuration component of a contemplated state vector, whether or
not it ever gets a deterministic completion.

The set of logical features {a_1, ..., a_n} provides a basis for generating
an n-dimensional "universe of discourse" that I denote as [a_1, ..., a_n].
It is useful to consider each universe of discourse as a unified categorical
object that incorporates both the set of points <|a_1, ... , a_n|> and the set
of propositions f : <|a_1, ..., a_n|> -> B that are implicit with the ordinary
picture of a venn diagram on n features.  Thus, we may regard the universe of
discourse [a_1, ..., a_n] as an ordered pair having the type (B^n, (B^n - >B)),
and we may abbreviate this last type designation as (B^n +-> B), or even more
succinctly as [B^n].  (NB.  I am using "<| ... |>" as "generator brackets".)

Table 2 exhibits the scheme of notation I use to formalize the domain
of propositional calculus, corresponding to the logical content of
truth tables and venn diagrams.  Although it overworks the square
brackets a bit, I also use either one of the equivalent notations
[n] or #n# to denote the data type of a finite set on n elements.

Table 2.  Fundamental Notations for Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol  | Notation          | Description       | Type              |
o---------o-------------------o-------------------o-------------------o
| !A!     | {a_1, ..., a_n}   | Alphabet          | [n]  =  #n#       |
o---------o-------------------o-------------------o-------------------o
|  A_i    | {(a_i), a_i}      | Dimension i       |  B                |
o---------o-------------------o-------------------o-------------------o
|  A      | <|!A!|>           | Set of cells,     |  B^n              |
|         | <|a_i, ..., a_n|> | coordinate tuples,|                   |
|         | {<a_i, ..., a_n>} | interpretations,  |                   |
|         | A_1 x ... x A_n   | points, or vectors|                   |
|         | Prod_i A_i        | in the universe   |                   |
o---------o-------------------o-------------------o-------------------o
|  A*     | (hom : A -> B)    | Linear functions  | (B^n)*  =  B^n    |
o---------o-------------------o-------------------o-------------------o
|  A^     | (A -> B)          | Boolean functions |  B^n -> B         |
o---------o-------------------o-------------------o-------------------o
|  A%     | [!A!]             | Universe of Disc. | (B^n, (B^n -> B)) |
|         | (A, A^)           | based on features | (B^n +-> B)       |
|         | (A +-> B)         | {a_1, ..., a_n}   | [B^n]             |
|         | (A, (A -> B))     |                   |                   |
|         | [a_1, ..., a_n]   |                   |                   |
o---------o-------------------o-------------------o-------------------o

Diff Log D 2004 • Note 3


Qualitative Logic and Quantitative Analogy

| Logical, however, is used in a third sense, which is at once more
| vital and more practical;  to denote, namely, the systematic care,
| negative and positive, taken to safeguard reflection so that it
| may yield the best results under the given conditions.
|
| John Dewey, 'How We Think', [Dew, 56]

These concepts and notations can now be explained in greater detail.
In order to begin as simply as possible, I distinguish two levels of
analysis and set out initially on the easier path.  On the first level
of analysis, I take spaces like B, B^n, and (B^n->B) at face value and
treat them as the primary objects of interest.  On the second level of
analysis, I use these spaces as coordinate charts for talking about
points and functions in more fundamental spaces.

A pair of spaces, of types B^n and (B^n->B), give typical expression
to everything that we commonly associate with the ordinary picture of
a venn diagram.  The dimension, n, counts the number of "circles" or
simple closed curves that are inscribed in the universe of discourse,
corresponding to its relevant logical features or basic propositions.
Elements of type B^n correspond to what are often called propositional
"interpretations" in logic, that is, the different assignments of truth
values to sentence letters.  Relative to a given universe of discourse,
these interpretations are visualized as its "cells", in other words,
the smallest enclosed areas or undivided regions of the venn diagram.
The functions f : B^n -> B correspond to the different ways of shading
the venn diagram to indicate arbitrary propositions, regions, or sets.
Regions included under a shading indicate the "models", and regions
excluded represent the "non-models" of a proposition.  To recognize
and formalize the natural cohesion of these two layers of concepts
into a single universe of discourse, I introduce the type notations
[B^n] = B^n +-> B to stand for the pair of types (B^n, (B^n -> B)).
The resulting "stereotype" serves to frame the universe of discourse
as a unified categorical object, and makes it subject to prescribed
sets of evaluations and transformations (categorical morphisms or
"arrows") that affect the universe of discourse as an integrated
whole.

Most of the time we can serve the algebraic, geometric, and logical interests
of our study without worrying about their occasional conflicts and incidental
divergences.  The conventions and definitions already set down will continue
to cover most of the algebraic and functional aspects of our discussion, but
to handle the logical and qualitative aspects we will need to add a few more.
In general, abstract sets may be denoted by gothic, greek, or script capital
variants of A, B, C, and so on, with elements denoted by a corresponding set
of subscripted letters in plain lower case, for example, !A! = {a_i}.  Most
of the time, a set such as !A! = {a_i} will be employed as the "alphabet" of
a formal language.  These alphabet letters serve to name the logical features
(properties or proposition) that generate a particular universe of discourse.
When we want to discuss the particular features of a universe of discourse,
beyond the abstract designation of a type like (B^n +-> B), then we may use
the following notations.  If !A! = {a_1, ..., a_n} is an alphabet of logical
features, then A = <|!A!|> = <|a_1, ..., a_n|> is the set of interpretations,
A^ = (A -> B) is the set of propositions, and A% = [!A!] = [a_1, ..., a_n] is
the combination of these interpretations and propositions into the universe of
discourse that is based on the features {a_1, ..., a_n}.

As always, especially in concrete examples, these rules may be dropped whenever
necessary, reverting to a free assortment of feature labels.  However, when we
need to talk about the logical aspects of a space that is already named as a
vector space, it will be necessary to make special provisions.  At any rate,
these elaborations can be deferred until actually needed.

Diff Log D 2004 • Note 4


Philosophy of Notation:  Formal Terms and Flexible Types

| Where number is irrelevant, regimented mathematical technique has hitherto
| tended to be lacking.  Thus it is that the progress of natural science has
| depended so largely upon the discernment of measurable quantity of one sort
| or another.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7]

For much of our discussion propositions and boolean functions are treated as the
same formal objects, or as different interpretations of the same formal calculus.
This rule of has exceptions, though.  There is a distinctively logical interest
in the use of propositional calculus that is not exhausted by its functional
interpretation.  It is part of our task in this study to deal with these
uniquely logical characteristics as they present themselves both in our
subject matter and in our formal calculus.  Just to provide a hint of
what's at stake:  In logic, as opposed to the more imaginative realms
of mathematics, we consider it a good thing to always know what we are
talking about.  Where mathematics encourages tolerance for uninterpreted
symbols as intermediate terms, logic exerts a keener effort to interpret
directly each oblique carrier of meaning, no matter how slight, and to
unfold the complicities of every indirection in the flow of information.
Translated into functional terms, this means that we want to maintain a
continual, immediate, and persistent sense of both the converse relation
f^(-1) c B x B^n, or what is the same thing, f^(-1) : B -> Pow(B^n), and
the "fibers" or inverse images f^(-1)(0) and f^(-1)(1), associated with
each boolean function f : B^n -> B that we use.  In practical terms, the
desired implementation of a propositional interpreter should incorporate
our intuitive recognition that the induced partition of the functional
domain into level sets f^(-1)(b), for b in B, is part and parcel of
understanding the denotative uses of each propositional function f.

Diff Log D 2004 • Note 5


Special Classes of Propositions

It is important to remember that the coordinate propositions {a_i},
besides being projection maps a_i : B^n -> B, are propositions on
an equal footing with all others, even though employed as a basis
in a particular moment.  This set of n propositions may sometimes
be referred to as the "basic" or "simple" propositions that found
the universe of discourse.  As typical and collective notations,
we may use the forms {a_i : B^n -> B} = (B^n -i-> B) = (B^n -:> B)
to indicate the adoption of a set of a_i as a basis for discourse.

Among the 2^2^n propositions or functions in (B^n -> B) are several
fundamental sets of 2^n propositions each that take on special forms
with respect to a given basis !A! = {a_i}.  Three of these forms are
especially common, the "linear", the "positive", and the "singular"
propositions.  Each set is naturally parameterized by the coordinate
vectors in B^n and falls into n+1 ranks, with a binomial coefficient
C(n, k) giving the number of propositions that have rank or weight k.

The "linear" propositions, {hom : B^n -> B} = (B^n -h-> B) = (B^n ++> B),
may be expressed as sums of the following form:

   Sum_i e_i    =   e_1 + ... + e_n   where   e_i = a_i   or   e_i = 0.

The "positive" propositions, {pos : B^n -> B} = (B^n -p-> B) = (B^n >=> B),
may be expressed as products of the following form:

   Prod_i e_i   =   e_1 * ... * e_n   where   e_i = a_i   or   e_i = 1.

The "singular" propositions, {x : B^n -> B} = (B^n -s-> B) = (B^n ::> B),
may be expressed as products of the following form:

   Prod_i e_i   =   e_1 * ... * e_n   where   e_i = a_i   or   e_i = (a_i).

In each case the rank k ranges from 0 to n and counts the number of
positive appearances of coordinate propositions a_i in the resulting
expression.  For example, for n = 3, the linear proposition of rank 0
is "0", the positive proposition of rank 0 is "1", and the singular
proposition of rank 0 is "(a_1)(a_2)(a_3)".

The coordinate projections or simple propositions a_i : B^n -> B are both
linear and positive.  So these two kinds of propositions, the linear or the
positive, may be viewed as two different ways of generalizing the class of
simple projections.  The linear and the positive propositions are generated
by taking boolean sums and products, respectively, over selected subsets of
the basic propositions in {a_i}.  Therefore, each set of functions can be
parameterized by the subsets J of the basic index set I = {1, ..., n}.
Let us define A_J as the subset of A that is given by {a_i : i in J}.
Then we may comprehend the action of the linear and the positive
propositions in the following terms:

1.  The linear proposition hom_J : B^n -> B evaluates each cell x of B^n
    by looking at x's coefficients with respect to the features that hom_J
    "likes", namely those in A_J, and then adds them up in B.  Thus, hom_J (x)
    computes the parity of the number of features that x has in A_J, yielding
    one for odd and zero for even.  Expressed in this idiom, hom_J (x) = 1
    says that x seems "odd" (or "oddly true") to A_J, whereas hom_J (x) = 0
    says that x seems "even" (or "evenly true") to A_J, so long as we recall
    that "zero times" is evenly often, too.

2.  The positive proposition pos_J : B^n -> B evaluates each cell x of B^n
    by looking at x's coefficients with regard to the features that pos_J
    "likes", namely those in A_J, and then takes their product in B.  Thus,
    pos_J (x) assesses the unanimity of the multitude of features that x has
    in A_J, yielding one for all and aught for else.  In these consensual or
    contractual terms, pos_J (x) = 1 means that x is "AOK" or congruent with
    all of the conditions of A_J, while pos_J (x) = 0 means that x defaults
    or dissents from some condition of A_J.

Diff Log D 2004 • Note 6


Basis Relativity and Type Ambiguity

Finally, two things are important to keep in mind with regard to the
simplicity, linearity, positivity, and singularity of propositions.

First, all of these properties are relative to a particular basis.
For example, a singular proposition with respect to a basis !A!
will not remain singular if A is extended by a number of new
and independent features.  Even if we stick to the original
set of pairwise options {a_i} |_| {(a_i)} to select a new
basis, the sets of linear and positive propositions are
determined by the choice of simple propositions, and
this determination is tantamount to the conventional
choice of a cell as origin.

Second, the singular propositions B ::> B, picking out as they do a single cell
or a coordinate tuple of B^n, become the carriers or the vehicles of a certain
type-ambiguity that vacillates between the dual forms B^n and (B^n ::> B) and
infects the whole hierarchy of types built on them.  In plainer language, the
terms that signify the interpretations x : B^n and the singular propositions
x : B^n ::> B are fully equivalent in information, and this means that every
token of the type B^n can be reinterpreted as an appearance of the subtype
B^n ::> B.  And vice versa, the two types can be exchanged with each other
everywhere that they turn up.  In practical terms, this allows the use of
singular propositions as a way of denoting points, forming an alternative
to coordinate tuples.  For example, relative to the universe of discourse
[a_1, a_2, a_3] the singular proposition a_1 a_2 a_3 : B^3 ::> B could be
explicitly retyped as a_1 a_2 a_3 : B^3 to indicate the point <1, 1, 1>,
but in most cases the proper interpretation could be gathered from context.
Both notations remain dependent on a particular basis, but the code that is
generated under the singular option has the advantage in its self-commenting
features, in other words, it constantly reminds us of its basis in the process
of denoting points.  When the time comes to put a multiplicity of different bases
into play, and to search for objects and properties that remain invariant under the
transformations between them, this infinitesimal potential advantage may well evolve
into an overwhelming practical necessity.

Diff Log D 2004 • Note 7


The Analogy Between Real and Boolean Types

| Measurement consists in correlating our subject matter with the series of
| real numbers;  and such correlations are desirable because, once they are
| set up, all the well-worked theory of numerical mathematics lies ready at
| hand as a tool for our further reasoning.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7]

There are two further reasons why I am spending so much time on a careful
treatment of types, and they both have to do with our being able to take
full computational advantage of certain dimensions of flexibility in the
types that apply to terms.  First, the domains of differential geometry
and logic programming are connected by analogies between real and boolean
types of the same pattern.  Second, the types involved in these patterns
have important isomorphisms connecting them that apply on both the real
and the boolean sides of the picture.

Amazingly enough, these isomorphisms are themselves schematized by the
axioms and theorems of propositional logic.  This fact is known as the
"propositions as types" analogy or the Curry-Howard isomorphism [How].
In another formulation it says that terms are to types as proofs are
to propositions.  This principle seems to have more implications for
our subject than I can fully comprehend at present, though I sense
that they must be crucial.  (Cf. [LaS, 42-46] and [SeH] for a good
discussion and further references.)  To anticipate the bearing of
these issues on our immediate topic, Table 3 sketches a partial
overview of the Real to Boolean analogy that may serve to
illustrate the paradigm that I have in mind.

Table 3.  Analogy of Real and Boolean Types
o-------------------------o-------------------------o-------------------------o
|      Real Domain R      |           <->           |    Boolean Domain B     |
o-------------------------o-------------------------o-------------------------o
|           R^n           |       Basic Space       |           B^n           |
o-------------------------o-------------------------o-------------------------o
|        R^n -> R         |     Function Space      |        B^n -> B         |
o-------------------------o-------------------------o-------------------------o
|     (R^n -> R) -> R     |     Tangent Vector      |     (B^n -> B) -> B     |
o-------------------------o-------------------------o-------------------------o
| R^n -> ((R^n -> R) -> R)|      Vector Field       | B^n -> ((B^n -> B) -> B)|
o-------------------------o-------------------------o-------------------------o
| (R^n x (R^n -> R)) -> R |          ditto          | (B^n x (B^n -> B)) -> B |
o-------------------------o-------------------------o-------------------------o
| ((R^n -> R) x R^n) -> R |          ditto          | ((B^n -> B) x B^n) -> B |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^n -> R)|       Derivation        | (B^n -> B) -> (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
|        R^n -> R^m       |  Basic Transformation   |        B^n -> B^m       |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)|
o-------------------------o-------------------------o-------------------------o
|           ...           |           ...           |           ...           |
o-------------------------o-------------------------o-------------------------o

The Table exhibits a sample of likely parallels between the real and boolean domains.
The central column gives a selection of terminology that I borrow from typical usage
in differential geometry and extend in its meaning to the logical side of the Table.
These are the varieties of spaces that come up when we turn to analyzing the dynamics
of processes that followe their courses through the states of an arbitrary space X.
Moreover, when it becomes necessary to approach situations of overwhelming dynamic
complexity in a succession of qualitative reaches, then the methods of logic that
are afforded by the boolean domains, with their declarative means of synthesis
and deductive modes of analysis, supply a natural battery of tools for the task.

It is usually expedient to take these spaces two at a time, in dual pairs of the
form X and (X -> K).  In general, one creates pairs of type schemas by replacing
any space X with its dual (X -> K), for example, pairing the type X -> Y with the
type (X -> K) -> (Y -> K), and X x Y with (X -> K) x (Y -> K).  Here, I am using
the word "dual" in its larger sense to mean all of the functionals, not just the
linear ones.  Given any function f : X -> K, the "converse" or inverse relation
corresponding to f is denoted as f^(-1), and the subsets of X that are defined
by f^(-1)(k), taken over k in K, are called the "fibers" or the "level sets"
of the function f.

Diff Log D 2004 • Note 8


Theory of Control and Control of Theory

| You will hardly know who I am or what I mean,
| But I shall be good health to you nevertheless,
| And filter and fibre your blood.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]

In the boolean context, a function f : X -> B is tantamount to a "proposition"
about elements of X, and the elements of X constitute the "interpretations" of
that proposition.  The fiber f^(-1)(1) comprises the set of "models" of f, or
examples of elements in X satisfying the proposition f.  The fiber f^(-1)(0)
collects the complementary set of "anti-models", or the exceptions to the
proposition f that exist in X.  Of course, the space of functions (X -> B)
is isomorphic to the set of all subsets of X, called the "power set" of X
and often denoted as Pow(X) or 2^X.

The operation of replacing X by (X -> B) in a type schema corresponds
to a certain shift of attitude towards the space X, in which one passes
from a focus on the ostensibly individual elements of X to a concern with
the states of information and uncertainty that one possesses about objects
and situations in X.  The conceptual obstacles in the path of this transition
can be smoothed over by using singular functions (X ::> B) as stepping stones.
First of all, it's an easy step from an element x of type B^n to the equivalent
information of a singular proposition x : X ::> B, and then only a small jump of
generalization remains to reach the type of an arbitrary proposition f : X -> B,
perhaps understood to indicate a relaxed constraint on the singularity of points
or a neighborhood circumscribing the original x.  I have frequently discovered
this to be a useful transformation, communicating between the "objective" and
the "intentional" perspectives, in spite perhaps of the open objection that
this distinction is transient in the mean time and ultimately superficial.

It is hoped that this measure of flexibility, allowing us to stretch a point
into a proposition, can be useful in the examination of inquiry driven systems,
where the differences between empirical, intentional, and theoretical propositions
constitute the discrepancies and the distributions that drive experimental activity.
I can give this model of inquiry a cybernetic cast by realizing that theory change
and theory evolution, as well as the choice and the evaluation of experiments, are
actions that are taken by a system or its agent in response to the differences
that are detected between observational contents and theoretical coverage.

All of the above notwithstanding, there are several points that distinguish
these two tasks, namely, the "theory of control" and the "control of theory",
features that are often obscured by too much precipitation in the quickness
with which we understand their similarities.  In the control of uncertainty
through inquiry, some of the actuators that we need to be concerned with are
axiom changers and theory modifiers, operators with the power to compile and
to revise the theories that generate expectations and predictions, effectors
that form and edit our grammars for the languages of observational data, and
agencies that rework the proposed model to fit the actual sequences of events
and the realized relationships of values that are observed in the environment.
Moreover, when steps must be taken to carry out an experimental action, there
must be something about the particular shape of our uncertainty that guides us
in choosing what directions to explore, and this impression is more than likely
influenced by previous accumulations of experience.  Thus it must be anticipated
that much of what goes into scientific progress, or any sustainable effort toward
a goal of knowledge, is necessarily predicated on long term observation and modal
expectations, not only on the more local or short term prediction and correction.

Diff Log D 2004 • Note 9


Propositions as Types and Higher Order Types

The arrangement of types collected in Table 3 can
serve as a good introduction to several ideas about
"higher order propositional expressions" (HOPE's) and
also about the "propositions as types" (PAT) isomorphism.

Table 3.  Analogy of Real and Boolean Types
o-------------------------o-------------------------o-------------------------o
|      Real Domain R      |           <->           |    Boolean Domain B     |
o-------------------------o-------------------------o-------------------------o
|           R^n           |       Basic Space       |           B^n           |
o-------------------------o-------------------------o-------------------------o
|        R^n -> R         |     Function Space      |        B^n -> B         |
o-------------------------o-------------------------o-------------------------o
|     (R^n -> R) -> R     |     Tangent Vector      |     (B^n -> B) -> B     |
o-------------------------o-------------------------o-------------------------o
| R^n -> ((R^n -> R) -> R)|      Vector Field       | B^n -> ((B^n -> B) -> B)|
o-------------------------o-------------------------o-------------------------o
| (R^n x (R^n -> R)) -> R |          ditto          | (B^n x (B^n -> B)) -> B |
o-------------------------o-------------------------o-------------------------o
| ((R^n -> R) x R^n) -> R |          ditto          | ((B^n -> B) x B^n) -> B |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^n -> R)|       Derivation        | (B^n -> B) -> (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
|        R^n -> R^m       |  Basic Transformation   |        B^n -> B^m       |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)|
o-------------------------o-------------------------o-------------------------o
|           ...           |           ...           |           ...           |
o-------------------------o-------------------------o-------------------------o

First, observe that the type of a "Tangent Vector at a Point",
also known as a "directional derivative" at that point, has the
form (K^n -> K) -> K, where K is the chosen ground field, in the
present case either R or B.  At a point in a space of type K^n,
a directional derivative operator !q! takes a function on that
space, an f of type (K^n -> K), and maps it to a ground field
value of type K.  This value is known as the "derivative" of
f in the direction !q! [Che46, 76-77].  In the boolean case,
!q! : (B^n -> B) -> B has the form of a proposition about
propositions, in other words, a proposition of the next
higher type.

Next, by way of illustrating the propositions as types theme, consider
a proposition of the form X => (Y => Z).  One knows from propositional
calculus that this is logically equivalent to a proposition of the form
(X & Y) => Z.  But this equivalence should remind us of the functional
isomorphism that exists between a construction of the type X -> (Y -> Z)
and a construction of the type (X x Y) -> Z.  The propositions as types
analogy permits us to take a functional type like this and, under the
right conditions, replace the functional arrows "->" and products "x"
with the respective logical arrows "=>" and products "&".  Accordingly,
viewing the result as a proposition, we can employ axioms and theorems
of propositional calculus to suggest appropriate isomorphisms among the
categorical and functional constructions.

Finally, examine the middle four rows of Table 3.  These display
a series of isomorphic types that stretch from the categories that
are labeled "Vector Field" to those that are labeled "Derivation".
A "vector field", also known as an "infinitesimal transformation",
associates a tangent vector at a point with each point of a space.
In symbols, a vector field is a function !X! : X -> |_| !X!_x that
assigns to each point x of the space X a tangent vector !X!_x to X
at that point [Che46, 82-83].  If X is of type K^n, then !X! is of
type K^n -> ((K^n -> K) -> K).  This has the pattern X -> (Y -> Z),
with X = K^n, Y = (K^n -> K), and Z = K.

Applying the propositions as types analogy, one can follow this pattern
through a series of metamorphoses from the type of a vector field to the
type of a derivation, as traced out in Table 4.  Observe how the function
f : X -> K, associated with the place of Y in the pattern, moves through
its paces from the second to the first position.  In this way, the vector
field !X!, initially viewed as attaching each tangent vector !X!_x to the
site x where it acts in X, now comes to be seen as acting on each scalar
potential f : X -> K like a generalized species of differentiation,
producing another function !X!f : X -> K of the same type.

Table 4.  An Equivalence Based on the Propositions as Types Analogy
o-------------------------o------------------------o--------------------------o
|         Pattern         |      Construction      |        Instance          |
o-------------------------o------------------------o--------------------------o
|      X -> (Y -> Z)      |      Vector Field      | K^n -> ((K^n -> K) -> K) |
o-------------------------o------------------------o--------------------------o
|     (X x Y)  -> Z       |                        | (K^n x (K^n -> K)) -> K  |
o-------------------------o------------------------o--------------------------o
|     (Y x X)  -> Z       |                        | ((K^n -> K) x K^n) -> K  |
o-------------------------o------------------------o--------------------------o
|      Y -> (X -> Z)      |       Derivation       | (K^n -> K) -> (K^n -> K) |
o-------------------------o------------------------o--------------------------o

Diff Log D 2004 • Note 10


Reality at the Threshold of Logic

| But no science can rest entirely on measurement, and many
| scientific investigations are quite out of reach of that
| device.  To the scientist longing for non-quantitative
| techniques, then, mathematical logic brings hope.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7]

Table 5 accumulates an array of notation that I hope will not be too distracting.
Some of it is rarely needed, but has been filled in for the sake of completeness.
Its purpose is simple, to give literal expression to the visual intuitions that
come with venn diagrams, and to help build a bridge between our qualitative
and quantitative outlooks on dynamic systems.

NB.  I'm trying to keep the Asciification of the text as simple as possible
this time around, so I will use emphasis bars !...! around characters in
a context-dependent way, sometimes for Gothic and sometimes for Greek.
Thus, I will rely on the reader's own recognizance to discriminate
between entities like a "Script X" alphabet !X! = {x_1, ..., x_n}
and a "Greek Chi" vector field !X!.  Also, the original version
of the Table below used underlined variants of the characters
in the middle column, to suggest quantities rising above the
relevant threshold.  In this copy, I use grave markers `...`
around the thresheld symbols instead of underscoring them.
Finally, there does not seem to be any way to avoid the
clash of symbols between the stars, that is, the '*'
that is used in algebra to denote the dual space of
linear functionals, and the '*' that is used in
formal language theory to denote the set of
all finite sequences over an alphabet.

Table 5.  A Bridge Over Troubled Waters
o-------------------------o-------------------------o-------------------------o
|      Linear Space       |      Liminal Space      |      Logical Space      |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| !X!                     | !`X`!                   | !A!                     |
|                         |                         |                         |
| {x_1, ..., x_n}         | {`x`_1, ..., `x`_n}     | {a_1, ..., a_n}         |
|                         |                         |                         |
| cardinality n           | cardinality n           | cardinality n           |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X_i                     | `X`_i                   | A_i                     |
|                         |                         |                         |
| <|x_i|>                 | {(`x`_i), `x`_i}        | {(a_i), a_i}            |
|                         |                         |                         |
| isomorphic to K         | isomorphic to B         | isomorphic to B         |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X                       | `X`                     | A                       |
|                         |                         |                         |
| <|!X!|>                 | <|!`X`!|>               | <|!A!|>                 |
|                         |                         |                         |
| <|x_1, ..., x_n|>       | <|`x`_1, ..., `x`_n|>   | <|a_1, ..., a_n|>       |
|                         |                         |                         |
| {<x_1, ..., x_n>}       | {<`x`_1, ..., `x`_n>}   | {<a_1, ..., a_n>}       |
|                         |                         |                         |
| X_1 x ... x X_n         | `X`_1 x ... x `X`_n     | A_1 x ... x A_n         |
|                         |                         |                         |
| Prod_i X_i              | Prod_i `X`_i            | Prod_i A_i              |
|                         |                         |                         |
| isomorphic to K^n       | isomorphic to B^n       | isomorphic to B^n       |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X*                      | `X`*                    | A*                      |
|                         |                         |                         |
| (hom : X -> K)          | (hom : `X` -> B)        | (hom : A -> B)          |
|                         |                         |                         |
| isomorphic to K^n       | isomorphic to B^n       | isomorphic to B^n       |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X^                      | `X`^                    | A^                      |
|                         |                         |                         |
| (X -> K)                | (`X` -> B)              | (A -> B)                |
|                         |                         |                         |
| isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X%                      | `X`%                    | A%                      |
|                         |                         |                         |
| [!X!]                   | [!`X`!]                 | [!A!]                   |
|                         |                         |                         |
| [x_1, ..., x_n]         | [`x`_1, ..., `x`_n]     | [a_1, ..., a_n]         |
|                         |                         |                         |
| (X, X^)                 | (`X`, `X`^)             | (A, A^)                 |
|                         |                         |                         |
| (X +-> K)               | (`X` +-> B)             | (A +-> B)               |
|                         |                         |                         |
| (X, (X -> K))           | (`X`, (`X` -> B))       | (A, (A -> B))           |
|                         |                         |                         |
| isomorphic to:          | isomorphic to:          | isomorphic to:          |
|                         |                         |                         |
| (K^n, (K^n -> K)        | (B^n, (B^n -> B)        | (B^n, (B^n -> K)        |
|                         |                         |                         |
| (K^n +-> K)             | (B^n +-> B)             | (B^n +-> B)             |
|                         |                         |                         |
| [K^n]                   | [B^n]                   | [B^n]                   |
o-------------------------o-------------------------o-------------------------o

The left side of the Table collects mostly standard notation
for an n-dimensional vector space over a field K.  The right
side of the table repeats the first elements of a notation
that I sketched above, to be used in further developments
of propositional calculus.  (I plan to use this notation
in the logical analysis of neural network systems.)  The
middle column of the table is designed as a transitional
step from the case of an arbitrary field K, with a special
interest in the continuous line R, to the qualitative and
discrete situations that are instanced and typified by B.

I now proceed to explain these concepts in more detail.
The two most important ideas developed in the table are:

1.  The idea of a universe of discourse, which includes both
    a space of "points" and a space of "maps" on those points.

2.  The idea of passing from a more complex universe to
    a simpler universe by a process of "thresholding"
    each dimension of variation down to a single bit
    of information.

For the sake of concreteness, let us suppose that we start with a continuous
n-dimensional vector space like X = <|x_1, ..., x_n|> ~=~ R^n.  The coordinate
system !X! = {x_i} is a set of maps x_i : R^n -> R, also known as the coordinate
projections.  Given a "dataset" of points x in R^n, there are numerous ways of
sensibly reducing the data down to one bit for each dimension.  One strategy
that is general enough for our present purposes is as follows.  For each i
we choose an n-ary relation L_i on R, that is, a subset of R^n, and then
we define the i^th threshold map, or "limen" `x`_i as follows:

   `x`_i : R^n -> B such that:

   `x`_i (x) = 1 if x in L_i,

   `x`_i (x) = 0 if otherwise.

In other notations that are sometimes used, the operator <chi>( ) or the
corner brackets |^...^| can be used to denote a "characteristic function",
that is, a mapping from statements to their truth values, given as elements
of B.  Finally, it is not uncommon to use the name of the relation itself as
a predicate that maps n-tuples into truth values.  In each of these notations,
the above definition could be expressed as follows:

   `x`_i (x)  =  <chi>(x in L_i)  =  |^ x in L_i ^|  =  L_i (x).

Notice that, as defined here, there need be no actual relation between the
n-dimensional subsets {L_i} and the coordinate axes corresponding to {x_i},
aside from the circumstance that the two sets have the same cardinality.
In concrete cases, though, one usually has some reason for associating
these "volumes" with these "lines", for instance, L_i is bounded by
some hyperplane that intersects the i^th axis at a unique threshold
value r_i in R.  Often, the hyperplane is chosen normal to the axis.
In recognition of this motive, let us make the following convention.
When the set L_i has points on the i^th axis, that is, points of the
form <0, ..., 0, r_i, 0, ..., 0> where only the x_i coordinate is
possibly non-zero, we may pick any one of these coordinate values
as a parametric index of the relation.  In this case we say that
the indexing is "real", otherwise the indexing is "imaginary".
For a knowledge based system X, this should serve once again
to mark the distinction between "acquaintance" and "opinion".

States of knowledge about the location of a system or about the distribution of
a population of systems in a state space X = R^n can now be expressed by taking
the set !`X`! = {`x`_i} as a basis of logical features.  In picturesque terms,
one may think of the underscore [here, the grave accents] and the subscript as
combining to form a subtextual spelling for the i^th threshold map.  This can
help to remind us that the "threshold operator" `( )`_i acts on x by setting up
a kind of a "hurdle" for it.  In this interpretation, the coordinate proposition
`x`_i asserts that the representative point x resides "above" the i^th threshold.

Primitive assertions of the form `x`_i (x) can then be negated and
joined by means of propositional connectives in the usual ways to
provide information about the state x of a contemplated system or
a statistical ensemble of systems.  Parentheses "( )" may be used
to indicate negation.  Eventually one discovers the usefulness of
the k-ary "just one false" operators of the form "( , , , )", as
treated in earlier reports.  This much tackle generates a space
of points (cells, interpretations), `X` = <|!`X`!|> ~=~ B^n, and
a space of functions (regions, propositions), `X`^ ~=~ (B^n -> B).
Together these form a new universe of discourse `X`% = [!`X`!] of
the type (B^n, (B^n -> B)), which we may abbreviate as B^n +-> B,
or most succinctly as [B^n].

The square brackets have been chosen to recall the rectangular frame
of a venn diagram.  In thinking about a universe of discourse it is
a good idea to keep this picture in mind, where we constantly think
of the elementary cells `x`, the defining features `x`_i, and the
potential shadings f : `X` -> B, all at the same time, remaining
aware of the arbitrariness of the way that we choose to inscribe
our distinctions in the medium of a continuous space.  Finally,
let X* denote the space of linear functions, (hom : X -> K),
which in the finite case has the same dimensionality as X,
and let the same notation be extended across the table.

We have just gone through a lot of work, apparently doing nothing more
substantial than spinning a complex spell of notational devices through
a labyrinth of baffled spaces and baffling maps.  The reason for doing
this was to bind together and to constitute the intuitive concept of
a universe of discourse into a coherent categorical object, the kind
of thing, once grasped, which can be turned over in the mind and
considered in all its manifold changes and facets.  The effort
invested in these preliminary measures is intended to pay off
later, when we need to consider the state transformations
and the time evolution of neural network systems.

Diff Log D 2004 • Note 11


Tables of Propositional Forms

| To the scientist longing for non-quantitative techniques, then,
| mathematical logic brings hope.  It provides explicit techniques
| for manipulating the most basic ingredients of discourse.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7-8]

To prepare for the next phase of discussion, Tables 6 and 7 collect
and summarize all of the propositional forms on one and two variables.
These propositional forms are represented over bases of boolean variables
as complete sets of boolean-valued functions.  Adjacent to their names and
specifications are listed what are roughly the simplest expressions in the
"cactus language", the particular syntax for propositional calculus that
I use in formal and computational contexts.  For the sake of orientation,
the English paraphrases and the more common notations are listed in the
last two columns.  As simple and circumscribed as these low-dimensional
universes may appear to be, a careful exploration of their differential
extensions will involve us in complexities sufficient to demand our
attention for some time to come.

Propositional forms on one variable correspond to boolean functions f : B^1 -> B.
In Table 6 these functions are listed in a variant form of truth table, one which
rotates the axes of the usual arrangement.  Each function f_i is indexed by the
string of values that it takes on the points of the universe X% = [x] ~=~ B^1.
The binary index generated in this way is converted to its decimal equivalent,
and these are used as conventional names for the f_i, as shown in the first
column of the Table.  In their own right the 2^1 points of the universe X%
are coordinated as a space of type B^1, this in light of the universe X%
being a functional domain where the coordinate projection x takes on
its values in B.

Table 6.  Propositional Forms on One Variable
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x :   1 0   |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     |  f_00   |   0 0   |   ( )    | false            |    0     |
|         |         |         |          |                  |          |
| f_1     |  f_01   |   0 1   |   (x)    | not x            |   ~x     |
|         |         |         |          |                  |          |
| f_2     |  f_10   |   1 0   |    x     | x                |    x     |
|         |         |         |          |                  |          |
| f_3     |  f_11   |   1 1   |  (( ))   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o

Propositional forms on two variables correspond to boolean functions f : B^2 -> B.
In Table 7 each function f_i is indexed by the values that it takes on the points
of the universe X% = [x, y] ~=~ B^2.  Converting the binary index thus generated
to a decimal equivalent, we obtain the functional nicknames that are listed in
the first column.  The 2^2 points of the universe X% are coordinated as a space
of type B^2, as indicated under the heading of the Table, where the coordinate
projections x and y run through the various combinations of their values in B.

Table 7.  Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x : 1 1 0 0 |          |                  |          |
|         |       y : 1 0 1 0 |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
|         |         |         |          |                  |          |
| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
|         |         |         |          |                  |          |
| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
|         |         |         |          |                  |          |
| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
|         |         |         |          |                  |          |
| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
|         |         |         |          |                  |          |
| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
|         |         |         |          |                  |          |
| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
|         |         |         |          |                  |          |
| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
|         |         |         |          |                  |          |
| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
|         |         |         |          |                  |          |
| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
|         |         |         |          |                  |          |
| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
|         |         |         |          |                  |          |
| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
|         |         |         |          |                  |          |
| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
|         |         |         |          |                  |          |
| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
|         |         |         |          |                  |          |
| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
|         |         |         |          |                  |          |
| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o

Diff Log D 2004 • Note 12


| Fire over water:
| The image of the condition before transition.
| Thus the superior man is careful
| In the differentiation of things,
| So that each finds its place.
|
| 'I Ching', Hexagram 64, [Wil, 249]

A Differential Extension of Propositional Calculus

This much preparation is enough to begin introducing my subject, if I excuse
myself from giving full arguments for my definitional choices until some later
stage.  I am trying to develop a "differential theory of qualitative equations"
that parallels the application of differential geometry to dynamic systems.  The
idea of a tangent vector is key to this work and a major goal is to find the right
logical analogues of tangent spaces, bundles, and functors.  The strategy is taken
of looking for the simplest versions of these constructions that can be discovered
within the realm of propositional calculus, so long as they serve to fill out the
general theme.

Diff Log D 2004 • Note 13


Differential Propositions:  The Qualitative Analogue of Differential Equations

In order to define the differential extension of a universe of discourse [!A!],
the initial alphabet !A! must be extended to include a collection of symbols for
"differential features", or basic "changes" that are capable of occurring in [!A!].
Intuitively, these symbols may be construed as denoting primitive features of change,
qualitative attributes of motion, or propositions about how things or points in [!A!]
may change or move with respect to the features that are noted in the initial alphabet.

Hence, let us define the corresponding "differential alphabet" or "tangent alphabet"
as d!A! = {da_1, ... , da_n}, in principle, just an arbitrary alphabet of symbols,
disjoint from the initial alphaber !A! = {a_1, ..., a_n}, that is intended to be
interpreted in the way just indicated.  It only remains to be understood that
the precise interpretation of the symbols in d!A! is often conceived to be
changeable from point to point of the underlying space A.  (For all we know,
the state space A might well be the state space of a language interpreter,
one that is concerned, among other things, with the idiomatic meanings of
the dialect generated by !A! and d!A!.)

In the above terms, a typical "tangent space of A at a point x", frequently
denoted as T_x (A), can be characterized as having the generic construction
dA = <|d!A!|> = <|da_1, ..., da_n|>.  Strictly speaking, the name "cotangent
space" is probably more correct for this construction, but the fact that we
take up spaces and their duals in pairs to form our universes of discourse
allows our language to be pliable here.

Proceeding as we did before with the base space A, we can analyze the
individual tangent space at a point of A as a product of distinct and
independent factors:

   dA  =  Prod_i dA_i  =  dA_1 x ... x dA_n.

Here, dA_i is an alphabet of two symbols, dA_i = {(da_i), da_i},
where (da_i) is a symbol with the logical value of "not da_i".
Each component dA_i has the type B, under the correspondence
{(da_i), da_i} ~=~ {0, 1}.  However, clarity is often served
by acknowledging this differential usage with a superficially
distinct type D, whose intension may be indicated as follows:

   D  =  {(da_i), da_i} = {same, different} = {stay, change} =  {stop, step}.

Viewed within a coordinate representation, spaces of type B^n and D^n may
appear to be identical sets of binary vectors, but taking a view at this
level of abstraction would be like ignoring the qualitative units and
the diverse dimensions that distinguish position and momentum, or
the different roles of quantity and impulse.

Diff Log D 2004 • Note 14


An Interlude on the Path

| There would have been no beginnings:
| instead, speech would proceed from me,
| while I stood in its path - a slender gap -
| the point of its possible disappearance.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]

It may help to get a sense of the relation between B and D by considering the
"path classifier" (or equivalence class of curves) approach to tangent vectors.
As if by reflex, the thought of physical motion makes us cross over to a universe
marked by the nominal character [!X!].  Given the boolean value system, a path in
the space X = <|!X!|> is a map q : B -> X.  In this case, the set of paths (B -> X)
is isomorphic to the cartesian square X^2 = X x X, or the set of ordered pairs from X.

We may analyze X^2 = {<u, v> : u, v in X} into two parts,
specifically, the pairs that lie on and off the diagonal:

   X^2  =  {<u, v> : u = v}  |_|  {<u, v> : u =/= v}.

In symbolic terms, this partition may be expressed as:

   X^2 ~=~ Diag(X) + 2 * Comb(X, 2),

where:

   Diag(X)  =  {<x, x> : x in X},

and where:

   Comb(X, k)  =  "X choose k"  =  {k-sets from X},

so that:

   Comb(X, 2)  =  {{u, v} : u, v in X}.

We can now use the features in d!X! = {dx_1, ... , dx_n} to classify the paths
of (B -> X) by way of the pairs in X^2.  If X ~=~ B^n then a path in X has the
form q : (B -> B^n) ~=~ B^n x B^n ~=~ B^2n ~=~ (B^2)^n.  Intuitively, we want
to map this (B^2)^n onto D^n by mapping each component B^2 onto a copy of D.
But in our current situation "D" is just a name we give, or an accidental
quality we attribute, to coefficient values in B when they are attached
to features in d!X!.

Therefore, define dx_i : X^2 -> B such that:

   dx_i (<u, v>)  =  (| x_i (u) , x_i (v) |)

                  =  x_i (u)  +  x_i (v)

                  =  x_i (v)  -  x_i (u).

NB.  In the above transcription, "(| ... , ... |)" is a "cactus lobe",
signifying "just one false", in this case among two boolean variables,
while "+" is boolean addition in the proper sense of addition in GF(2),
and thus equivalent to "-", in the sense of adding the additive inverse.

The above definition is equivalent to defining dx_i : (B -> X) -> B such that:

   dx_i (q)  =  (| x_i (q_0) , x_i (q_1) |)

             =  x_i (q_0)  +  x_i (q_1)

             =  x_i (q_1)  -  x_i (q_0),

where q_b = q(b), for each b in B.  Thus, the proposition dx_i
is true of the path q = <u, v> exactly if the terms of q, the
endpoints u and v, lie on different sides of the question x_i.

Now we can use the language of features in <|d!X!|>, indeed the whole calculus
of propositions in [d!X!], to classify paths and sets of paths.  In other words,
the paths can be taken as models of the propositions g : dX -> B.  For example,
the paths corresponding to Diag(X) fall under the description (dx_1)...(dx_n),
which says that nothing changes among the set of features {x_1, ..., x_n}.

Finally, a few words of explanation may be in order.  If this concept of a path
appears to be described in a roundabout fashion, it is because I am trying to
avoid using any assumption of vector space properties for the space X which
contains its range.  In many ways the treatment is still unsatisfactory,
but improvements will have to wait for the introduction of substitution
operators acting on singular propositions.

Diff Log D 2004 • Note 15


The Extended Universe of Discourse

| At the moment of speaking, I would like to have perceived a nameless voice,
| long preceding me, leaving me merely to enmesh myself in it, taking up its
| cadence, and to lodge myself, when no one was looking, in its interstices
| as if it had paused an instant, in suspense, to beckon to me.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]

Next, we define the so-called "extended alphabet" or "bundled alphabet" E!A! as:

   E!A!  =  !A! |_| d!A!  =  {a_1, ..., a_n,  da_1, ..., da_n}.

This supplies enough material to construct the "differential extension" EA,
or the "tangent bundle" over the initial space A, in the following fashion:

   EA  =  A x dA  =  <|E!A!|>  =  <|!A! |_| d!A!|>

       =  <|a_1, ..., a_n,  da_1, ..., da_n|>,

thus giving EA the type B^n x D^n.

Finally, the tangent universe EA% = [E!A!] is constituted from the totality
of points and maps, or interpretations and propositions, which are based on
the extended set of features E!A!:

   EA%  =  [E!A!]  =  [a_1, ..., a_n,  da_1, ..., da_n],

thus giving the tangent universe E!A! the type
(B^n x D^n +-> B) = (B^n x D^n, (B^n x D^n -> B).

A proposition in the tangent universe [E!A!] is
called a "differential proposition" and forms the
analogue of a system of differential equations,
constraints, or relations in ordinary calculus.

With these constructions, to be specific, the differential extension EA
and the differential proposition h : EA -> B, we have arrived, in concept
at least, at one of the major subgoals of this study.  At this juncture,
I pause by way of summary to set another Table with the current crop of
mathematical produce (Table 8).

Table 8.  Notation for the Differential Extension of Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol  | Notation          | Description       | Type              |
o---------o-------------------o-------------------o-------------------o
| d!A!    | {da_1, ..., da_n} | Alphabet of       | [n]  =  #n#       |
|         |                   | differential      |                   |
|         |                   | features          |                   |
o---------o-------------------o-------------------o-------------------o
| dA_i    | {(da_i), da_i}    | Differential      |  D                |
|         |                   | dimension i       |                   |
o---------o-------------------o-------------------o-------------------o
| dA      | <|d!A!|>          | Tangent space     |  D^n              |
|         | <|da_i,...,da_n|> | at a point:       |                   |
|         | {<da_i,...,da_n>} | Set of changes,   |                   |
|         | dA_1 x ... x dA_n | motions, steps,   |                   |
|         | Prod_i dA_i       | tangent vectors   |                   |
|         |                   | at a point        |                   |
o---------o-------------------o-------------------o-------------------o
| dA*     | (hom : dA -> B)   | Linear functions  | (D^n)*  ~=~  D^n  |
|         |                   | on dA             |                   |
o---------o-------------------o-------------------o-------------------o
| dA^     | (dA -> B)         | Boolean functions |  D^n -> B         |
|         |                   | on dA             |                   |
o---------o-------------------o-------------------o-------------------o
| dA%     | [d!A!]            | Tangent universe  | (D^n, (D^n -> B)) |
|         | (dA, dA^)         | at a point of A%, | (D^n +-> B)       |
|         | (dA +-> B)        | based on the      | [D^n]             |
|         | (dA, (dA -> B))   | tangent features  |                   |
|         | [da_1, ..., da_n] | {da_1, ..., da_n} |                   |
o---------o-------------------o-------------------o-------------------o

The adjectives "differential" or "tangent" are systematically attached to
every construct based on the differential alphabet d!A!, taken by itself.
Strictly speaking, we probably ought to call d!A! the set of "cotangent"
features derived from !A!, but the only time this distinction really
seems to matter is when we need to distinguish the tangent vectors
as maps of type (B^n -> B) -> B from cotangent vectors as elements
of type D^n.  In like fashion, having defined E!A! = !A! |_| d!A!,
we can systematically attach the adjective "extended" or the
substantive "bundle" to all the constructs associated with
this full complement of 2n features.

Eventually we may want to extend our basic alphabet even further,
to allow for discussion of higher order differential expressions.
For those who want to run ahead, and would like to play through,
I submit the following gamut of notation (Table 9).

Table 9.  Higher Order Differential Features
o----------------------------------------o----------------------------------------o
|                                        |                                        |
| !A!   = d^0.!A! = {a_1, ..., a_n}      | E^0.!A!  = d^0.!A!                     |
|                                        |                                        |
| d!A!  = d^1.!A! = {da_1, ..., da_n}    | E^1.!A!  = d^0.!A! |_| d^1.!A!         |
|                                        |                                        |
|         d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A!  = d^0.!A! |_| ... |_| d^k.!A! |
|                                        |                                        |
| d*!A! = {d^0.!A!, ..., d^k.!A!, ...}   | E^oo.!A! = d*!A!                       |
|                                        |                                        |
o----------------------------------------o----------------------------------------o

Diff Log D 2004 • Note 16


Intentional Propositions

| Do you guess I have some intricate purpose?
| Well I have . . . . for the April rain has, and the mica on
|     the side of a rock has.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 45]

In order to analyze the behavior of a system at successive moments in time,
while staying within the limitations of propositional logic, it is necessary
to create independent alphabets of logical features for each moment of time
that we contemplate using in our discussion.  These moments have reference
to typical instances and relative intervals, not actual or absolute times.
For example, to discuss "velocities" (first order rates of change) we need
to consider points of time in pairs.  There are a number of natural ways of
doing this.  Given an initial alphabet, we could use its symbols as a lexical
basis to generate successive alphabets of compound symbols, say, with temporal
markers appended as suffixes.

As a standard way of dealing with these situations, I produce the following
scheme of notation, which extends any alphabet of logical features through
as many temporal moments as a particular order of analysis may demand.
The lexical operators p^k and Q^k are convenient in many contexts
where the accumulation of prime symbols and union symbols
would otherwise be cumbersome.

Table 10.  A Realm of Intentional Features
o---------------------------------------o----------------------------------------o
|                                       |                                        |
| p^0.!A!  =  !A!  =  {a_1, ..., a_n}   | Q^0.!A!  =  !A!                        |
|                                       |                                        |
| p^1.!A!  =  !A!' =  {a_1', ..., a_n'} | Q^1.!A!  =  !A! |_| !A!'               |
|                                       |                                        |
| p^2.!A!  =  !A!" =  {a_1", ..., a_n"} | Q^2.!A!  =  !A! |_| !A!' |_| !A!"      |
|                                       |                                        |
| ...         ...     ...               | ...         ...                        |
|                                       |                                        |
| p^k.!A!  =  {p^k.a_1, ..., p^k.a_n}   | Q^k.!A!  =  !A! |_| ... |_| p^k.!A!    |
|                                       |                                        |
o---------------------------------------o----------------------------------------o

The resulting augmentations of our logical basis found a series of
discursive universes that may be called the "intentional extension"
of propositional calculus.  The pattern of this extension is analogous
to that of the differential extension, which was developed in terms of
the operators d^k and E^k, and there is an obvious and natural relation
between these two extensions that falls within our purview to explore.
In contexts displaying this regular pattern, where a series of domains
stretches up from an anchoring domain X through an indefinite number
of higher reaches, I refer to a particular collection of domains
based on X as "a realm of X", and when the succession exhibits
a temporal aspect, "a reign of X".

For the purposes of this discussion, let us define an "intentional proposition"
as a proposition in the universe of discourse QX% = [Q!X!], in other words,
a map q : QX -> B.  The sense of this definition may be seen if we consider
the following facts.  First, the equivalence QX = X x X' motivates the
following chain of isomorphisms between spaces:

   (QX -> B)  ~=~  (X x X' -> B)  ~=~  (X -> (X' -> B))  ~=~  (X' -> (X -> B)).

Viewed in this light, an intentional proposition q may be rephrased as a map
q : X x X' -> B, which judges the juxtaposition of states in X from one moment
to the next.  Alternatively, q may be parsed in two stages in two different ways,
as q : X -> (X' -> B) and q : X' -> (X -> B), which associate to each point of
X or X' a proposition about states in X' or X, respectively.  In this way, an
intentional proposition embodies a type of value system, in effect, a proposal
that places a value on a collection of ends-in-view, or a project that evaluates
a set of goals as regarded from each point of view in the state space of a system.

In sum, the intentional proposition q indicates a method for the systematic
selection of local goals.  As a general form of description, we may refer to
a map of the type q : Q^i.X -> B as an "i^th order intentional proposition".
Naturally, when we speak of intentional propositions without qualification,
we usually mean first order intentions.

Many different realms of discourse have the same structure as the extensions that
have been indicated here.  From a strictly logical point of view, each new layer
of terms is composed of independent logical variables that are no different in
kind from those that go before, and each further course of logical atoms is
treated like so many individual, but otherwise indifferent bricks by the
prototype computer program that I use as a propositional interpreter.
Thus, the names that I use to single out the differential and the
intentional extensions, and the lexical paradigms that I follow
to construct them, are meant to suggest the interpretations
that I have in mind, but they can only hint at the extra
meanings that human communicators may pack into their
terms and inflections.

As applied here, the word "intentional" is drawn from common use
and may have little bearing on its technical use in other, more
properly philosophical, contexts.  I am merely using the complex
of intentional concepts -- aims, ends, goals, objectives, purposes,
and so on -- metaphorically to flesh out and vividly to represent
any situation where one needs to contemplate a system in multiple
aspects of state and destination, that is, its being in certain
states and at the same time acting as if headed through certain
states.  If confusion arises, more neutral words like conative,
contingent, discretionary, experimental, kinetic, progressive,
tentative, or trial would probably serve as well.

Diff Log D 2004 • Note 17


Life on Easy Street

| Failing to fetch me at first keep encouraged,
| Missing me one place search another,
| I stop some where waiting for you
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]

The finite character of the extended universe [E!A!] makes the problem
of solving differential propositions relatively straightforward, at least,
in principle.  The solution set of the differential proposition q : EA -> B
is the set of models (q^(-1))(1) in EA.  Finding all of the models of q, the
extended interpretations in EA that satisfy q, can be carried out by a finite
search.  Being in possession of complete algorithms for propositional calculus
theorem proving makes the analytic task fairly simple in principle, though the
question of efficiency in the face of arbitrary complexity may always remain
another matter entirely.  While the fact that propositional satisfiability
is NP-complete may be discouraging for the prospects of a single efficient
algorithm that covers the whole space of [E!A!] with equal facility, there
appears to be much room for improvement in classifying special forms and
in developing algorithms that are tailored to their practical processing.

In view of these constraints and contingencies, my focus shifts to the tasks of
approximation and interpretation that support intuition, especially in dealing
with the natural kinds of differential propositions that arise in applications,
and in the effort to understand, in succinct and adaptive forms, their dynamic
implications.  In the absence of direct insights, these tasks are partially
carried out by forging analogies with the familiar situations and customary
routines of ordinary calculus.  But the indirect approach, going by way of
specious analogy and intuitive habit, forces us to remain on guard against
the circumstance that occurs when the word "forging" takes on its shadier
nuance, indicting the constant risk of a counterfeit in the proportion.

Diff Log D 2004 • Note 18


| I would have preferred to be enveloped in words,
| borne way beyond all possible beginnings.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]

Back to the Beginning:  Some Exemplary Universes

To anchor our understanding of differential logic, let us look at how the
various concepts apply in the simplest possible concrete cases, where the
initial dimension is only 1 or 2.  In spite of the obvious simplicity of
these cases, it is possible to observe how central difficulties of the
subject begin to arise already at this stage.

Diff Log D 2004 • Note 19


A One-Dimensional Universe

| There was never any more inception than there is now,
| Nor any more youth or age than there is now;
| And will never be any more perfection than there is now,
| Nor any more heaven or hell than there is now.
|
| Walt Whitman, Leaves of Grass, [Whi, 28]

Let !X! = {x_1} = {A} be an alphabet that represents one boolean variable or
a single logical feature.  In this example I am using the capital letter "A"
in a more usual informal way, to name a feature and not a space, at variance
with my formerly stated formal conventions.  At any rate, the basis element
A = x_1 may be interpreted as a simple proposition or a coordinate projection
A = x_1 : B^1 -:> B.  The space X = <|A|> = {(A), A} of points (cells, vectors,
interpretations) has cardinality 2^n = 2^1 = 2 and is isomorphic to B = {0, 1}.
Moreover, X may be identified with the set of singular propositions {x : B ::> B}.
The space of linear propositions X* = {hom : B ++> B} = {0, A} is algebraically
dual to X and also has cardinality 2.  Here, "0" is interpreted as denoting the
constant function 0 : B -> B, amounting to the linear proposition of rank 0,
while A is the linear proposition of rank 1.  Last but not least we have the
positive propositions {pos : B oo> B} = {A, 1}, of rank 1 and 0, respectively,
where "1" is understood as denoting the constant function 1 : B -> B.  In sum,
there are 2^2^n = 2^2^1 = 4 propositions altogether in the universe of discourse,
comprising the set X^ = {f : X -> B} = {0, (A), A, 1} ~=~ (B -> B).

The first order differential extension of !X! is E!X! = {x_1, dx_1} = {A, dA}.
If the feature "A" is understood as applying to some object or state, then the
feature "dA" may be interpreted as an attribute of the same object or state that
says that it is changing "significantly" with respect to the property A, or that
it has an "escape velocity" with respect to the state A.  In practice, differential
features acquire their logical meaning through a class of "temporal inference rules".

For example, relative to a frame of observation that is left implicit for now,
one is permitted to make the following sorts of inference:  from the fact that
A and dA are true at a given moment one may infer that (A) will be true in the
next moment of observation.  Altogether in the present instance, there is the
fourfold scheme of inference that is shown below:

o-------------------------------------------------o
|                                                 |
|      From  (A) & (dA)  infer  (A)  next.        |
|                                                 |
|      From  (A) &  dA   infer   A   next.        |
|                                                 |
|      From   A  & (dA)  infer   A   next.        |
|                                                 |
|      From   A  &  dA   infer  (A)  next.        |
|                                                 |
o-------------------------------------------------o

It might be thought that we need to bring in an independent time variable
at this point, but an insight of fundamental importance appears to be that
the idea of process is more basic than the notion of time.  A time variable
is actually a reference to a "clock", that is, a canonical or a convenient
process that is established or accepted as a standard of measurement, but
in essence no different than any other process.  This raises the question
of how different subsystems in a more global process can be brought into
comparison, and what it means for one process to serve the function of
a local standard for others.  But these inquiries only wrap up puzzles
in further riddles, and are obviously too involved to be handled at
our current level of approximation.

| The clock indicates the moment . . . . but what does
|    eternity indicate?
|
| Walt Whitman, 'Leaves of Grass', [Whi, 79]

Observe that the secular inference rules, used by themselves,
involve a loss of information, since nothing in them can tell
us whether the momenta {(dA), dA} are preserved or changed in
the next instance.  In order to know this, we would have to
determine d^2.A, and so on, pursuing an infinite regress.
Ultimately, in order to rest with a finitely determinate
system, it is necessary to make an infinite assumption,
for example, that d^k.A = 0 for all k greater than some
fixed value M.  Another way to escape the regress is
through the provision of a dynamic law, in typical
form making higher order differentials dependent
on lower degrees and estates.

Diff Log D 2004 • Note 20


Example 1.  A Square Rigging

| Urge and urge and urge,
| Always the procreant urge of the world.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 28]

By way of example, suppose that we are given the initial condition A = dA and
the law d^2.A = (A).  Then, since "A = dA" <=> "A dA or (A)(dA)", we may infer
two possible trajectories, as displayed in Table 11.  In either of these cases,
the state A (dA)(d^2.A) is a stable attractor or a terminal condition for both
starting points.

Table 11.  A Pair of Commodious Trajectories
o---------o-------------------o-------------------o
| Time    | Trajectory 1      | Trajectory 2      |
o---------o-------------------o-------------------o
|         |                   |                   |
| 0       |  A   dA  (d^2.A)  | (A) (dA)  d^2.A   |
|         |                   |                   |
| 1       | (A)  dA   d^2.A   | (A)  dA   d^2.A   |
|         |                   |                   |
| 2       |  A  (dA) (d^2.A)  |  A  (dA) (d^2.A)  |
|         |                   |                   |
| 3       |  A  (dA) (d^2.A)  |  A  (dA) (d^2.A)  |
|         |                   |                   |
| 4       |  "    "    "      |  "    "    "      |
|         |                   |                   |
o---------o-------------------o-------------------o

Because the initial space X = <|A|> is one-dimensional, we can easily fit
the second order extension E^2.X = <|A, dA, d^2.A|> within the compass of
a single venn diagram, charting the couple of converging trajectories as
shown in Figure 12.

o-------------------------------------------------o
| E^2.X                                           |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /        A        \               |
|              /                   \              |
|             /         ->-         \             |
|            o         /   \         o            |
|            |         \   /         |            |
|            |          -o-          |            |
|            |           ^           |            |
|        o---o---------o | o---------o---o        |
|       /     \         \|/         /     \       |
|      /       \    o    |         /       \      |
|     /         \   |   /|\       /         \     |
|    /           \  |  / | \     /           \    |
|   o             o-|-o--|--o---o             o   |
|   |               | |  |  |                 |   |
|   |               ---->o<----o              |   |
|   |                 |     |                 |   |
|   o       dA        o     o      d^2.A      o   |
|    \                 \   /                 /    |
|     \                 \ /                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 12.  The Anchor

If we eliminate from view the regions of E^2.X that are ruled out
by the dynamic law d^2.A = (A), then what remains is the quotient
structure that is shown in Figure 13.  This picture makes it easy
to see that the dynamically allowable portion of the universe is
partitioned between the properties A and d^2.A.  As it happens,
this fact might have been expressed "right off the bat" by an
equivalent formulation of the differential law, one that uses
the exclusive disjunction to state the law as (A, d^2.A).

o-------------------------------------------------o
|                                                 |
|                                   ->-           |
|                                  /   \          |
|                                  \   /          |
|                 o-------------o   -o-           |
|                /               \  ^             |
|               /       dA        \/         A    |
|              /                  /\              |
|             /                  /  \             |
|            o    o             /    o            |
|            |     \           /     |            |
|            |      \         /      |            |
o------------|-------\-------/-------|------------o
|            |        \     /        |            |
|            |         \   /         |            |
|            o          v /          o            |
|             \          o          /             |
|              \         ^         /              |
|               \        |        /        d^2.A  |
|                \       |       /                |
|                 o------|------o                 |
|                        |                        |
|                        |                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o
Figure 13.  The Tiller

What we have achieved in this example is to give a differential description of
a simple dynamic process.  In effect, we did this by embedding a directed graph,
which can be taken to represent the state transitions of a finite automaton, in
a dynamically allotted quotient structure that is created from a boolean lattice
or an n-cube by nullifying all of the regions that the dynamics outlaws.  With
growth in the dimensions of our contemplated universes, it becomes essential,
both for human comprehension and for computer implementation, that the dynamic
structures of interest to us be represented not actually, by acquaintance, but
virtually, by description.  In our present study, we are using the language of
propositional calculus to express the relevant descriptions, and to comprehend
the structure that is implicit in the subsets of a n-cube without necessarily
being forced to actualize all of its points.

One of the reasons for engaging in this kind of extremely reduced, but explicitly
controlled case study is to throw light on the general study of languages, formal
and natural, in their full array of syntactic, semantic, and pragmatic aspects.
Propositional calculus is one of the last points of departure where we can view
these three aspects interacting in a non-trivial way without being immediately
and totally overwhelmed by the complexity they generate.  Often this complexity
causes investigators of formal and natural languages to adopt the strategy of
focusing on a single aspect and to abandon all hope of understanding the whole,
whether it's the still living natural language or the dynamics of inquiry that
lies crystallized in formal logic.

From the perspective that I find most useful here, a language is a syntactic
system that is designed or evolved in part to express a set of descriptions.
When the explicit symbols of a language have extensions in its object world
that are actually infinite, or when the implicit categories and generative
devices of a linguistic theory have extensions in its subject matter that
are potentially infinite, then the finite characters of terms, statements,
arguments, grammars, logics, and rhetorics force an excess of intension
to reside in all these symbols and functions, across the spectrum from
the object language to the metalinguistic uses.  In the aphorism from
W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30]
and [Cho93, 49], language requires "the infinite use of finite means".
This is necessarily true when the extensions are infinite, when the
referential symbols and grammatical categories of a language possess
infinite sets of models and instances.  But it also voices a practical
truth when the extensions, though finite at every stage, tend to grow
at exponential rates.

This consequence of dealing with extensions that are "practically infinite"
becomes crucial when one tries to build neural network systems that learn,
since the learning competence of any intelligent system is limited to the
objects and domains that it is able to represent.  If we want to design
systems that operate intelligently with the full deck of propositions
dealt by intact universes of discourse, then we must supply them with
succinct representations and efficient transformations in this domain.
Furthermore, in the project of constructing inquiry driven systems,
we find ourselves forced to contemplate the level of generality
that is embodied in propositions, because the dynamic evolution
of these systems is driven by the measurable discrepancies that
occur among their expectations, intentions, and observations,
and because each of these subsystems or components of knowledge
constitutes a propositional modality that can take on the fully
generic character of an empirical summary or an axiomatic theory.

A compression scheme by any other name is a symbolic representation,
and this is what the differential extension of propositional calculus,
through all of its many universes of discourse, is intended to supply.
Why is this particular program of mental calisthenics worth carrying out
in general?  By providing a uniform logical medium for describing dynamic
systems we can make the task of understanding complex systems much easier,
both in looking for invariant representations of individual cases and in
finding points of comparison among diverse structures that would otherwise
appear as isolated systems.  All of this goes to facilitate the search for
compact knowledge and to adapt what is learned from individual cases to
the general realm.

Diff Log D 2004 • Note 21


Back to the Feature

| I guess it must be the flag of my disposition, out of hopeful
|     green stuff woven.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 31]

Let us assume that the sense intended for differential features is well enough
established in the intuition, for now, that I may continue with outlining the
structure of the differential extension [E!X!] = [A, dA].  Over the extended
alphabet E!X! = {x_1, dx_1} = {A, dA}, of cardinality 2^n = 2, we generate
the set of points, EX, of cardinality 2^2n = 4, that bears the following
chain of equivalent descriptions:

   EX  =  <|A, dA|>

       =  {(A), A} x {(dA), dA}

       =  {(A)(dA), (A) dA,  A (dA), A dA}.

The space EX may be assigned the mnemonic type B x D, which is really
no different than B x B = B^2.  An individual element of EX may be
regarded as a "disposition at a point" or a "situated direction",
in effect, a singular mode of change occurring at a single point
in the universe of discourse.  In applications, the modality of
this change can be interpreted in various ways, for example,
as an expectation, an intention, or an observation with
respect the behavior of a system.

To complete the construction of the extended universe of discourse
EX% = [x_1, dx_1] = [A, dA], one must add the set of differential
propositions EX^ = {g : EX -> B} ~=~ (B x D -> B) to the set of
dispositions in EX.  There are 2^2^2n = 16 propositions in EX^,
as detailed in Table 14.

Table 14.  Differential Propositions
o-------o--------o---------o-----------o-------------------o----------o
|       |      A : 1 1 0 0 |           |                   |          |
|       |     dA : 1 0 1 0 |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
| f_0   | g_0    | 0 0 0 0 |    ()     | False             |    0     |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_1    | 0 0 0 1 |  (A)(dA)  | Neither A nor dA  | ~A & ~dA |
|       |        |         |           |                   |          |
|       | g_2    | 0 0 1 0 |  (A) dA   | Not A but dA      | ~A &  dA |
|       |        |         |           |                   |          |
|       | g_4    | 0 1 0 0 |   A (dA)  | A but not dA      |  A & ~dA |
|       |        |         |           |                   |          |
|       | g_8    | 1 0 0 0 |   A  dA   | A and dA          |  A &  dA |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
| f_1   | g_3    | 0 0 1 1 |  (A)      | Not A             | ~A       |
|       |        |         |           |                   |          |
| f_2   | g_12   | 1 1 0 0 |   A       | A                 |  A       |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_6    | 0 1 1 0 |  (A, dA)  | A not equal to dA |  A + dA  |
|       |        |         |           |                   |          |
|       | g_9    | 1 0 0 1 | ((A, dA)) | A equal to dA     |  A = dA  |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_5    | 0 1 0 1 |     (dA)  | Not dA            |      ~dA |
|       |        |         |           |                   |          |
|       | g_10   | 1 0 1 0 |      dA   | dA                |       dA |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_7    | 0 1 1 1 |  (A  dA)  | Not both A and dA | ~A v ~dA |
|       |        |         |           |                   |          |
|       | g_11   | 1 0 1 1 |  (A (dA)) | Not A without dA  |  A => dA |
|       |        |         |           |                   |          |
|       | g_13   | 1 1 0 1 | ((A) dA)  | Not dA without A  |  A <= dA |
|       |        |         |           |                   |          |
|       | g_14   | 1 1 1 0 | ((A)(dA)) | A or dA           |  A v  dA |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
| f_3   | g_15   | 1 1 1 1 |   (())    | True              |    1     |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o

Aside from changing the names of variables and shuffling the order of rows,
this Table follows the format that was used previously for boolean functions
of two variables.  The rows are grouped to reflect natural similarity classes
among the propositions.  In a future discussion, these classes will be given
additional explanation and motivation as the orbits of a certain transformation
group acting on the set of 16 propositions.  Notice that four of the propositions,
in their logical expressions, resemble those given in the table for X^.  Thus the
first set of propositions {f_i} is automatically embedded in the present set {g_j},
and the corresponding inclusions are indicated at the far left margin of the table.

Diff Log D 2004 • Note 22


Tacit Extensions

| I would really like to have slipped imperceptibly into this lecture, as
| into all the others I shall be delivering, perhaps over the years ahead.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]

Strictly speaking, however, there is a subtle distinction in type between the
function f_i : X -> B and the corresponding function g_j : EX -> B, even though
they share the same logical expression.  Being human, we insist on preserving all
the aesthetic delights afforded by the abstractly unified form of the "cake" while
giving up none of the diverse contents that its substantive consummation can provide.
In short, we want to maintain the logical equivalence of expressions that represent
the same proposition, while appreciating the full diversity of that proposition's
functional and typical representatives.  Both perspectives, and all the levels of
abstraction extending through them, have their reasons, as will develop in time.

Because this special circumstance points up an important general theme,
it is a good idea to discuss it more carefully.  Whenever there arises
a situation like this, where one alphabet !X! is a subset of another
alphabet !Y!, then we say that any proposition f : <|!X!|> -> B has
a "tacit extension" to a proposition !e!f : <|!Y!|> -> B, and that
the space (<|!X!|> -> B) has an "automatic embedding" within the
space (<|!Y!|> -> B).  The extension is defined in such a way
that !e!f puts the same constraint on the variables of X that
are contained in Y as the proposition f initially did, while
it puts no constraint on the variables of Y outside of X,
in effect, conjoining the two constraints.

If the variables in question are indexed as !X! = {x_1, ..., x_n}
and !Y! = {x_1, ..., x_n, ..., x_n+k}, then the definition of the
tacit extension from !X! to !Y! may be expressed in the form of
an equation:

   !e!f(x_1, ..., x_n, ..., x_n+k)  =  f(x_1, ..., x_n).

On formal occasions, such as the present context of definition,
the tacit extension from !X! to !Y! is explicitly symbolized by
the operator !e! : (<|!X!|> -> B) -> (<|!Y!|> -> B), where the
appropriate alphabets !X! and !Y! are understood from context,
but normally one may leave the "!e!" silent.

Let's explore what this means for the present Example.
Here, !X! = {A} and !Y! = E!X! = {A, dA}.  For each of
the propositions f_i over X, specifically, those whose
expression e_i lies in the collection {0, (A), A, 1},
the tacit extension !e!f of f to EX can be phrased as
a logical conjunction of two factors, f_i = e_i.!t!,
where !t! is a logical tautology that uses all the
variables of !Y! - !X!.  Working in these terms,
the tacit extensions !e!f of f to EX may be
explicated as shown in Table 15.

Table 15.  Tacit Extensions of [A] to [A, dA]
o---------------------------------------------------------------------o
|                                                                     |
|    0    =      0  . ((dA), dA)        =              0              |
|                                                                     |
|   (A)   =     (A) . ((dA), dA)        =      (A)(dA) + (A) dA       |
|                                                                     |
|    A    =      A  . ((dA), dA)        =       A (dA) +  A  dA       |
|                                                                     |
|    1    =      1  . ((dA), dA)        =              1              |
|                                                                     |
o---------------------------------------------------------------------o

In its effect on the singular propositions over X, this analysis has an
interesting interpretation.  The tacit extension takes us from thinking
about a particular state, like A or (A), to considering the collection
of outcomes, the outgoing changes or the singular dispositions, that
spring from that state.

Diff Log D 2004 • Note 23


Example 2.  Drives and Their Vicissitudes

| I open my scuttle at night and see the far-sprinkled systems,
| And all I see, multiplied as high as I can cipher, edge but
|    the rim of the farther systems.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 81]

Before we leave the one-feature case let's look at a more substantial example,
one that illustrates a general class of curves that can be charted through
the extended feature spaces and that provides an opportunity to discuss
a number of important themes concerning their structure and dynamics.

Again, let !X! = {x_1} = {A}.  In the discussion that follows I will consider
a class of trajectories having the property that d^k.A = 0 for all k greater
than some fixed m, and I indulge in the use of some picturesque terms that
describe salient classes of such curves.  Given the finite order condition,
there is a highest order non-zero difference d^m.A exhibited at each point
in the course of any determinate trajectory that one may wish to consider.
With respect to any point of the corresponding orbit or curve let us call
this highest order differential feature d^m.A the "drive" at that point.
Curves of constant drive d^m.A are then referred to as "m^th gear curves".

Scholium.  The fact that a difference calculus can be developed
for boolean functions is well known [Fuji], [Koh, sec. 8-4] and
was probably familiar to Boole, who was an expert in difference
equations before he turned to logic.  And of course there is the
strange but true story of how the Turin machines of the 1840's
prefigured the Turing machines of the 1940's [Men, 225-297].
At the very outset of general purpose, mechanized computing
we find that the motive power driving the Analytical Engine
of Babbage, the kernel of an idea behind all of his wheels,
was exactly his notion that difference operations, suitably
trained, can serve as universal joints for any conceivable
computation [M&M], [Mel, ch. 4].

Given this language, the particular Example that I take up here can be described
as the family of 4^th gear curves through E^4.X = <|A, dA, d^2.A, d^3.A, d^4.A|>.
These are the trajectories generated subject to the dynamic law d^4.A = 1, where
it is understood in such a statement that all higher order differences are equal
to 0.  Since d^4.A and all higher d^k.A are fixed, the temporal or transitional
conditions (initial, mediate, terminal -- transient or stable states) vary only
with respect to their projections as points of E^3.X = <|A, dA, d^2.A, d^3.A|>.
Thus, there is just enough space in a planar venn diagram to plot all of these
orbits and to show how they partition the points of E^3.X.  It turns out that
there are exactly two possible orbits, of eight points each, as illustrated
in Figures 16-a and 16-b.  (NB.  I leave it as an exercise for the reader
to connect the dots in the second figure.)

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                       / \                       |
|                      /   \                      |
|                     /     \                     |
|                    /       \                    |
|                   o         o                   |
|                  / \       / \                  |
|                 /   \     /   \                 |
|                /     \   /     \                |
|               /       \ /       \               |
|              o         o         o              |
|             / \       / \       / \             |
|            /   \     /   \     /   \            |
|           /     \   /     \   /     \           |
|          /       \ /       \ /       \          |
|         o    5    o    7    o         o         |
|        / \  ^|   / \  ^|   / \       / \        |
|       /   \/ |  /   \/ |  /   \     /   \       |
|      /    /\ | /    /\ | /     \   /     \      |
|     /    /  \|/    /  \|/       \ /       \     |
|    o    4<---|----/----|----3    o         o    |
|    |\       /|\  /    /|\  ^    / \       /|    |
|    | \     / | \/    / | \/    /   \     / |    |
|    |  \   /  | /\   /  | /\   /     \   /  |    |
|    |   \ /   v/  \ /   |/  \ /       \ /   |    |
|    |    o    6    o    |    o         o    |    |
|    |    |\       / \  /|   / \       /|    |    |
|    |    | \     /   \/ |  /   \     / |    |    |
|    |    |  \   /    /\ | /     \   /  |    |    |
|    | d^0.A  \ /    /  \|/       \ /  d^1.A |    |
|    o----+----o    2<---|----1    o----+----o    |
|         |     \       /|\  ^    /     |         |
|         |      \     / | \/    /      |         |
|         |       \   /  | /\   /       |         |
|         | d^2.A  \ /   v/  \ /  d^3.A |         |
|         o---------o    0    o---------o         |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        o                        |
|                                                 |
o-------------------------------------------------o
Figure 16-a.  A Couple of Fourth Gear Orbits:  1

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                       / \                       |
|                      /   \                      |
|                     /     \                     |
|                    /       \                    |
|                   o    0    o                   |
|                  / \       / \                  |
|                 /   \     /   \                 |
|                /     \   /     \                |
|               /       \ /       \               |
|              o    5    o    2    o              |
|             / \       / \       / \             |
|            /   \     /   \     /   \            |
|           /     \   /     \   /     \           |
|          /       \ /       \ /       \          |
|         o         o         o    6    o         |
|        / \       / \       / \       / \        |
|       /   \     /   \     /   \     /   \       |
|      /     \   /     \   /     \   /     \      |
|     /       \ /       \ /       \ /       \     |
|    o         o    7    o         o    4    o    |
|    |\       / \       / \       / \       /|    |
|    | \     /   \     /   \     /   \     / |    |
|    |  \   /     \   /     \   /     \   /  |    |
|    |   \ /       \ /       \ /       \ /   |    |
|    |    o         o    3    o    1    o    |    |
|    |    |\       / \       / \       /|    |    |
|    |    | \     /   \     /   \     / |    |    |
|    |    |  \   /     \   /     \   /  |    |    |
|    | d^0.A  \ /       \ /       \ /  d^1.A |    |
|    o----+----o         o         o----+----o    |
|         |     \       / \       /     |         |
|         |      \     /   \     /      |         |
|         |       \   /     \   /       |         |
|         | d^2.A  \ /       \ /  d^3.A |         |
|         o---------o         o---------o         |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        o                        |
|                                                 |
o-------------------------------------------------o
Figure 16-b.  A Couple of Fourth Gear Orbits:  2

Diff Log D 2004 • Note 24


Example 2.  Drives and Their Vicissitudes (concl.)

With a little thought it is possible to devise an indexing scheme for
the general run of dynamic states that allows for comparing universes
of discourse that weigh in on different scales of observation.  With
this end in sight, let us index the states q in E^m.X with the dyadic
rationals (or the binary fractions) in the half-open interval [0, 2).
Formally and canonically, a state q_r is indexed by a fraction r = s/t
whose denominator is the power of two t = 2^m and whose numerator is a
binary numeral that is formed from the coefficients of state in a manner
to be described next.  The "differential coefficients" of the state q are
just the values d^k.A(q), for k = 0 to m, where d^0.A is defined as being
identical to A.  To form the binary index d_0 . d_1 ... d_m of the state q
the coefficient d^k.A(q) is read off as the binary digit d_k associated with
the place value 2^(-k).  Expressed by way of algebraic formulas, the rational
index r of the state q can be given by the following equivalent formulations:

o-------------------------------------------------------------------------------o
|                                                                               |
|  r(q)    =   Sum_k d_k . 2^(-k)          =   Sum_k d^k.A(q) . 2^(-k)          |
|                                                                               |
|  =                                                                            |
|                                                                               |
|  s(q)/t  =  (Sum_k d_k . 2^(m-k)) / 2^m  =  (Sum_k d^k.A(q) . 2^(m-k)) / 2^m  |
|                                                                               |
o-------------------------------------------------------------------------------o

Applied to the example of fourth gear curves, this scheme results in the data
of Tables 17-a and 17-b, which exhibit one period for each orbit.  The states
in each orbit are listed as ordered pairs <p_i, q_j>, where p_i may be read
as a temporal parameter that indicates the present time of the state, and
where j is the decimal equivalent of the binary numeral 's'.  Informally
and more casually, the Tables exhibit the states q_s as subscripted
with the numerators of their rational indices, taking for granted
the constant denominators of 2^m = 2^4 = 16.  Within this set-up,
the temporal successions of states can be reckoned as given by
a kind of "parallel round-up rule".  That is, if <d_k, d_(k+1)>
is any pair of adjacent digits in the state index r, then the
value of d_k in the next state is (d_k)' = d_k + d_(k+1).

Table 17-a.  A Couple of Orbits in Fourth Gear:  Orbit 1
o---------o---------o---------o---------o---------o---------o---------o
| Time    | State   |    A    |   dA    |         |         |         |
|  p_i    |  q_j    |  d^0.A  |  d^1.A  |  d^2.A  |  d^3.A  |  d^4.A  |
o---------o---------o---------o---------o---------o---------o---------o
|         |         |                                                 |
|  p_0    |  q_01   |    0.        0         0         0         1    |
|         |         |                                                 |
|  p_1    |  q_03   |    0.        0         0         1         1    |
|         |         |                                                 |
|  p_2    |  q_05   |    0.        0         1         0         1    |
|         |         |                                                 |
|  p_3    |  q_15   |    0.        1         1         1         1    |
|         |         |                                                 |
|  p_4    |  q_17   |    1.        0         0         0         1    |
|         |         |                                                 |
|  p_5    |  q_19   |    1.        0         0         1         1    |
|         |         |                                                 |
|  p_6    |  q_21   |    1.        0         1         0         1    |
|         |         |                                                 |
|  p_7    |  q_31   |    1.        1         1         1         1    |
|         |         |                                                 |
o---------o---------o---------o---------o---------o---------o---------o

Table 17-b.  A Couple of Orbits in Fourth Gear:  Orbit 2
o---------o---------o---------o---------o---------o---------o---------o
| Time    | State   |    A    |   dA    |         |         |         |
|  p_i    |  q_j    |  d^0.A  |  d^1.A  |  d^2.A  |  d^3.A  |  d^4.A  |
o---------o---------o---------o---------o---------o---------o---------o
|         |         |                                                 |
|  p_0    |  q_25   |    1.        1         0         0         1    |
|         |         |                                                 |
|  p_1    |  q_11   |    0.        1         0         1         1    |
|         |         |                                                 |
|  p_2    |  q_29   |    1.        1         1         0         1    |
|         |         |                                                 |
|  p_3    |  q_07   |    0.        0         1         1         1    |
|         |         |                                                 |
|  p_4    |  q_09   |    0.        1         0         0         1    |
|         |         |                                                 |
|  p_5    |  q_27   |    1.        1         0         1         1    |
|         |         |                                                 |
|  p_6    |  q_13   |    0.        1         1         0         1    |
|         |         |                                                 |
|  p_7    |  q_23   |    1.        0         1         1         1    |
|         |         |                                                 |
o---------o---------o---------o---------o---------o---------o---------o

Diff Log D 2004 • Note 25


| It is understandable that an engineer should be completely absorbed in his
| speciality, instead of pouring himself out into the freedom and vastness
| of the world of thought, even though his machines are being sent off
| to the ends of the earth;  for he no more needs to be capable of
| applying to his own personal soul what is daring and new
| in the soul of his subject than a machine is in fact
| capable of applying to itself the differential
| calculus on which it is based.  The same
| thing cannot, however, be said about
| mathematics;  for here we have
| the new method of thought,
| pure intellect, the very
| well-spring of the times,
| the 'fons et origo' of an
| unfathomable transformation.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 39]

Transformations of Discourse

In this section we take up the general study of logical transformations,
or maps that relate one universe of discourse to another.  In many ways,
and especially as applied to the subject of intelligent dynamic systems,
my argument develops the antithesis of the statement just quoted.  Along
the way, if incidental to my ends, I hope this essay can pose a fittingly
irenic epitaph to the frankly ironic epigram I have inscribed at its head.

My goal in this section is to answer a single question:
What is a propositional tangent functor?  In other words,
my aim is to develop a clear conception of what manner of
thing would pass in the logical realm for a genuine analogue
of the tangent functor, an object conceived to generalize as
far as possible in the abstract terms of category theory the
ordinary notions of functional differentiation and the all
too familiar operations of taking derivatives.

As a first step I discuss the kinds of transformations that
we already know as "extensions" and "projections", and I use
these special cases to illustrate several different styles of
logical and visual representation that will figure heavily in
the sequel.

Diff Log D 2004 • Note 26


| And, despite the care which she took to look behind her at every moment,
| she failed to see a shadow which followed her like her own shadow, which
| stopped when she stopped, which started again when she did, and which made
| no more noise than a well-conducted shadow should.
|
| Gaston Leroux, 'The Phantom of the Opera', [Ler, 126]

Foreshadowing Transformations:  Extensions and Projections of Discourse

Many times in our discussion we have occasion to place one universe of discourse
in the context of a larger universe of discourse.  An embedding of the general
type [!X!] -> [!Y!] is implied any time that we make use of one alphabet !X!
that happens to be included in another alphabet !Y!.  When we are discussing
differential issues we usually have in mind that the extended alphabet !Y!
has a special construction or a specific lexical relation with respect to
the initial alphabet !X!, one that is marked by characteristic types of
accents, indices, or inflected forms.

Extension from 1 to 2 Dimensions

Figure 18-a lays out the "angular form" of venn diagram for universes
of 1 and 2 dimensions, indicating the embedding map of type B^1 -> B^2
and detailing the coordinates that are associated with individual cells.
Because all points, cells, or logical interpretations are represented
as connected geometric areas, we can say that these pictures provide
us with an "areal view" of each universe of discourse.

o-----------------------------------------------------------o
|                                                           |
|              o                             o              |
|             / \                           / \             |
|            /   \                         /   \            |
|           /     \                       /     \           |
|          /       \                     /       \          |
|         /         o                   o   1 1   o         |
|        /         / \                 / \       / \        |
|       /         /   \               /   \     /   \       |
|      /    1    /     \             /     \   /     \      |
|     /         /       \    !e!    /       \ /       \     |
|    o         /         o  ---->  o   1 0   o   0 1   o    |
|    |\       /         /          |\       / \       /|    |
|    | \     /    0    /           | \     /   \     / |    |
|    |  \   /         /            |  \   /     \   /  |    |
|    |x_1\ /         /             |x_1\ /       \ /x_2|    |
|    o----o         /              o----o   0 0   o----o    |
|          \       /                     \       /          |
|           \     /                       \     /           |
|            \   /                         \   /            |
|             \ /                           \ /             |
|              o                             o              |
|                                                           |
o-----------------------------------------------------------o
Figure 18-a.  Extension from 1 to 2 Dimensions:  Areal

Figure 18-b shows the differential extension from X% = [x] to EX% = [x, dx] in
a "bundle of boxes" form of venn diagram.  As awkward as it may seem at first,
this type of picture is often the most natural and the most easily available
representation when we want to conceptualize the localized information or
momentary knowledge of an intelligent dynamic system.  It gives a ready
picture of a "proposition at a point", in the present instance, of a
proposition about changing states which is itself associated with a
particular dynamic state of a system.  It is easy to see how this
application might be extended to conceive of more general types
of instantaneous knowledge that are possessed by a system.

o-----------------------------o         o-------------------o
|                             |         |                   |
|                             |         |     o-------o     |
|         o---------o         |         |    /         \    |
|        /           \        |         |   o           o   |
|       /      o------------------------|   |    dx     |   |
|      /               \      |         |   o           o   |
|     /                 \     |         |    \         /    |
|    o                   o    |         |     o-------o     |
|    |                   |    |         |                   |
|    |                   |    |         o-------------------o
|    |         x         |    |
|    |                   |    |         o-------------------o
|    |                   |    |         |                   |
|    o                   o    |         |     o-------o     |
|     \                 /     |         |    /         \    |
|      \               /      |         |   o           o   |
|       \             /    o------------|   |    dx     |   |
|        \           /        |         |   o           o   |
|         o---------o         |         |    \         /    |
|                             |         |     o-------o     |
|                             |         |                   |
o-----------------------------o         o-------------------o
Figure 18-b.  Extension from 1 to 2 Dimensions:  Bundle

Figure 18-c shows the same extension in a "compact" style of venn diagram,
where the differential features at each position are represented by arrows
extending from that position that cross or do not cross, as the case may be,
the corresponding feature boundaries.

o-----------------------------------------------------------o
|                                                           |
|                                                           |
|               o-----------------o                         |
|              /         o         \                        |
|             /    (dx) / \         \ dx                    |
|            /         v   o--------------------->o         |
|           /           \ /           \                     |
|          /             o             \                    |
|         o                             o                   |
|         |                             |                   |
|         |                             |                   |
|         |              x              |        (x)        |
|         |                             |                   |
|         |                             |                   |
|         o                             o                   |
|          \                           /          o         |
|           \                         /          / \        |
|            \           o<---------------------o   v       |
|             \                     / dx         \ / (dx)   |
|              \                   /              o         |
|               o-----------------o                         |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 18-c.  Extension from 1 to 2 Dimensions:  Compact

Figure 18-d compresses the picture of the differential extension even further,
yielding a directed graph or "digraph" form of representation.  (Notice that
my definition of a digraph allows for loops or "slings" at individual points,
in addition to arcs or "arrows" between the points.)

o-----------------------------------------------------------o
|                                                           |
|                                                           |
|                            dx                             |
|           .--.   .---------->----------.   .--.           |
|           |   \ /                       \ /   |           |
|     (dx)  ^    @  x                 (x)  @    v  (dx)     |
|           |   / \                       / \   |           |
|           *--*   *----------<----------*   *--*           |
|                             dx                            |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 18-d.  Extension from 1 to 2 Dimensions:  Digraph

Diff Log D 2004 • Note 27


Extension from 2 to 4 Dimensions

Figure 19-a lays out the "areal view" or the "angular form" of venn diagram for
universes of 2 and 4 dimensions, indicating the embedding map of type B^2 -> B^4.
In many ways these pictures are the best kind there is, giving full canvass to an
ideal vista.  Their style allows the clearest, the fairest, and the plainest view
that we can form of a universe of discourse, affording equal representation to all
dispositions and maintaining a balance with respect to ordinary and differential
features.  If only we could extend this view!  Unluckily, an obvious difficulty
beclouds this prospect, and that is how precipitately we run into the limits
of our plane and visual intuitions.  Even within the scope of the spare few
dimensions that we have scanned up to this point subtle discrepancies have
crept in already.  The circumstances that bind us and the frameworks that
block us, the flat distortion of the planar projection and the inevitable
ineffability that precludes us from wrapping its rhomb figure into rings
around a torus, all of these factors disguise the underlying but true
connectivity of the universe of discourse.

o-------------------------------------------------------------------------------o
|                                                                               |
|                   o                                       o                   |
|                  / \                                     / \                  |
|                 /   \                                   /   \                 |
|                /     \                                 /     \                |
|               /       \                               o 1100  o               |
|              /         \                             / \     / \              |
|             /           \                           /   \   /   \             |
|            /             \           !e!           /     \ /     \            |
|           o      1 1      o         ---->         o 1101  o 1110  o           |
|          / \             / \                     / \     / \     / \          |
|         /   \           /   \                   /   \   /   \   /   \         |
|        /     \         /     \                 /     \ /     \ /     \        |
|       /       \       /       \               o 1001  o 1111  o 0110  o       |
|      /         \     /         \             / \     / \     / \     / \      |
|     /           \   /           \           /   \   /   \   /   \   /   \     |
|    /             \ /             \         /     \ /     \ /     \ /     \    |
|   o      1 0      o      0 1      o       o 1000  o 1011  o 0111  o 0100  o   |
|   |\             / \             /|       |\     / \     / \     / \     /|   |
|   | \           /   \           / |       | \   /   \   /   \   /   \   / |   |
|   |  \         /     \         /  |       |  \ /     \ /     \ /     \ /  |   |
|   |   \       /       \       /   |       |   o 1010  o 0011  o 0101  o   |   |
|   |    \     /         \     /    |       |   |\     / \     / \     /|   |   |
|   |     \   /           \   /     |       |   | \   /   \   /   \   / |   |   |
|   | x_1  \ /             \ /  x_2 |       |x_1|  \ /     \ /     \ /  |x_2|   |
|   o-------o      0 0      o-------o       o---+---o 0010  o 0001  o---+---o   |
|            \             /                    |    \     / \     /    |       |
|             \           /                     |     \   /   \   /     |       |
|              \         /                      | x_3  \ /     \ /  x_4 |       |
|               \       /                       o-------o 0000  o-------o       |
|                \     /                                 \     /                |
|                 \   /                                   \   /                 |
|                  \ /                                     \ /                  |
|                   o                                       o                   |
|                                                                               |
o-------------------------------------------------------------------------------o
Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal

Figure 19-b shows the differential extension
from U% = [u, v] to EU% = [u, v, du, dv] in
the "bundle of boxes" form of venn diagram.

                                                  o-----------------------------o
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         / \         \   |
                                                  |  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                 /|  o         o   o         o  |
                                                / |   \         \ /         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     o-----------------------------o
                                           /
o-----------------------------------------/---o   o-----------------------------o
|                                        /    |   |      o-----o   o-----o      |
|                                       @     |   |     /       \ /       \     |
|          o---------o   o---------o          |   |    /         o         \    |
|         /           \ /           \         |   |   /         / \         \   |
|        /             o             \        |   |  o         o   o         o  |
|       /             / \     @-------\-----------@  |   du    |   |    dv   |  |
|      /             / @ \             \      |   |  o         o   o         o  |
|     /             /   \ \             \     |   |   \         \ /         /   |
|    /             /     \ \             \    |   |    \         o         /    |
|   o             o       \ o             o   |   |     \       / \       /     |
|   |             |        \|             |   |   |      o-----o   o-----o      |
|   |             |         |             |   |   o-----------------------------o
|   |      u      |         |\     v      |   |
|   |             |         | \           |   |   o-----------------------------o
|   |             |         |  \          |   |   |      o-----o   o-----o      |
|   o             o         o   \         o   |   |     /       \ /       \     |
|    \             \       /     \       /    |   |    /         o         \    |
|     \             \     /       \     /     |   |   /         / \         \   |
|      \             \   /         \   /      |   |  o         o   o         o  |
|       \       @-----\-/-----------\-------------@  |   du    |   |    dv   |  |
|        \             o             /        |   |  o         o   o         o  |
|         \           / \           / \       |   |   \         \ /         /   |
|          o---------o   o---------o   \      |   |    \         o         /    |
|                                       \     |   |     \       / \       /     |
|                                        \    |   |      o-----o   o-----o      |
o-----------------------------------------\---o   o-----------------------------o
                                           \
                                            \     o-----------------------------o
                                             \    |      o-----o   o-----o      |
                                              \   |     /       \ /       \     |
                                               \  |    /         o         \    |
                                                \ |   /         / \         \   |
                                                 \|  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                  |  o         o   o         o  |
                                                  |   \         \ /         /   |
                                                  |    \         o         /    |
                                                  |     \       / \       /     |
                                                  |      o-----o   o-----o      |
                                                  o-----------------------------o
Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle

As dimensions increase, this factorization of the extended universe
along the lines that are marked out by the bundle picture begins to
look more and more like a practical necessity.  But whenever we use
a propositional model to address a real situation in the context of
nature we need to remain aware that this articulation into factors,
affecting our description, may be wholly artificial in nature and
cleave to nothing, no joint in nature, nor any juncture in time
to be in or out of joint.

Figure 19-c illustrates the extension from 2 to 4 dimensions in the "compact" style
of venn diagram.  Here, just the changes with respect to the center cell are shown.

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (du).(dv) o                       o    |
|    |                       |   -->--   |                       |    |
|    |                       |   \   /   |                       |    |
|    |              dv .(du) |    \ /    | du .(dv)              |    |
|    |      u      <---------------@--------------->      v      |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  V                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact

Figure 19-d gives the "digraph" form of representation for the
differential extension U% -> EU%, where the 4 nodes marked "@"
are the cells uv, u(v), (u)v, (u)(v), respectively, and where
a 2-headed arc counts as two arcs of the differential digraph.

o-----------------------------------------------------------o
|                                                           |
|                           .->-.                           |
|                           |   |                           |
|                           *   *                           |
|                            \ /                            |
|                       .-->--@--<--.                       |
|                      /     / \     \                      |
|                     /     /   \     \                     |
|                    /     .     .     \                    |
|                   /      |     |      \                   |
|                  /       |     |       \                  |
|                 /        |     |        \                 |
|                .         |     |         .                |
|                |         |     |         |                |
|                v         |     |         v                |
|           .--. | .---------->----------. | .--.           |
|           |   \|/        |     |        \|/   |           |
|           ^    @         ^     v         @    v           |
|           |   /|\        |     |        /|\   |           |
|           *--* | *----------<----------* | *--*           |
|                ^         |     |         ^                |
|                |         |     |         |                |
|                *         |     |         *                |
|                 \        |     |        /                 |
|                  \       |     |       /                  |
|                   \      |     |      /                   |
|                    \     .     .     /                    |
|                     \     \   /     /                     |
|                      \     \ /     /                      |
|                       *-->--@--<--*                       |
|                            / \                            |
|                           .   .                           |
|                           |   |                           |
|                           *-<-*                           |
|                                                           |
o-----------------------------------------------------------o
Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph

Diff Log D 2004 • Note 28


| And as imagination bodies forth
| The forms of things unknown, the poet's pen
| Turns them to shapes, and gives to airy nothing
| A local habitation and a name.
|
| 'A Midsummer Night's Dream', 5.1.18

Thematization of Functions:  And a Declaration of Independence for Variables

In the representation of propositions as functions it is possible to notice
different degrees of explicitness in the way their functional character is
symbolized.  To indicate what I mean by this, the next series of Figures
illustrates a set of graphic conventions that will be put to frequent
use in the remainder of this discussion, both to mark the relevant
distinctions and to help us convert between related expressions
at different levels of explicitness in their functionality.

Diff Log D 2004 • Note 29


Thematization:  Venn Diagrams

| The known universe has one complete lover and that is the greatest poet.
| He consumes an eternal passion and is indifferent which chance happens
| and which possible contingency of fortune or misfortune and persuades
| daily and hourly his delicious pay.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 11-12]

Figure 20-i traces the first couple of steps in this order of "thematic" progression,
that will gradually run the gamut through a complete series of degrees of functional
explicitness in the expression of logical propositions.  The first venn diagram
represents a situation where the function is indicated by a shaded figure and
a logical expression.  At this stage one may be thinking of the proposition
only as expressed by a formula in a particular language and its content
only as a subset of the universe of discourse, as when one considers
the proposition u.v in [u, v].

The second venn diagram depicts a situation in which two significant
steps have been taken.  First, one has taken the trouble to give the
proposition u.v a distinctive functional name "J".  Second, one has
come to think explicitly about the target domain that contains the
functional values of J, as when one writes J : <|u, v|> -> B.

o-------------------------------o     o-------------------------------o
|                               |     |                               |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|      /       \ /       \      |     |      /       \ /       \      |
|     /         o         \     |     |     /         o         \     |
|    /         /`\         \    |     |    /         /`\         \    |
|   o         o```o         o   |     |   o         o```o         o   |
|   |    u    |```|    v    |   |     |   |    u    |```|    v    |   |
|   o         o```o         o   |     |   o         o```o         o   |
|    \         \`/         /    |     |    \         \`/         /    |
|     \         o         /     |     |     \         o         /     |
|      \       / \       /      |     |      \       / \       /      |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
                                       \                             /
                                         \                         /
                                           \                     /
               u v                           \        J        /
                                               \             /
                                                 \         /
                                                   \     /
                                                     \ /
                                                      o
Figure 20-i.  Thematization of Conjunction (Stage 1)

In Figure 20-ii the proposition J is viewed explicitly as
a transformation from one universe of discourse to another.

o-------------------------------o     o-------------------------------o
|                               |     |                               |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|      /       \ /       \      |     |      /       \ /       \      |
|     /         o         \     |     |     /         o         \     |
|    /         /`\         \    |     |    /         /`\         \    |
|   o         o```o         o   |     |   o         o```o         o   |
|   |    u    |```|    v    |   |     |   |    u    |```|    v    |   |
|   o         o```o         o   |     |   o         o```o         o   |
|    \         \`/         /    |     |    \         \`/         /    |
|     \         o         /     |     |     \         o         /     |
|      \       / \       /      |     |      \       / \       /      |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
 \                             /       \                             /
   \                         /           \                         /
     \                     /               \          J          /
       \                 /                   \                 /
         \             /                       \             /
o----------\---------/----------o     o----------\---------/----------o
|            \     /            |     |            \     /            |
|              \ /              |     |              \ /              |
|         o-----@-----o         |     |         o-----@-----o         |
|        /`````````````\        |     |        /`````````````\        |
|       /```````````````\       |     |       /```````````````\       |
|      /`````````````````\      |     |      /`````````````````\      |
|     o```````````````````o     |     |     o```````````````````o     |
|     |```````````````````|     |     |     |```````````````````|     |
|     |```````` J ````````|     |     |     |```````` x ````````|     |
|     |```````````````````|     |     |     |```````````````````|     |
|     o```````````````````o     |     |     o```````````````````o     |
|      \`````````````````/      |     |      \`````````````````/      |
|       \```````````````/       |     |       \```````````````/       |
|        \`````````````/        |     |        \`````````````/        |
|         o-----------o         |     |         o-----------o         |
|                               |     |                               |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
             J = u v                             x = J<u, v>

Figure 20-ii.  Thematization of Conjunction (Stage 2)

In the first venn diagram the name that is assigned to a composite proposition,
function, or region in the source universe is delegated to a simple feature in
the target universe.  This can result in a single character or term exceeding
the responsibilities it can carry off well.  Allowing the name of a function
J : <|u, v|> -> B to serve as the name of its dependent variable J : B does
not mean that one has to confuse a function with any of its values, but it
does put one at risk for a number of obvious problems, and we should not
be surprised, on numerous and limiting occasions, when quibbling arises
from the attempts of a too original syntax to serve these two masters.

The second venn diagram circumvents these difficulties by introducing a new
variable name for each basic feature of the target universe, as when one
writes J : <|u, v|> -> <|x|> and thereby assigns a concrete type <|x|>
to the abstract codomain B.  To make this induction of variables more
formal one can append subscripts, as in x_J, to indicate the origin
or the derivation of these parvenu characters.  However, it is not
always convenient to keep inventing new variable names in this way.
For use at these times, I introduce a lexical operator "¢", read
"cents" or "obelus", that converts a function name into a variable
name.  For example, one may think of x = x_J = ¢(J) = J¢ = J^¢ as
"the cache variable of J", "J circumscript", "J made circumstantial",
or "J considered as a contingent variable".

In Figure 20-iii we arrive at a stage where the functional equations,
J = u.v  and  x = u.v, are regarded as propositions in their own right,
reigning in and ruling over 3-feature universes of discourse, [u, v, J]
and [u, v, x], respectively.  Subject to the cautions already noted, the
function name "J" can be reinterpreted as the name of a feature J^¢, and
the equation J = u.v can be read as the logical equivalence ((J, u v)).
To give it a generic name let us call this newly expressed, collateral
proposition the "thematization" or the "thematic extension" of the
original proposition J.

o-------------------------------o     o-------------------------------o
|                               |     |```````````````````````````````|
|                               |     |````````````o-----o````````````|
|                               |     |```````````/       \```````````|
|                               |     |``````````/         \``````````|
|                               |     |`````````/           \`````````|
|                               |     |````````/             \````````|
|               J               |     |```````o       x       o```````|
|                               |     |```````|               |```````|
|                               |     |```````|               |```````|
|                               |     |```````|               |```````|
|       o-----o   o-----o       |     |```````o-----o   o-----o```````|
|      /       \ /       \      |     |``````/`\     \ /     /`\``````|
|     /         o         \     |     |`````/```\     o     /```\`````|
|    /         /`\         \    |     |````/`````\   /`\   /`````\````|
|   /         /```\         \   |     |```/```````\ /```\ /```````\```|
|  o         o`````o         o  |     |``o`````````o-----o`````````o``|
|  |    u    |`````|    v    |  |     |``|`````````|     |`````````|``|
o--o---------o-----o---------o--o     |``|``` u ```|     |``` v ```|``|
|``|`````````|     |`````````|``|     |``|`````````|     |`````````|``|
|``o`````````o     o`````````o``|     |``o`````````o     o`````````o``|
|```\`````````\   /`````````/```|     |```\`````````\   /`````````/```|
|````\`````````\ /`````````/````|     |````\`````````\ /`````````/````|
|`````\`````````o`````````/`````|     |`````\`````````o`````````/`````|
|``````\```````/`\```````/``````|     |``````\```````/`\```````/``````|
|```````o-----o```o-----o```````|     |```````o-----o```o-----o```````|
|```````````````````````````````|     |```````````````````````````````|
o-------------------------------o     o-------------------------------o
                                       \                             /
                                         \                         /
          J   =   u v                      \                     /
                                             \       !j!       /
                                               \             /
         !j!  =   (( x , u v ))                  \         /
                                                   \     /
                                                     \ /
                                                      @
Figure 20-iii.  Thematization of Conjunction (Stage 3)

The first venn diagram represents the thematization of the conjunction J
with shading in the appropriate regions of the universe [u, v, J].  Also,
it illustrates a quick way of constructing a thematic extension.  First,
draw a line, in practice or the imagination, that bisects every cell of
the original universe, placing half of each cell under the aegis of the
thematized proposition and the other half under its antithesis.  Next,
on the scene where the theme applies leave the shade wherever it lies,
and off the stage, where it plays otherwise, stagger the pattern in
a harlequin guise.

In the final venn diagram of this sequence the thematic progression comes
full circle and completes one round of its development.  The ambiguities
that were occasioned by the changing role of the name "J" are resolved
by introducing a new variable name "x" to take the place of J^¢, and
the region that represents this fresh featured x is circumscribed
in a more conventional symmetry of form and placement.  Just as
we once gave the name "J" to the proposition u.v, we now give
the name "!j!" to its thematization ((x, u v)).  Already,
again, at this culminating stage of reflection, we begin
to think of the newly named proposition as a distinctive
individual, a particular function !j! : <|u, v, x|> -> B.

From now on, the terms "thematic extension" and "thematization" will be used to
describe both the process and degree of explication that progresses through this
series of pictures, both the operation of increasingly explicit symbolization and
the dimension of variation that is swept out by it.  To speak of this change in
general, that takes us in our current example from J to !j!, I introduce a class
of operators symbolized by the Greek letter theta, writing !j! = theta(J) in the
present instance.  The operator theta, in the present situation bearing the type
theta : [u, v] -> [u, v, x], provides us with a convenient way of recapitulating
and summarizing the complete cycle of thematic developments.

Diff Log D 2004 • Note 30


Thematization:  Venn Diagrams (concl.)

Figure 21 shows how the thematic extension operator theta acts on two
further examples, the disjunction ((u)(v)) and the equality ((u, v)).
Referring to the disjunction as f<u, v> and the equality as g<u, v>,
I write the thematic extensions as !f! = theta(f) and !g! = theta(g).

                f                                     g
o-------------------------------o     o-------------------------------o
|                               |     |```````````````````````````````|
|       o-----o   o-----o       |     |```````o-----o```o-----o```````|
|      /```````\ /```````\      |     |``````/       \`/       \``````|
|     /`````````o`````````\     |     |`````/         o         \`````|
|    /`````````/`\`````````\    |     |````/         /`\         \````|
|   /`````````/```\`````````\   |     |```/         /```\         \```|
|  o`````````o`````o```````` o  |     |``o         o`````o         o``|
|  |`````````|`````|`````````|  |     |``|         |`````|         |``|
|  |``` u ```|`````|``` v ```|  |     |``|    u    |`````|    v    |``|
|  |`````````|`````|`````````|  |     |``|         |`````|         |``|
|  o`````````o`````o`````````o  |     |``o         o`````o         o``|
|   \`````````\```/`````````/   |     |```\         \```/         /```|
|    \`````````\`/`````````/    |     |````\         \`/         /````|
|     \`````````o`````````/     |     |`````\         o         /`````|
|      \```````/ \```````/      |     |``````\       /`\       /``````|
|       o-----o   o-----o       |     |```````o-----o```o-----o```````|
|                               |     |```````````````````````````````|
o-------------------------------o     o-------------------------------o
            ((u)(v))                              ((u , v))

                |                                     |
                |                                     |
              theta                                 theta
                |                                     |
                |                                     |
                v                                     v

               !f!                                   !g!
o-------------------------------o     o-------------------------------o
|```````````````````````````````|     |                               |
|````````````o-----o````````````|     |            o-----o            |
|```````````/       \```````````|     |           /```````\           |
|``````````/         \``````````|     |          /`````````\          |
|`````````/           \`````````|     |         /```````````\         |
|````````/             \````````|     |        /`````````````\        |
|```````o       f       o```````|     |       o`````` g ``````o       |
|```````|               |```````|     |       |```````````````|       |
|```````|               |```````|     |       |```````````````|       |
|```````|               |```````|     |       |```````````````|       |
|```````o-----o   o-----o```````|     |       o-----o```o-----o       |
|``````/ \`````\ /`````/ \``````|     |      /`\     \`/     /`\      |
|`````/   \`````o`````/   \`````|     |     /```\     o     /```\     |
|````/     \```/`\```/     \````|     |    /`````\   /`\   /`````\    |
|```/       \`/```\`/       \```|     |   /```````\ /```\ /```````\   |
|``o         o-----o         o``|     |  o`````````o-----o`````````o  |
|``|         |     |         |``|     |  |`````````|     |`````````|  |
|``|    u    |     |    v    |``|     |  |``` u ```|     |``` v ```|  |
|``|         |     |         |``|     |  |`````````|     |`````````|  |
|``o         o     o         o``|     |  o`````````o     o`````````o  |
|```\         \   /         /```|     |   \`````````\   /`````````/   |
|````\         \ /         /````|     |    \`````````\ /`````````/    |
|`````\         o         /`````|     |     \`````````o`````````/     |
|``````\       /`\       /``````|     |      \```````/ \```````/      |
|```````o-----o```o-----o```````|     |       o-----o   o-----o       |
|```````````````````````````````|     |                               |
o-------------------------------o     o-------------------------------o
        ((f , ((u)(v)) ))                    ((g , ((u , v)) ))

Figure 21.  Thematization of Disjunction and Equality

Diff Log D 2004 • Note 31


Thematization:  Truth Tables

| That which distorts honest shapes or which creates unearthly
| beings or places or contingencies is a nuisance and a revolt.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 19]

Tables 22 through 25 outline a method for computing
the thematic extensions of propositions in terms of
their coordinate values.

A preliminary step, as illustrated in Table 22, is to write out
the truth table representations of the propositional forms whose
thematic extensions one wants to compute, in the present instance,
the functions f<u, v> = ((u)(v)) and g<u, v> = ((u, v)).

Table 22.  Disjunction f and Equality g
o-------------------o-------------------o
|    u         v    |    f         g    |
o-------------------o-------------------o
|                   |                   |
|    0         0    |    0         1    |
|                   |                   |
|    0         1    |    1         0    |
|                   |                   |
|    1         0    |    1         0    |
|                   |                   |
|    1         1    |    1         1    |
|                   |                   |
o-------------------o-------------------o

Next, each propositional form is individually represented in the fashion
shown in Tables 23-i and 23-ii, using "f" and "g" as function names and
creating new variables x and y to hold the associated functional values.
This pair of Tables outlines the first stage in the transition from the
2-dimensional universes of f and g to the 3-dimensional universes of
theta(f) and theta(g).  The top halves of the Tables replicate the
truth table patterns for f and g in the form f : [u, v] -> [x]
and g : [u, v] -> [y].  The bottom halves of the tables print
the negatives of these pictures, as it were, and paste the
truth tables for (f) and (g) under the copies for f and g.
At this stage, the columns for theta(f) and theta(g) are
appended almost as afterthoughts, amounting to indicator
functions for the sets of ordered triples that make up
the functions f and g.

Tables 23-i and 23-ii.  Thematics of Disjunction and Equality (1)
o-----------------o-----------o         o-----------------o-----------o
|  u     v     f  |  x    !f! |         |  u     v     g  |  y    !g! |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0    --> |  0     1  |         |  0     0    --> |  1     1  |
|                 |           |         |                 |           |
|  0     1    --> |  1     1  |         |  0     1    --> |  0     1  |
|                 |           |         |                 |           |
|  1     0    --> |  1     1  |         |  1     0    --> |  0     1  |
|                 |           |         |                 |           |
|  1     1    --> |  1     1  |         |  1     1    --> |  1     1  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0        |  1     0  |         |  0     0        |  0     0  |
|                 |           |         |                 |           |
|  0     1        |  0     0  |         |  0     1        |  1     0  |
|                 |           |         |                 |           |
|  1     0        |  0     0  |         |  1     0        |  1     0  |
|                 |           |         |                 |           |
|  1     1        |  0     0  |         |  1     1        |  0     0  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o

All the data is now in place to give the truth tables for theta(f) and theta(g).
In the remaining steps all we do is to permute the rows and change the roles of
x and y from dependent to independent variables.  In Tables 24-i and 24-ii the
rows are arranged in such a way as to put the 3-tuples <u, v, x> and <u, v, y>
in binary numerical order, suitable for viewing as the arguments of the maps
theta(f) = !f! : [u, v, x] -> B and theta(g) = !g! : [u, v, y]->B.  Moreover,
the structure of the tables is altered slightly, allowing the now vestigial
functions f and g to be passed over without further attention and shifting
the heavy vertical bars a notch to the right.  In effect, this clinches
the fact that the thematic variables x := f^¢ and y := g^¢ are now to
be regarded as independent variables.

Tables 24-i and 24-ii.  Thematics of Disjunction and Equality (2)
o-----------------o-----------o         o-----------------o-----------o
|  u     v     f  |  x    !f! |         |  u     v     g  |  y    !g! |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0    --> |  0     1  |         |  0     0        |  0     0  |
|                 |           |         |                 |           |
|  0     0        |  1     0  |         |  0     0    --> |  1     1  |
|                 |           |         |                 |           |
|  0     1        |  0     0  |         |  0     1    --> |  0     1  |
|                 |           |         |                 |           |
|  0     1    --> |  1     1  |         |  0     1        |  1     0  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  1     0        |  0     0  |         |  1     0    --> |  0     1  |
|                 |           |         |                 |           |
|  1     0    --> |  1     1  |         |  1     0        |  1     0  |
|                 |           |         |                 |           |
|  1     1        |  0     0  |         |  1     1        |  0     0  |
|                 |           |         |                 |           |
|  1     1    --> |  1     1  |         |  1     1    --> |  1     1  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o

An optional reshuffling of the rows brings additional features of the thematic
extensions to light.  Leaving the columns in place for the sake of comparison,
Tables 25-i and 25-ii sorts the rows in a different order, in effect treating
x and y as the primary variables in their respective 3-tuples.  Regarding the
thematic extensions in the form !f! : [x, u, v] -> B and !g! : [y, u, v] -> B
makes it easier to see in this tabular setting a property that was graphically
obvious in the venn diagrams above.  Specifically, when the thematic variable
F^¢ is true then theta(F) exhibits the pattern of the original F, and when
F^¢ is false then theta(F) exhibits the pattern of its negation (F).

Tables 25-i and 25-ii.  Thematics of Disjunction and Equality (3)
o-----------------o-----------o         o-----------------o-----------o
|  u     v     f  |  x    !f! |         |  u     v     g  |  y    !g! |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0    --> |  0     1  |         |  0     0        |  0     0  |
|                 |           |         |                 |           |
|  0     1        |  0     0  |         |  0     1    --> |  0     1  |
|                 |           |         |                 |           |
|  1     0        |  0     0  |         |  1     0    --> |  0     1  |
|                 |           |         |                 |           |
|  1     1        |  0     0  |         |  1     1        |  0     0  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0        |  1     0  |         |  0     0    --> |  1     1  |
|                 |           |         |                 |           |
|  0     1    --> |  1     1  |         |  0     1        |  1     0  |
|                 |           |         |                 |           |
|  1     0    --> |  1     1  |         |  1     0        |  1     0  |
|                 |           |         |                 |           |
|  1     1    --> |  1     1  |         |  1     1    --> |  1     1  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o

Finally, Tables 26-i and 26-ii compare the tacit extensions
!e! : [u, v] -> [u, v, x] and !e! : [u, v]->[u, v, y] with
the thematic extensions of the same types, as applied to
the propositions f and g, respectively.

Tables 26-i and 26-ii.  Tacit Extension and Thematization
o-----------------o-----------o         o-----------------o-----------o
|  u     v     x  | !e!f  !f! |         |  u     v     y  | !e!g  !g! |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0     0  |  0     1  |         |  0     0     0  |  1     0  |
|                 |           |         |                 |           |
|  0     0     1  |  0     0  |         |  0     0     1  |  1     1  |
|                 |           |         |                 |           |
|  0     1     0  |  1     0  |         |  0     1     0  |  0     1  |
|                 |           |         |                 |           |
|  0     1     1  |  1     1  |         |  0     1     1  |  0     0  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  1     0     0  |  1     0  |         |  1     0     0  |  0     1  |
|                 |           |         |                 |           |
|  1     0     1  |  1     1  |         |  1     0     1  |  0     0  |
|                 |           |         |                 |           |
|  1     1     0  |  1     0  |         |  1     1     0  |  1     0  |
|                 |           |         |                 |           |
|  1     1     1  |  1     1  |         |  1     1     1  |  1     1  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o

Diff Log D 2004 • Note 32


Thematization:  Truth Tables (cont.)

Table 27 summarizes the thematic extensions of all propositions on two variables.
Column 4 lists the equations of form (( f^¢ , f^¢ <u, v> )) and Column 5 simplifies
these equations into the form of algebraic expressions.  (As always, "+" refers to
exclusive disjunction, and "f" should be read as "[f_i]^¢" in the body of the Table.)

Table 27.  Thematization of Bivariate Propositions
o---------o---------o----------o--------------------o--------------------o
|       u : 1 1 0 0 |    f     |     theta (f)      |     theta (f)      |
|       v : 1 0 1 0 |          |                    |                    |
o---------o---------o----------o--------------------o--------------------o
|         |         |          |                    |                    |
| f_0     | 0 0 0 0 |    ()    | (( f ,    ()    )) | f              + 1 |
|         |         |          |                    |                    |
| f_1     | 0 0 0 1 |  (u)(v)  | (( f ,  (u)(v)  )) | f + u + v + uv     |
|         |         |          |                    |                    |
| f_2     | 0 0 1 0 |  (u) v   | (( f ,  (u) v   )) | f     + v + uv + 1 |
|         |         |          |                    |                    |
| f_3     | 0 0 1 1 |  (u)     | (( f ,  (u)     )) | f + u              |
|         |         |          |                    |                    |
| f_4     | 0 1 0 0 |   u (v)  | (( f ,   u (v)  )) | f + u     + uv + 1 |
|         |         |          |                    |                    |
| f_5     | 0 1 0 1 |     (v)  | (( f ,     (v)  )) | f     + v          |
|         |         |          |                    |                    |
| f_6     | 0 1 1 0 |  (u, v)  | (( f ,  (u, v)  )) | f + u + v      + 1 |
|         |         |          |                    |                    |
| f_7     | 0 1 1 1 |  (u  v)  | (( f ,  (u  v)  )) | f         + uv     |
|         |         |          |                    |                    |
o---------o---------o----------o--------------------o--------------------o
|         |         |          |                    |                    |
| f_8     | 1 0 0 0 |   u  v   | (( f ,   u  v   )) | f         + uv + 1 |
|         |         |          |                    |                    |
| f_9     | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v          |
|         |         |          |                    |                    |
| f_10    | 1 0 1 0 |      v   | (( f ,      v   )) | f     + v      + 1 |
|         |         |          |                    |                    |
| f_11    | 1 0 1 1 |  (u (v)) | (( f ,  (u (v)) )) | f + u     + uv     |
|         |         |          |                    |                    |
| f_12    | 1 1 0 0 |   u      | (( f ,   u      )) | f + u          + 1 |
|         |         |          |                    |                    |
| f_13    | 1 1 0 1 | ((u) v)  | (( f , ((u) v)  )) | f     + v + uv     |
|         |         |          |                    |                    |
| f_14    | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 |
|         |         |          |                    |                    |
| f_15    | 1 1 1 1 |   (())   | (( f ,   (())   )) | f                  |
|         |         |          |                    |                    |
o---------o---------o----------o--------------------o--------------------o

Diff Log D 2004 • Note 33


Thematization:  Truth Tables (concl.)

In order to show what all of the thematic extensions from two dimensions
to three dimensions look like in terms of coordinates, Tables 28 and 29
present ordinary truth tables for the functions f_i : B^2 -> B and for
the corresponding thematizations theta(f_i) = !f!_i : B^3 -> B.

Table 28.  Propositions on Two Variables
o-------o-----o----------------------------------------------------------------o
| u   v |     | f   f   f   f   f   f   f   f   f   f   f   f   f   f   f   f  |
|       |     | 00  01  02  03  04  05  06  07  08  09  10  11  12  13  14  15 |
o-------o-----o----------------------------------------------------------------o
|       |     |                                                                |
| 0   0 |     | 0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1  |
|       |     |                                                                |
| 0   1 |     | 0   0   1   1   0   0   1   1   0   0   1   1   0   0   1   1  |
|       |     |                                                                |
| 1   0 |     | 0   0   0   0   1   1   1   1   0   0   0   0   1   1   1   1  |
|       |     |                                                                |
| 1   1 |     | 0   0   0   0   0   0   0   0   1   1   1   1   1   1   1   1  |
|       |     |                                                                |
o-------o-----o----------------------------------------------------------------o

Table 29.  Thematic Extensions of Bivariate Propositions
o-------o-----o----------------------------------------------------------------o
| u   v | f^¢ |!f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! |
|       |     | 00  01  02  03  04  05  06  07  08  09  10  11  12  13  14  15 |
o-------o-----o----------------------------------------------------------------o
|       |     |                                                                |
| 0  0  |  0  | 1   0   1   0   1   0   1   0   1   0   1   0   1   0   1   0  |
|       |     |                                                                |
| 0  0  |  1  | 0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1  |
|       |     |                                                                |
| 0  1  |  0  | 1   1   0   0   1   1   0   0   1   1   0   0   1   1   0   0  |
|       |     |                                                                |
| 0  1  |  1  | 0   0   1   1   0   0   1   1   0   0   1   1   0   0   1   1  |
|       |     |                                                                |
| 1  0  |  0  | 1   1   1   1   0   0   0   0   1   1   1   1   0   0   0   0  |
|       |     |                                                                |
| 1  0  |  1  | 0   0   0   0   1   1   1   1   0   0   0   0   1   1   1   1  |
|       |     |                                                                |
| 1  1  |  0  | 1   1   1   1   1   1   1   1   0   0   0   0   0   0   0   0  |
|       |     |                                                                |
| 1  1  |  1  | 0   0   0   0   0   0   0   0   1   1   1   1   1   1   1   1  |
|       |     |                                                                |
o-------o-----o----------------------------------------------------------------o

Diff Log D 2004 • Note 34


| If only the word 'artificial' were associated with
| the idea of 'art', or expert skill gained through
| voluntary apprenticeship (instead of suggesting
| the factitious and unreal), we might say that
| 'logical' refers to artificial thought.
|
| John Dewey, 'How We Think', [Dew, 56-57]

Propositional Transformations

In this Subdivision I develop a comprehensive set of concepts for
dealing with transformations between universes of discourse.  In
this most general context the source and the target universes of
a transformation are allowed to be distinct, but may also be one
and the same.  When these concepts are applied to dynamic systems
one focuses on the important special cases of transformations that
map a universe into itself, and transformations of this shape may be
interpreted as the state transitions of a discrete dynamical process,
as these take place among the myriad ways that a universe of discourse
might change, and by that change turn into itself.

Diff Log D 2004 • Note 35


Alias and Alibi Transformations

There are customarily two modes of understanding a transformation,
at least, when we try to interpret its relevance to something in
reality.  A transformation always refers to a changing prospect,
to say it in a unified but equivocal way, but this can be taken
to mean either a subjective change in the interpreting observer's
point of view or an objective change in the systematic subject of
discussion.  In practice these variant uses of the transformation
concept are distinguished in the following terms:

1.  A "perspectival" or "alias" transformation refers to
    a shift in perspective or a change in language that
    takes place in the observer's frame of reference.

2.  A "transitional" or "alibi" transformation refers to
    a change of position or an alteration of state that
    occurs in the object system as it falls under study.

(For a recent discussion of the "alias vs. alibi" issue, as
it relates to linear transformations in vector spaces and to
other issues of an algebraic nature, see [MaB, 256, 582-4].)

Naturally, when we are concerned with the dynamical properties of a system,
the transitional aspect of transformation is the factor that comes to the
fore, and this involves us in contemplating all of the ways of changing
a universe into itself while remaining under the rule of established
dynamical laws.  In the prospective application to dynamic systems,
and to neural networks viewed in this light, our interest lies
chiefly with the transformations of a state space into itself
that constitute the state transitions of a discrete dynamic
process.  Nevertheless, many important properties of these
transformations, and some constructions that we need to
see most clearly, are independent of the transitional
interpretation and are likely to be confounded with
irrelevant features if presented first and only
in that association.

In addition, and in partial contrast, intelligent systems are exactly that
species of dynamic agents that have the capacity to have a point of view,
and we cannot do justice to their peculiar properties without examining
their ability to form and transform their own frames of reference in
exposure to the elements of their own experience.  In this setting,
the perspectival aspect of transformation is the facet that shines
most brightly, perhaps too often leaving us fascinated with mere
glimmerings of its actual potential.  It needs to be emphasized
that nothing of the ordinary sort needs be moved in carrying out
a transformation under the alias interpretation, that it may only
involve a change in the forms of address, an amendment of the terms
which are customed to approach and fashioned to describe the very same
things in the very same world.  But again, working within a discipline of
realistic computation, we know how formidably complex and resource-consuming
such transformations of perspective can be to implement in practice, much less
to endow in the self-governed form of a nascently intelligent dynamical system.

Diff Log D 2004 • Note 36


Transformations of General Type

| 'Es ist passiert', "it just sort of happened", people said there
| when other people in other places thought heaven knows what had
| occurred.  It was a peculiar phrase, not known in this sense to
| the Germans and with no equivalent in other languages, the very
| breath of it transforming facts and the bludgeonings of fate
| into something light as eiderdown, as thought itself.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 34]

Consider the situation illustrated in Figure 30, where the alphabets
!U! = {u, v} and !X! = {x, y, z} are used to label basic features in
two different logical universes, U% = [u, v] and X% = [x, y, z].

             o-------------------------------------------------------o
             | U                                                     |
             |                                                       |
             |             o-----------o   o-----------o             |
             |            /             \ /             \            |
             |           /               o               \           |
             |          /               / \               \          |
             |         /               /   \               \         |
             |        o               o     o               o        |
             |        |               |     |               |        |
             |        |       u       |     |       v       |        |
             |        |               |     |               |        |
             |        o               o     o               o        |
             |         \               \   /               /         |
             |          \               \ /               /          |
             |           \               o               /           |
             |            \             / \             /            |
             |             o-----------o   o-----------o             |
             |                                                       |
             |                                                       |
             o---------------------------o---------------------------o
            / \                         / \                         / \
           /   \                       /   \                       /   \
          /     \                     /     \                     /     \
         /       \                   /       \                   /       \
        /         \                 /         \                 /         \
       /           \               /           \               /           \
      /             \             /             \             /             \
     /               \           /               \           /               \
    /                 \         /                 \         /                 \
   /                   \       /                   \       /                   \
  /                     \     /                     \     /                     \
 /                       \   /                       \   /                       \
o-------------------------o o-------------------------o o-------------------------o
| U                       | | U                       | | U                       |
|      o---o   o---o      | |      o---o   o---o      | |      o---o   o---o      |
|     /     \ /     \     | |     /     \ /     \     | |     /     \ /     \     |
|    /       o       \    | |    /       o       \    | |    /       o       \    |
|   /       / \       \   | |   /       / \       \   | |   /       / \       \   |
|  o       o   o       o  | |  o       o   o       o  | |  o       o   o       o  |
|  |   u   |   |   v   |  | |  |   u   |   |   v   |  | |  |   u   |   |   v   |  |
|  o       o   o       o  | |  o       o   o       o  | |  o       o   o       o  |
|   \       \ /       /   | |   \       \ /       /   | |   \       \ /       /   |
|    \       o       /    | |    \       o       /    | |    \       o       /    |
|     \     / \     /     | |     \     / \     /     | |     \     / \     /     |
|      o---o   o---o      | |      o---o   o---o      | |      o---o   o---o      |
|                         | |                         | |                         |
o-------------------------o o-------------------------o o-------------------------o
 \                        |  \                       /  |                        /
  \                       |   \                     /   |                       /
   \                      |    \                   /    |                      /
    \                     |     \                 /     |                     /
     \       g            |      \       f       /      |            h       /
      \                   |       \             /       |                   /
       \                  |        \           /        |                  /
        \                 |         \         /         |                 /
         \                |          \       /          |                /
          \    o----------|-----------\-----/-----------|----------o    /
           \   | X        |            \   /            |          |   /
            \  |          |             \ /             |          |  /
             \ |          |        o-----o-----o        |          | /
              \|          |       /             \       |          |/
               \          |      /               \      |          /
               |\         |     /                 \     |         /|
               | \        |    /                   \    |        / |
               |  \       |   /                     \   |       /  |
               |   \      |  o           x           o  |      /   |
               |    \     |  |                       |  |     /    |
               |     \    |  |                       |  |    /     |
               |      \   |  |                       |  |   /      |
               |       \  |  |                       |  |  /       |
               |        \ |  |                       |  | /        |
               |         \|  |                       |  |/         |
               |          o--o--------o     o--------o--o          |
               |         /    \        \   /        /    \         |
               |        /      \        \ /        /      \        |
               |       /        \        o        /        \       |
               |      /          \      / \      /          \      |
               |     /            \    /   \    /            \     |
               |    o              o--o-----o--o              o    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    |        y        |     |        z        |    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    o                 o     o                 o    |
               |     \                 \   /                 /     |
               |      \                 \ /                 /      |
               |       \                 o                 /       |
               |        \               / \               /        |
               |         \             /   \             /         |
               |          o-----------o     o-----------o          |
               |                                                   |
               |                                                   |
               o---------------------------------------------------o
                \                                                 /
                  \                                             /
                    \                                         /
                      \                                     /
                        \                                 /
                          \            p , q            /
                            \                         /
                              \                     /
                                \                 /
                                  \             /
                                    \         /
                                      \     /
                                        \ /
                                         o

Figure 30.  Generic Frame of a Logical Transformation

Enter the picture, as we usually do, in the middle of things, with features
like x, y, z that present themselves to be simple enough in their own right
and that form a satisfactory, if a temporary, foundation to provide a basis
for discussion.  In this universe and on these terms we find expression for
various propositions and questions of principal interest to ourselves, as
indicated by the maps p, q : X -> B.  Then we discover that the simple
features {x, y, z} are really more complex than we thought at first,
and it becomes useful to regard them as functions {f, g, h} of other
features {u, v}, that we place in a preface to our original discourse,
or suppose as topics of a preliminary universe of discourse U% = [u, v].
It may happen that these late-blooming but pre-ambling features are found
to lie closer, in a sense that may be our job to determine, to the central
nature of the situation of interest, in which case they earn our regard as
being more fundamental, but these functions and features are only required
to supply a critical stance on the universe of discourse or an alternate
perspective on the nature of things in order to be preserved as useful.

A particular transformation F : [u, v] -> [x, y, z] may be expressed
by a system of equations, as shown below.  Here, F is defined by its
component maps F = <F_1, F_2, F_3> = <f, g, h>, where each component
map in {f, g, h} is a proposition of type B^n -> B^1.

o-------------------------------------------------o
|                                                 |
|         x              =           f<u, v>      |
|                                                 |
|         y              =           g<u, v>      |
|                                                 |
|         z              =           h<u, v>      |
|                                                 |
o-------------------------------------------------o

Regarded as a logical statement, this system of equations expresses a relation
between a collection of freely chosen propositions {f, g, h} in one universe
of discourse and the special collection of simple propositions {x, y, z} on
which are founded another universe of discourse.  Growing familiarity with
a particular transformation of discourse, and the desire to achieve a ready
understanding of its implications, requires that we be able to convert this
information about generals and simples into information about all the main
subtypes of propositions, including the linear and singular propositions.

Diff Log D 2004 • Note 37


| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| objects of your 'conception' to have.  Then,
| your 'conception' of those effects is the
| whole of your 'conception' of the object.
|
| C.S. Peirce, "The Maxim of Pragmatism", CP 5.438

Analytic Expansions:  Operators and Functors

Given the barest idea of a logical transformation,
as suggested by the sketch in Figure 30, and having
conceptualized the universe of discourse, with all of
its points and propositions, as a beginning object of
discussion, we are ready to enter the next phase of
our investigation.

Diff Log D 2004 • Note 38


Operators on Propositions and Transformations

The next step is naturally inclined toward objects of the next higher order,
namely, with operators that take in argument lists of logical transformations
and that give back specified types of logical transformations as their results.
For our present aims, we do not need to consider the most general class of such
operators, nor any one of them for its own sake.  Rather, we are interested in
the special sorts of operators that arise in the study and analysis of logical
transformations.  Figuratively speaking, these operators serve as instruments
for the live tomography (and hopefully not the vivisection) of the forms of
change under view.  Beyond that, they open up ways to implement the changes
of view that we need to grasp all the variations on a transformational theme,
or to appreciate enough of its significant features to "get the drift" of the
change occurring, to form a passing acquaintance or a synthetic comprehension
of its general character and disposition.

The simplest type of operator is one that takes a single transformation
as an argument and returns a single transformation as a result, and most
of the operators that I will explicitly consider here are of this kind.
Figure 31 illustrates the typical situation.

o---------------------------------------o
|                                       |
|                                       |
|      U%           F           X%      |
|         o------------------>o         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|     !W! |                   | !W!     |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         v                   v         |
|         o------------------>o         |
|   !W!U%         !W!F          !W!X%   |
|                                       |
|                                       |
o---------------------------------------o
Figure 31.  Operator Diagram (1)

In this Figure, "!W!" serves as a generic name for an operator, in this case
one that takes a logical transformation F of type (U% -> X%) into a logical
transformation !W!F of the type (!W!U% -> !W!X%).  Thus, the operator !W!
must be viewed as making assignments for both families of objects that we
have previously considered, both for universes of discourse like U% and X%
and for logical transformations like F.

NB.  Strictly speaking, an operator like !W! works between two whole categories
of universes and transformations, which we call the "source" and the "target"
categories of !W!.  Given this setting, !W! specifies for each universe U%
in its source category a definite universe !W!U% in its target category,
and to each transformation F in its source category it assigns a unique
transformation !W!F in its target category.  Naturally, this only works
if !W! takes the source U% and the target X% of the map F over to the
source !W!U% and the target !W!X% of the map !W!F.  With luck or care
enough, we can avoid ever having to put anything like that in words
again, letting diagrams do the work.  In the situations of present
concern we are usually focused on a single transformation F, and
thus we can take it for granted that the assignment of universes
under !W! is defined appropriately at the source and the target
ends of F.  It is not always the case, though, that we need to
use the particular names (like "!W!U%" and "!W!X%") that !W!
assigns by default to its operative image universes.  In most
contexts we will usually have a prior acquaintance with these
universes under other names, and it is only necessary that we
can tell from the information associated with an operator !W!
what universes they are.

In Figure 31 the maps F and !W!F are displayed horizontally, the way that one
normally orients functional arrows in a written text, and !W! rolls the map F
downward into the images that are associated with !W!F.  In Figure 32 the same
information is redrawn so that the maps F and !W!F flow down the page, and !W!
unfurls the map F rightward into domains that are the eminent purview of !W!F.

o---------------------------------------o
|                                       |
|                                       |
|      U%          !W!          !W!U%   |
|         o------------------>o         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|      F  |                   | !W!F    |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         v                   v         |
|         o------------------>o         |
|      X%          !W!          !W!X%   |
|                                       |
|                                       |
o---------------------------------------o
Figure 32.  Operator Diagram (2)

The latter arrangement, as it appears in Figure 32, is more congruent with the
thinking about operators that we shall be doing in the rest of this discussion,
since all logical transformations from here on out will be pictured vertically,
after the fashion of Figure 30.

Diff Log D 2004 • Note 39


Differential Analysis of Propositions and Transformations

| The resultant metaphysical problem now is this:
|
| 'Does the man go round the squirrel or not?'
|
| William James, 'Pragmatism', [Jam, 43]

The approach to the differential analysis of logical propositions and
transformations of discourse that will be pursued here is carried out
in terms of particular operators !W! that act on propositions F or on
transformations F to yield the corresponding operator maps !W!F.  The
operator results then become the subject of a series of further stages
of analysis, which take them apart into their propositional components,
rendering them as a set of purely logical constituents.  After this is
done, all the parts are then re-integrated to reconstruct the original
object in the light of a more complete understanding, at least in ways
that enable one to appreciate certain aspects of it with fresh insight.

NB.  Remark on Strategy.  At this point I run into a set of conceptual
difficulties that force me to make a strategic choice in how I proceed.
Part of the problem can be remedied by extending my discussion of tacit
extensions to the transformational context.  But the troubles that remain
are much more obstinate and lead me to try two different types of solution.
The approach that I develop first makes use of a variant type of extension
operator, the "trope extension", to be defined below.  This method is more
conservative and requires less preparation, but has features which make it
seem unsatisfactory in the long run.  A more radical approach, but one with
a better hope of long term success, makes use of the notion of "contingency
spaces".  These are an even more generous type of extended universe than the
kind I currently use, but are defined subject to certain internal constraints.
The extra work needed to set up this method forces me to put it off to a later
stage.  However, as a compromise, and to prepare the ground for the next pass,
I call attention to the various conceptual difficulties as they arise along
the way and try to give an honest estimate of how well my first approach
deals with them.

I now describe in general terms the particular operators that are
instrumental to this form of analysis.  The main series of operators
all have the form !W! : (U% -> X%) -> (EU% -> EX%).  If we assume that
the source universe U% and the target universe X% have finite dimensions
n and k, respectively, then each operator !W! is encompassed by the same
abstract type:

   !W!  :  ([B^n] -> [B^k])  ->  ([B^n x D^n] -> [B^k x D^k]).

Since the range features of the operator result !W!F : [B^n x D^n] -> [B^k x D^k]
can be sorted out by their ordinary versus their differential qualities and the
component maps can be examined independently, the complete operator !W! can be
separated accordingly into two components, in the form !W! = <!e!, W>.  Given
a fixed context of source and target universes of discourse, !e! is always
the same type of operator, a multiple component elaboration of the tacit
extension operators that were articulated earlier.  In this context !e!
has the shape:

   Concrete type.  !e!  :  ( U%   ->  X%  )  ->  (    EU%     ->  X%  )

   Abstract type.  !e!  :  ([B^n] -> [B^k])  ->  ([B^n x D^n] -> [B^k])

On the other hand, the operator W is specific to
each !W!.  In this context W always has the form:

   Concrete type.   W   :  ( U%   ->  X%  )  ->  (    EU%     ->  dX% )

   Abstract type.   W   :  ([B^n] -> [B^k])  ->  ([B^n x D^n] -> [D^k])

In the types just assigned to !e! and W, and implicitly to their results
!e!F and WF, I have listed the most restrictive ranges defined for them,
rather than the more expansive target spaces that subsume these ranges.
When there is need to recognize both, we may use type indications like
the following:

   !e!F  :  (EU% ->  X% c EX%)  ~=~  ([B^n x D^n] -> [B^k] c [B^k x D^k])

     WF  :  (EU% -> dX% c EX%)  ~=~  ([B^n x D^n] -> [D^k] c [B^k x D^k])

Hopefully, though, a general appreciation of these subsumptions will prevent
us from having to make such declarations more often than absolutely necessary.

In giving names to these operators I am attempting to preserve as much of the
traditional nomenclature and as many of the classical associations as possible.
The chief difficulty in doing this is occasioned by the distinction between the
"sans serif" operators !W! and their "serified" components W, which forces me
to find two distinct but parallel sets of terminology.  Here is the plan that
I have settled on.  First, the component operators W are named by analogy
with the corresponding operators in the classical difference calculus.
Next, the complete operators !W! = <!e!, W> are assigned their titles
according to their roles in a geometric or trigonometric allegory,
if only to ensure that the tangent functor, that belongs to this
family and whose exposition I am still working toward, comes out
fit with its customary name.  Finally, the operator results !W!F
and WF can be fixed in this frame of reference by tethering the
operative adjective for !W! or W to the anchoring epithet "map",
in conformity with an already standard practice.

Diff Log D 2004 • Note 40


The Secant Operator:  $E$

| Mr. Peirce, after pointing out that our beliefs are really
| rules for action, said that, to develop a thought's meaning,
| we need only determine what conduct it is fitted to produce:
| that conduct is for us its sole significance.
|
| William James, 'Pragmatism', [Jam, 46]

Figures 33-i and 33-ii depict two stages in the form of analysis that will be
applied to transformations throughout the remainder of this study.  From now
on our interest is staked on an operator denoted "$E$", which receives the
principal investment of analytic attention, and on the constituent parts
of $E$, which derive their shares of significance as developed by the
analysis.  In the sequel, I refer to $E$ as the "secant operator",
taking it for granted that a context has been chosen that defines
its type.  The secant operator has the component description
$E$ = <!e!, E>, and its active ingredient E is known as the
"enlargement operator".  (Here, I have named E after the
literal ancestor of the shift operator in the calculus
of finite differences, defined so that Ef(x) = f(x+1)
for any suitable function f, though, of course the
logical analogue that we take up here must have
a rather different definition.)

U%          $E$      $E$U%        $E$U%        $E$U%
   o------------------>o============o============o
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
F  |                   | $E$F   =   | $d$^0.F  + | $r$^0.F
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
   v                   v            v            v
   o------------------>o============o============o
X%          $E$      $E$X%        $E$X%        $E$X%

Figure 33-i.  Analytic Diagram (1)

U%          $E$      $E$U%        $E$U%        $E$U%        $E$U%
   o------------------>o============o============o============o
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
F  |                   | $E$F   =   | $d$^0.F  + | $d$^1.F  + | $r$^1.F
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
   v                   v            v            v            v
   o------------------>o============o============o============o
X%          $E$      $E$X%        $E$X%        $E$X%        $E$X%

Figure 33-ii.  Analytic Diagram (2)

In its action on universes $E$ yields the same result as E, a fact that can
be expressed in equational form by writing $E$U% = EU% for any universe U%.
Notice that the extended universes across the top and bottom of the diagram
are indicated to be strictly identical, rather than requiring a corresponding
decomposition for them.  In a certain sense, the functional parts of $E$F are
partitioned into separate contexts that have to be re-integrated again, but the
best image to use is that of making transparent copies of each universe and then
overlapping their functional contents once more at the conclusion of the analysis,
as suggested by the graphic conventions that are used at the top of Figure 30.

Acting on a transformation F from universe U% to universe X%, the operator $E$
determines a transformation $E$F from $E$U% to $E$X%.  The map $E$F forms the
main body of evidence to be investigated in performing a differential analysis
of F.  Because we shall frequently be focusing on small pieces of this map for
considerable lengths of time, and consequently lose sight of the "big picture",
it is critically important to emphasize that the map $E$F is a transformation
that determines a relation from one extended universe into another.  This means
that we should not be satisfied with our understanding of a transformation F
until we can lay out the full "parts diagram" of $E$F along the lines of the
generic frame in Figure 30.

Diff Log D 2004 • Note 41


The Secant Operator:  $E$ (concl.)

If one is working within the confines of propositional calculus,
it is possible to give an elementary definition of $E$F by means
of a system of propositional equations, as will now be described.

Given a transformation:

   F  =  <F_1, ..., F_k>  :  B^n -> B^k

of concrete type:

   F  :  [u_1, ..., u_n]  ->  [x_1, ..., x_k]

the transformation:

   $E$F  =  <F_1, ..., F_k,  EF_1, ..., EF_k>  :  B^n x D^n -> B^k x D^k

of concrete type:

   $E$F  :  [u_1, ..., u_n,  du_1, ..., du_n]  ->  [x_1, ..., x_k,  dx_1, ..., dx_k]

is defined by means of the following system of logical equations:

o--------------------------------------------------------------------------------------o
|                                                                                      |
|  x_1  =  !e!F_1 <u_1, ..., u_n, du_1, ..., du_n>  =  F_1 <u_1, ..., u_n>             |
|                                                                                      |
|  ...                                                                                 |
|                                                                                      |
|  x_k  =  !e!F_k <u_1, ..., u_n, du_1, ..., du_n>  =  F_k <u_1, ..., u_n>             |
|                                                                                      |
|                                                                                      |
| dx_1  =  EF_1 <u_1, ..., u_n, du_1, ..., du_n>  =  F_1 <u_1 + du_1, ..., u_n + du_n> |
|                                                                                      |
|  ...                                                                                 |
|                                                                                      |
| dx_k  =  EF_k <u_1, ..., u_n, du_1, ..., du_n>  =  F_k <u_1 + du_1, ..., u_n + du_n> |
|                                                                                      |
o--------------------------------------------------------------------------------------o

It is important to note that this system of equations can be read as a conjunction
of equational propositions, in effect, as a single proposition in the universe of
discourse that is generated by all of the named variables.  Specifically, this
is the universe of discourse over 2(n + k) variables that is denoted by:

   E[!U! |_| !X!]  =  [u_1, ..., u_n, x_1, ..., x_k, du_1, ..., du_n, dx_1, ..., dx_k].

In this light, it should be clear that the system of equations defining $E$F embodies,
in a higher rank and in a differentially extended version, an analogy with the process
of thematization that was treated earlier for propositions of the type F : B^n -> B.

The entire collection of constraints that is represented in the above system
of equations may be abbreviated by writing $E$F = <!e!F, EF>, for any map F.
This is tantamount to regarding $E$ as a complex operator, $E$ = <!e!, E>,
with a form of application that distributes each component of the operator
to work on each component of the operand:

   $E$F  =  <!e!, E> F  =  <!e!F, EF>  =  <!e!F_1, ..., !e!F_k, EF_1, ..., EF_k>.

Quite a lot of "thematic infrastructure" or interpretive information
is being swept under the rug in the use of such abbreviations.  When
confusion arises about the meaning of such constructions, one always
has recourse to the defining system of equations, in its totality a
purely propositional expression.  This means that the angle brackets,
which were used in this context to build an image of multi-component
transformations, should not be expected to determine a well-defined
product in themselves, but only to serve as reminders of the prior
thematic decisions (choices of variable names, etc.) that have to
be made in order to determine one.  Accordingly, the angle bracket
notation < , > can be regarded as a kind of "thematic frame", an
interpretive storage device that preserves the proper associations
of concrete logical features between the extended universes at the
source and target of $E$F.

The generic notations $d$^0.F, $d$^1.F, ..., $d$^m.F in Figure 33 refer to
the increasing orders of differentials that are extracted in the course of
analyzing F.  When the analysis is halted at a partial stage of development,
notations like $r$^0.F, $r$^1.F, ..., $r$^m.F may be used to summarize the
contributions to $E$F that remain to be analyzed.  The Figure illustrates
a convention that renders the remainder term $r$^m.F, in effect, the sum
of all differentials of order strictly greater than m.

I next discuss the set of operators that figure into this form of analysis,
describing their effects on transformations.  In simplified or specialized
contexts these operators tend to take on a variety of different names and
notations, some of whose number I will introduce along the way.

Diff Log D 2004 • Note 42


The Radius Operator:  $e$

| And the tangible fact at the root of all our thought-distinctions,
| however subtle, is that there is no one of them so fine as to
| consist in anything but a possible difference of practice.
|
| William James, 'Pragmatism', [Jam, 46]

The operator identified as $d$^0 in the analytic diagram (Figure 33) has the
sole purpose of creating a proxy for F in the appropriately extended context.
Construed in terms of its broadest components, $d$^0 is equivalent to the
doubly tacit extension operator <!e!, !e!>, in recognition of which let
us redub it as "$e$".  Pursuing a geometric analogy, we may refer to
$e$ = <!e!, !e!> = $d$^0 as the "radius operator".  The operation
that is intended by all of these forms is defined by the equation:

   $e$F  =  <!e!, !e!> F

         =  <!e!F, !e!F>

         =  <!e!F_1, ..., !e!F_k,  !e!F_1, ..., !e!F_k>,

which is tantamount to the system of equations given below.

o--------------------------------------------------------------------------------o
|                                                                                |
|  x_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
|  x_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
|                                                                                |
| dx_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
| dx_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
o--------------------------------------------------------------------------------o

Diff Log D 2004 • Note 43


The Phantom of the Operators:  !h!

| I was wondering what the reason could be,
| when I myself raised my head and everything
| within me seemed drawn towards the Unseen,
| 'which was playing the most perfect music'!
|
| Gaston Leroux, 'The Phantom of the Opera', [Ler, 81]

I now describe an operator whose persistent but elusive action behind the scenes,
whose slightly twisted and ambivalent character, and whose fugitive disposition,
caught somewhere in flight between the arrantly negative and the positive but
errant intent, has cost me some painstaking trouble to detect.  In the end
I shall place it among the other extensions and projections, as a shade
among shadows, of muted tones and motley hue, that adumbrates its own
thematic frame and paradoxically lights the way toward a whole new
spectrum of values.

Given a transformation F : [u_1, ..., u_n] -> [x_1, ..., x_k], we often
need to make a separate treatment of a related family of transformations
of the form F* : [u_1, ..., u_n,  du_1, ..., du_n] -> [dx_1, ..., dx_k].
The operator !h! (Greek eta) is introduced to deal with the simplest one
of these maps:

   !h!F  :  [u_1, ..., u_n,  du_1, ..., du_n]  ->  [dx_1, ..., dx_k]

which is defined by the equations:

o--------------------------------------------------------------------------------o
|                                                                                |
| dx_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
| dx_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
o--------------------------------------------------------------------------------o

In effect, the operator !h! is nothing but the stand-alone version of
a procedure that is otherwise invoked subordinate to the work of the
radius operator $e$.  Operating independently, !h! achieves precisely
the same results that the second !e! in <!e!, !e!> accomplishes by
working within the context of its adjuvant thematic frame, "< , >".
From this point on, because the use of !e! and !h! in this setting
combines the aims of both the tacit and the thematic extensions,
and because !h! reflects in regard to !e! little more than the
application of a differential twist, a mere turn of phrase,
I refer to !h! as the "trope extension" operator.

Diff Log D 2004 • Note 44


The Chord Operator:  $D$

| What difference would it practically make to any one if this
| notion rather than that notion were true?  If no practical
| difference whatever can be traced, then the alternatives
| mean practically the same thing, and all dispute is idle.
|
| William James, 'Pragmatism', [Jam, 45]

Next I discuss an operator that is always immanent in this form of
analysis, and remains implicitly present in the entire proceeding.
It may appear once as a record:  a relic or revenant that reprises
the reminders of an earlier stage of development.  Or it may appear
always as a resource:  a reserve or redoubt that caches in advance
an echo of what remains to be played out, cleared up, and requited
in full at a future stage.  And all of this remains true whether or
not we recall the key at any time, and whether or not the subtending
theme is recited explicitly at any stage of play.

This is the operator that is referred to as $r$^0 in the initial stage
of analysis (Figure 33-i), and that is expanded as $d$^1 + $r$^1 in the
subsequent step (Figure 33-ii).  In congruence, but not quite harmony,
with my allusions of analogy that are not quite geometry, I call this
the "chord operator" and denote it $D$.  In the more casual terms that
are here introduced, $D$ is defined as the remainder of $E$ and $e$,
and it assigns a due measure to each undertone of accord or discord
that is struck between the note of enterprise $E$ and the bar of
exigency $e$.

The tension between these counterposed notions, in balance transient but
regular in stridence, may be refracted along familiar lines, though never
by any such fraction resolved.  In this style we may write $D$ = <!e!, D>,
calling D the "difference operator" and noting that it plays a role in this
realm of mutable and diverse discourse that is analogous to the part taken
by the discrete difference operator in the ordinary difference calculus.
Finally, we should note that the chord $D$ is not one that need be lost
at any stage of development.  At the m^th stage of play it can always
be reconstituted in the following form:

o-------------------------------------------------o
|                                                 |
| $D$   =   $E$ - $e$                             |
|                                                 |
|       =   $r$^0                                 |
|                                                 |
|       =   $d$^1  +  $r$^1                       |
|                                                 |
|       =   Sum_(i = 1 to m) $d$^i  +  $r$^m      |
|                                                 |
o-------------------------------------------------o

Diff Log D 2004 • Note 45


The Tangent Operator:  $T$

| They take part in scenes of whose significance they have no inkling.
| They are merely tangent to curves of history the beginnings and ends
| and forms of which pass wholly beyond their ken.  So we are tangent
| to the wider life of things.
|
| William James, 'Pragmatism', [Jam, 300]

The operator tagged as $d$^1 in the analytic diagram (Figure 33) is called
the "tangent operator", and is usually denoted in this text as $d$ or $T$.
Because it has the properties required to qualify as a functor, namely,
preserving the identity element of the composition operation and the
articulated form of every composure among transformations, it also
earns the title of a "tangent functor".  According to the custom
adopted here, we dissect it as $T$ = $d$ = <!e!, d>, where d is
the operator that yields the first order differential dF when
applied to a transformation F, and whose name is legion.

Figure 34 illustrates a stage of analysis where we ignore everything but the
tangent functor $T$, and attend to it chiefly as it bears on the first order
differential dF in the analytic expansion of F.  In this situation, we often
refer to the extended universes EU% and EX% under the equivalent designations
$T$U% and $T$X%, respectively.  The purpose of the tangent functor $T$ is to
extract the tangent map $T$F at each point of U%, and the tangent map $T$F =
<!e!, d> F tells us not only what the transformation F is doing at each point
of the universe U% but also what F is doing to states in the neighborhood of
that point, approximately, linearly, and relatively speaking.

U%          $T$      $T$U%        $T$U%
   o------------------>o============o
   |                   |            |
   |                   |            |
   |                   |            |
   |                   |            |
F  |                   | $T$F   =   | <!e!, d> F
   |                   |            |
   |                   |            |
   |                   |            |
   v                   v            v
   o------------------>o============o
X%          $T$      $T$X%        $T$X%

Figure 34.  Tangent Functor Diagram

NB.  There is one aspect of the preceding construction that remains especially
problematic.  Why did we define the operators W in {!h!, E, D, d, r} so that the
ranges of their resulting maps all fall within the realms of differential quality,
even fabricating a variant of the tacit extension operator to have that character?
Clearly, not all of the operator maps WF have equally good reasons for placing their
values in differential stocks.  The only explanation I can devise at present is that,
without doing this, I cannot justify the comparison and combination of their values
in the various analytic steps.  By default, only those values in the same functional
component can be brought into algebraic modes of interaction.  Up till now, the only
mechanism provided for their broader association has been a purely logical one, their
common placement in a target universe of discourse, but the task of converting this
logical circumstance into algebraic forms of application has not yet been taken up.

Diff Log D 2004 • Note 46


Transformations of Type B^2 -> B^1

To study the effects of these analytic operators in the simplest
possible situation, let us revert to a still more primitive case.
Consider the singular proposition J<u, v> = uv, regarded either
as the functional product of the maps u and v or as the logical
conjunction of the features u and v, a map whose fiber of truth
J^(-1)(1) picks out the single cell of that logical description
in the universe of discourse U%.  Thus J, or uv, may be treated
as a pseudonym for the point whose coordinates are <1, 1> in U%.

Diff Log D 2004 • Note 47


Analytic Expansion of Conjunction

| In her sufferings she read a great deal and discovered
| that she had lost something, the possession of which
| she had previously not been much aware of:  a soul.
|
| What is that?  It is easily defined negatively:
| it is simply what curls up and hides when there
| is any mention of algebraic series.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 118]

Figure 35 pictures the form of conjunction J : B^2 -> B as a transformation
from the 2-dimensional universe [u, v] to the 1-dimensional universe [x].
This is a subtle but significant change of viewpoint on the proposition,
attaching an arbitrary but concrete quality to its functional value.
Using the language introduced earlier, we can express this change by
saying that the proposition J : <|u, v|> -> B is being recast into
the thematized role of a transformation J : [u, v] -> [x], where
the new variable "x" takes the part of a thematic variable ¢(J).

o---------------------------------------o
|                                       |
|                                       |
|       o---------o   o---------o       |
|      /           \ /           \      |
|     /             o             \     |
|    /             /`\             \    |
|   /             /```\             \   |
|  o             o`````o             o  |
|  |             |`````|             |  |
|  |      u      |`````|      v      |  |
|  |             |`````|             |  |
|  o             o`````o             o  |
|   \             \```/             /   |
|    \             \`/             /    |
|     \             o             /     |
|      \           / \           /      |
|       o---------o   o---------o       |
|                                       |
|                                       |
o---------------------------------------o
 \                                     /
   \                                 /
     \                             /
       \            J            /
         \                     /
           \                 /
             \             /
o--------------\---------/--------------o
|                \     /                |
|                  \ /                  |
|            o------@------o            |
|           /```````````````\           |
|          /`````````````````\          |
|         /```````````````````\         |
|        /`````````````````````\        |
|       o```````````````````````o       |
|       |```````````````````````|       |
|       |`````````` x ``````````|       |
|       |```````````````````````|       |
|       o```````````````````````o       |
|        \`````````````````````/        |
|         \```````````````````/         |
|          \`````````````````/          |
|           \```````````````/           |
|            o-------------o            |
|                                       |
|                                       |
o---------------------------------------o
Figure 35.  Conjunction as Transformation

Diff Log D 2004 • Note 48


Tacit Extension of Conjunction

| I teach straying from me, yet who can stray from me?
| I follow you whoever you are from the present hour;
| My words itch at your ears till you understand them.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 83]

Earlier I defined the tacit extension operators !e! : X% -> Y%
as maps embedding each proposition of a given universe X% in a
more generously given universe Y% containing X%.  Of immediate
interest are the tacit extensions !e! : U% -> EU%, that locate
each proposition of U% in the enlarged context of EU%.  In its
application to the propositional conjunction J = u v in [u, v],
the tacit extension operator !e! produces the proposition !e!J
in EU% = [u, v, du, dv].  The extended proposition !e!J may be
computed according to the scheme in Table 36, in effect, doing
nothing more than conjoining a tautology of [du, dv] to J in U%.

Table 36.  Computation of !e!J
o---------------------------------------------------------------------o
|                                                                     |
| !e!J  =  J<u, v>                                                    |
|                                                                     |
|       =  u v                                                        |
|                                                                     |
|       =  u v (du)(dv)  +  u v (du) dv  +  u v du (dv)  +  u v du dv |
|                                                                     |
o---------------------------------------------------------------------o
|                                                                     |
| !e!J  =  u v (du)(dv)  +                                            |
|          u v (du) dv   +                                            |
|          u v  du (dv)  +                                            |
|          u v  du  dv                                                |
|                                                                     |
o---------------------------------------------------------------------o

The lower portion of the Table contains the dispositional features of !e!J
arranged in such a way that the variety of ordinary features spreads across
the rows and the variety of differential features runs through the columns.
This organization serves to facilitate pattern matching in the remainder
of our computations.  Again, the tacit extension is usually so trivial
a concern that we do not always bother to make an explicit note of it,
taking it for granted that any function F that is being employed in
a differential context is equivalent to !e!F, for a suitable !e!.

Diff Log D 2004 • Note 49


Tacit Extension of Conjunction (cont.)

Figures 37-a through 37-d present several pictures of the proposition J and
its tacit extension !e!J.  Notice in these Figures how !e!J in EU% visibly
extends J in U%, by annexing to the indicated cells of J all of the arcs
that exit from or flow out of them.  In effect, this extension attaches
to these cells all of the dispositions that spring from them, in other
words, it attributes to these cells all of the conceivable changes
that are their issue.

o---------------------------------------o
|                                       |
|                   o                   |
|                  /%\                  |
|                 /%%%\                 |
|                /%%%%%\                |
|               o%%%%%%%o               |
|              /%\%%%%%/%\              |
|             /%%%\%%%/%%%\             |
|            /%%%%%\%/%%%%%\            |
|           o%%%%%%%o%%%%%%%o           |
|          / \%%%%%/%\%%%%%/ \          |
|         /   \%%%/%%%\%%%/   \         |
|        /     \%/%%%%%\%/     \        |
|       o       o%%%%%%%o       o       |
|      / \     / \%%%%%/ \     / \      |
|     /   \   /   \%%%/   \   /   \     |
|    /     \ /     \%/     \ /     \    |
|   o       o       o       o       o   |
|   |\     / \     / \     / \     /|   |
|   | \   /   \   /   \   /   \   / |   |
|   |  \ /     \ /     \ /     \ /  |   |
|   |   o       o       o       o   |   |
|   |   |\     / \     / \     /|   |   |
|   |   | \   /   \   /   \   / |   |   |
|   | u |  \ /     \ /     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 37-a.  Tacit Extension of J (Areal)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         / \         \   |
                                                  |  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                 /|  o         o   o         o  |
                                                / |   \         \ /         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /       \ /       \     |
|          o---------o   o---------o          |   |    /         o         \    |
|         /           \ /           \         |   |   /         / \         \   |
|        /             o             \        |   |  o         o   o         o  |
|       /             /`\      @------\-----------@  |   du    |   |    dv   |  |
|      /             /```\             \      |   |  o         o   o         o  |
|     /             /`````\             \     |   |   \         \ /         /   |
|    /             /```````\             \    |   |    \         o         /    |
|   o             o`````````o             o   |   |     \       / \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /       \     |
|    \             \```````/   \         /    |   |    /         o         \    |
|     \             \`````/     \       /     |   |   /         / \         \   |
|      \             \```/       \     /      |   |  o         o   o         o  |
|       \      @------\-/---------\---------------@  |   du    |   |    dv   |  |
|        \             o           \ /        |   |  o         o   o         o  |
|         \           / \           /         |   |   \         \ /         /   |
|          o---------o   o---------o \        |   |    \         o         /    |
|                                     \       |   |     \       / \       /     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |`````````````````````````````|
                                             \    |````` o-----o```o-----o``````|
                                              \   |`````/```````\`/```````\`````|
                                               \  |````/`````````o`````````\````|
                                                \ |```/`````````/`\`````````\```|
                                                 \|``o`````````o```o`````````o``|
                                                  @``|```du````|```|````dv```|``|
                                                  |``o`````````o```o`````````o``|
                                                  |```\`````````\`/`````````/```|
                                                  |````\`````````o`````````/````|
                                                  |`````\```````/`\```````/`````|
                                                  |``````o-----o```o-----o``````|
                                                  |`````````````````````````````|
                                                  o-----------------------------o
Figure 37-b.  Tacit Extension of J (Bundle)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (du).(dv) o                       o    |
|    |                       |   -->--   |                       |    |
|    |                       |   \   /   |                       |    |
|    |              dv .(du) |    \ /    | du .(dv)              |    |
|    |      u      <---------------@--------------->      v      |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  V                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 37-c.  Tacit Extension of J (Compact)

o-----------------------------------------------------------o
|                                                           |
|                         (du).(dv)                         |
|                          --->---                          |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                           u @ v                           |
|                            /|\                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|               (du) dv /     |     \ du (dv)               |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  v          |          v                  |
|                 @           |           @                 |
|               u (v)         |         (u) v               |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du . dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             v                             |
|                             @                             |
|                                                           |
|                          (u).(v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 37-d.  Tacit Extension of J (Digraph)

Diff Log D 2004 • Note 50


Tacit Extension of Conjunction (concl.)

The computational scheme that was shown in Table 36 treated J
as a proposition in U% and formed !e!J as a proposition in EU%.
When J is regarded as a mapping J : U% -> X% then !e!J must be
obtained as a mapping !e!J : EU% -> X%.  By default, the tacit
extension of the map J : [u, v] -> [x] is naturally taken to be
a particular map, of the following form:

   !e!J  :  [u, v, du, dv]  ->  [x]  c  [x, dx]

This is the map that looks like J when painted in the frame of the
extended source universe and that takes the same thematic variable
in the extended target universe as the one that J already employs.

But the choice of a particular thematic variable, for example "x" for ¢(J),
is a shade more arbitrary than the initial choice of variable names {u, v}.
This means that the map I am calling the "trope extension", specifically:

   !h!J  :  [u, v, du, dv]  ->  [dx]  c  [x, dx]

since it looks just the same as !e!J in the way that its
fibers paint the source domain, belongs just as fully to
the family of tacit extensions, generically considered.

These considerations have the practical consequence that all of our
computations and illustrations of !e!J perform the double duty of
capturing an image of !h!J as well.  In other words, we are saved
the work of carrying out calculations and drawing figures for the
trope extension !h!J, because the exercise would be identical to
the work already done for !e!J.  Since the computations given for
!e!J are expressed solely in terms of the variables {u, v, du, dv},
these variables work equally well for finding !h!J.  Furthermore,
since each of the above Figures shows only how the level sets of
!e!J partition the extended source universe EU% = [u, v, du, dv],
all of them serve equally well as portraits of !h!J.

Diff Log D 2004 • Note 51


Enlargement Map of Conjunction

| No one could have established the existence of any details
| that might not just as well have existed in earlier times
| too;  but all the relations between things had shifted
| slightly.  Ideas that had once been of lean account
| grew fat.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 62]

The enlargement map EJ is computed from the proposition J by making
a particular class of formal substitutions for its variables, in this
case "u + du" for "u" and "v + dv" for "v", and subsequently expanding
the result in whatever way happens to be convenient for the end in view.

Table 38 shows a typical scheme of computation, following
a systematic method of exploiting boolean expansions over
selected variables, and ultimately developing EJ over the
cells of [u, v].  The critical step of this procedure uses
the facts that (0, x) = 0 + x = x and (1, x) = 1 + x = (x)
for any boolean variable x.

Table 38.  Computation of EJ (Method 1)
o-------------------------------------------------------------------------------o
|                                                                               |
| EJ  =  J<u + du, v + dv>                                                      |
|                                                                               |
|     =  (u, du)(v, dv)                                                         |
|                                                                               |
|     =   u  v  J<1 + du, 1 + dv>  +                                            |
|                                                                               |
|         u (v) J<1 + du, 0 + dv>  +                                            |
|                                                                               |
|        (u) v  J<0 + du, 1 + dv>  +                                            |
|                                                                               |
|        (u)(v) J<0 + du, 0 + dv>                                               |
|                                                                               |
|     =   u  v  J<(du), (dv)>  +                                                |
|                                                                               |
|         u (v) J<(du),  dv >  +                                                |
|                                                                               |
|        (u) v  J< du , (dv)>  +                                                |
|                                                                               |
|        (u)(v) J< du ,  dv >                                                   |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| EJ  =   u  v (du)(dv)                                                         |
|                        +   u (v)(du) dv                                       |
|                                           +  (u) v  du (dv)                   |
|                                                              +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 39 exhibits another method that happens to work quickly in
this particular case, using distributive laws to multiply things
out in an algebraic manner, arranging the notations of feature
and fluxion according to a scale of simple character and degree.
Proceeding this way leads through an intermediate step which,
in chiming the changes of ordinary calculus, should take on
a familiar ring.  Consequential properties of exclusive
disjunction then carry us on to the concluding line.

Table 39.  Computation of EJ (Method 2)
o-------------------------------------------------------------------------------o
|                                                                               |
| EJ  =  <u + du> <v + dv>                                                      |
|                                                                               |
|     =       u v        +       u dv       +       v du       +      du dv     |
|                                                                               |
| EJ  =   u  v (du)(dv)  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Diff Log D 2004 • Note 52


Enlargement Map of Conjunction (concl.)

Figures 40-a through 40-d present several views of the enlarged proposition EJ.

o---------------------------------------o
|                                       |
|                   o                   |
|                  /%\                  |
|                 /%%%\                 |
|                /%%%%%\                |
|               o%%%%%%%o               |
|              / \%%%%%/ \              |
|             /   \%%%/   \             |
|            /     \%/     \            |
|           o       o       o           |
|          /%\     / \     /%\          |
|         /%%%\   /   \   /%%%\         |
|        /%%%%%\ /     \ /%%%%%\        |
|       o%%%%%%%o       o%%%%%%%o       |
|      / \%%%%%/ \     / \%%%%%/ \      |
|     /   \%%%/   \   /   \%%%/   \     |
|    /     \%/     \ /     \%/     \    |
|   o       o       o       o       o   |
|   |\     / \     /%\     / \     /|   |
|   | \   /   \   /%%%\   /   \   / |   |
|   |  \ /     \ /%%%%%\ /     \ /  |   |
|   |   o       o%%%%%%%o       o   |   |
|   |   |\     / \%%%%%/ \     /|   |   |
|   |   | \   /   \%%%/   \   / |   |   |
|   | u |  \ /     \%/     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 40-a.  Enlargement of J (Areal)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         /%\         \   |
                                                  |  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                 /|  o         o%%%o         o  |
                                                / |   \         \%/         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /%%%%%%%\ /       \     |
|          o---------o   o---------o          |   |    /%%%%%%%%%o         \    |
|         /           \ /           \         |   |   /%%%%%%%%%/ \         \   |
|        /             o             \        |   |  o%%%%%%%%%o   o         o  |
|       /             /`\      @------\-----------@  |%% du %%%|   |    dv   |  |
|      /             /```\             \      |   |  o%%%%%%%%%o   o         o  |
|     /             /`````\             \     |   |   \%%%%%%%%%\ /         /   |
|    /             /```````\             \    |   |    \%%%%%%%%%o         /    |
|   o             o`````````o             o   |   |     \%%%%%%%/ \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /%%%%%%%\     |
|    \             \```````/   \         /    |   |    /         o%%%%%%%%%\    |
|     \             \`````/     \       /     |   |   /         / \%%%%%%%%%\   |
|      \             \```/       \     /      |   |  o         o   o%%%%%%%%%o  |
|       \      @------\-/---------\---------------@  |   du    |   |%%% dv %%|  |
|        \             o           \ /        |   |  o         o   o%%%%%%%%%o  |
|         \           / \           /         |   |   \         \ /%%%%%%%%%/   |
|          o---------o   o---------o \        |   |    \         o%%%%%%%%%/    |
|                                     \       |   |     \       / \%%%%%%%/     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
                                             \    |%%%%%%o-----o%%%o-----o%%%%%%|
                                              \   |%%%%%/       \%/       \%%%%%|
                                               \  |%%%%/         o         \%%%%|
                                                \ |%%%/         / \         \%%%|
                                                 \|%%o         o   o         o%%|
                                                  @%%|   du    |   |    dv   |%%|
                                                  |%%o         o   o         o%%|
                                                  |%%%\         \ /         /%%%|
                                                  |%%%%\         o         /%%%%|
                                                  |%%%%%\       /%\       /%%%%%|
                                                  |%%%%%%o-----o%%%o-----o%%%%%%|
                                                  |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
                                                  o-----------------------------o
Figure 40-b.  Enlargement of J (Bundle)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (du).(dv) o                       o    |
|    |                       |   -->--   |                       |    |
|    |                       |   \   /   |                       |    |
|    |              dv .(du) |    \ /    | du .(dv)              |    |
|    |     u     o---------------->@<----------------o     v     |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 40-c.  Enlargement of J (Compact)

o-----------------------------------------------------------o
|                                                           |
|                         (du).(dv)                         |
|                          --->---                          |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                           u @ v                           |
|                            ^^^                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|               (du) dv /     |     \ du (dv)               |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  /          |          \                  |
|                 @           |           @                 |
|               u (v)         |         (u) v               |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du . dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             @                             |
|                                                           |
|                          (u).(v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 40-d.  Enlargement of J (Digraph)

An intuitive reading of the proposition EJ becomes available at this point,
and may be useful.  Recall that propositions in the extended universe EU%
express the "dispositions" of system and the constraints that are placed
on them.  In other words, a differential proposition in EU% can be read
as referring to various changes that a system might undergo in and from
its various states.  In particular, we can understand EJ as a statement
that tells us what changes need to be made with regard to each state in
the universe of discourse in order to reach the "truth" of J, that is,
the region of the universe where J is true.  This interpretation is
visibly clear in the Figures above, and appeals to the imagination
in a satisfying way, but it has the added benefit of giving fresh
meaning to the original name of the shift operator E.  Namely, EJ
can be read as a proposition that "enlarges" on the meaning of J,
in the sense of explaining its practical bearings and clarifying
what it means in terms of the available options for differential
action and the consequential effects that result from each choice.

Treated this way, the enlargement EJ has strong ties to the normal use of J,
no matter whether it is understood as a proposition or a function, namely,
to act as a figurative device for indicating the models of J, in effect,
pointing to the interpretive elements in its fiber of truth J^(-1)(1).
It is this kind of "use" that is often compared with the "mention" of
a proposition, and thereby hangs a tale.

Diff Log D 2004 • Note 53


Digression:  Reflection on Use and Mention

| Reflection is turning a topic over in various aspects and in various lights
| so that nothing significant about it shall be overlooked -- almost as one
| might turn a stone over to see what its hidden side is like or what is
| covered by it.
|
| John Dewey, 'How We Think', [Dew, 57]

The contrast drawn in logic between the "use" and the "mention" of a proposition
corresponds to the difference that we observe in functional terms between using
"J" to indicatet the region J^(-1)(1) and using "J" to indicate the function J.
You may think that one of these uses ought to be proscribed, and logicians
are quick to prescribe against their confusion.  But there seems to be
no likelihood in practice that their interactions can be avoided.
If the name "J" is used as a sign of the function J, and if the
function J has its use in signifying something else, as would
constantly be the case when some future theory of signs has
given a functional meaning to every sign whatsoever, then
is not "J" by transitivity a sign of the thing itself?
There are, of course, two answers to this question.
Not every act of signifying or referring need be
transitive.  Not every warrant or guarantee or
certificate is automatically transferable,
indeed, not many.  Not every feature of
a feature is a feature of the featuree.
Otherwise we have an inference like:
If a buffalo is white, and white is
a color, then a buffalo is a color.
But a buffalo is not, only buff is.

The logical or pragmatic distinction between use and mention
is cogent and necessary, and so is the analogous functional
distinction between determining a value and determining what
determines that value, but so are the normal techniques that
we use to make these distinctions apply flexibly in practice.
The way that the hue and cry about use and mention is raised
in logical discussions, you might be led to think that this
single dimension of choices embraces the only kinds of use
worth mentioning and the only kinds of mention worth using.
It will constitute the expeditionary and taxonomic tasks of
that future theory of signs to explore and to classify the
many other constellations and dimensions of use and mention
that are yet to be opened up by the generative potential of
full-fledged sign relations.

| The well-known capacity that thoughts have -- as doctors have discovered --
| for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly
| entangled conflict that arise out of gloomy regions of the self probably rests
| on nothing other than their social and worldly nature, which links the individual
| being with other people and things;  but unfortunately what gives them their power
| of healing seems to be the same as what diminishes the quality of personal experience
| in them.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 130]

Diff Log D 2004 • Note 54


Difference Map of Conjunction

| "It doesn't matter what one does", the Man Without Qualities said to himself,
| shrugging his shoulders.  "In a tangle of forces like this it doesn't make a
| scrap of difference."  He turned away like a man who has learned renunciation,
| almost indeed like a sick man who shrinks from any intensity of contact.  And
| then, striding through his adjacent dressing-room, he passed a punching-ball
| that hung there;  he gave it a blow far swifter and harder than is usual in
| moods of resignation or states of weakness.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 8]

With the tacit extension map !e!J and the enlargement map EJ well in place,
the difference map DJ can be computed along the lines displayed in Table 41,
ending up, in this instance, with an expansion of DJ over the cells of [u, v].

Table 41.  Computation of DJ (Method 1)
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  EJ                 +  !e!J                                             |
|                                                                               |
|     =  J<u + du, v + dv>  +  J<u, v>                                          |
|                                                                               |
|     =  (u, du)(v, dv)     +  u v                                              |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =        0                                                                |
|                                                                               |
|     +   u  v (du) dv   +   u (v)(du) dv                                       |
|                                                                               |
|     +   u  v  du (dv)                     +  (u) v  du (dv)                   |
|                                                                               |
|     +   u  v  du  dv                                         +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  u v ((du)(dv))  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Alternatively, the difference map DJ can be expanded over the cells of [du, dv]
to arrive at the formulation shown in Table 42.  The same development would be
obtained from the previous Table by collecting terms in an alternate manner,
along the rows rather than the columns of the middle portion of the Table.

Table 42.  Computation of DJ (Method 2)
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  !e!J            +  EJ                                                  |
|                                                                               |
|     =  J<u, v>         +  J<u + du, v + dv>                                   |
|                                                                               |
|     =  u v             +  (u, du)(v, dv)                                      |
|                                                                               |
|     =  0               +  u dv            +  v du            +  du dv         |
|                                                                               |
|     =  0               +  u (du) dv       +  v du (dv)       + ((u, v)) du dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Even more simply, the same result is reached by matching up the propositional
coefficients of !e!J and EJ along the cells of [du, dv] and adding the pairs
under boolean sums (that is, "mod 2", where 1 + 1 = 0), as shown in Table 43.

Table 43.  Computation of DJ (Method 3)
o-------------------------------------------------------------------------------o
|                                                                               |
|  DJ  =  !e!J           +   EJ                                                 |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| !e!J =  u  v (du)(dv)  +   u  v (du) dv   +   u  v  du (dv)  +   u  v  du  dv |
|                                                                               |
|  EJ  =  u  v (du)(dv)  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
|  DJ  =   0 . (du)(dv)  +    u . (du) dv   +     v . du (dv)  + ((u, v)) du dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Diff Log D 2004 • Note 55


Difference Map of Conjunction (cont.)

The difference map DJ can also be given a "dispositional" interpretation.
First, recall that !e!J exhibits the dispositions to change from anywhere
in J to anywhere at all, and EJ enumerates the dispositions to change from
anywhere at all to anywhere in J.  Next, observe that each of these classes
of dispositions may be divided in accordance with the case of J versus (J)
that applies to their points of departure and destination, as shown below.
Then, since the dispositions corresponding to !e!J and EJ have in common
the dispositions to preserve J, their symmetric difference (!e!J, EJ) is
made up of all the remaining dispositions, which are in fact disposed to
cross the boundary of J in one direction or the other.  In other words,
we may conclude that DJ expresses the collective disposition to make
a definite change with respect to J, no matter what value it holds
in the current state of affairs.

o-------------------------------------------------------------------------------o
|                                                                               |
| !e!J  =  {Dispositions from  J  to  J }  +  {Dispositions from  J  to (J)}    |
|                                                                               |
|  EJ   =  {Dispositions from  J  to  J }  +  {Dispositions from (J) to  J }    |
|                                                                               |
|  DJ   =  (!e!J, EJ)                                                           |
|                                                                               |
|  DJ   =  {Dispositions from  J  to (J)}  +  {Dispositions from (J) to  J }    |
|                                                                               |
o-------------------------------------------------------------------------------o

Diff Log D 2004 • Note 56


Difference Map of Conjunction (concl.)

Figures 44-a through 44-d illustrate the difference proposition DJ.

o---------------------------------------o
|                                       |
|                   o                   |
|                  / \                  |
|                 /   \                 |
|                /     \                |
|               o       o               |
|              /%\     /%\              |
|             /%%%\   /%%%\             |
|            /%%%%%\ /%%%%%\            |
|           o%%%%%%%o%%%%%%%o           |
|          /%\%%%%%/%\%%%%%/%\          |
|         /%%%\%%%/%%%\%%%/%%%\         |
|        /%%%%%\%/%%%%%\%/%%%%%\        |
|       o%%%%%%%o%%%%%%%o%%%%%%%o       |
|      / \%%%%%/ \%%%%%/ \%%%%%/ \      |
|     /   \%%%/   \%%%/   \%%%/   \     |
|    /     \%/     \%/     \%/     \    |
|   o       o       o       o       o   |
|   |\     / \     /%\     / \     /|   |
|   | \   /   \   /%%%\   /   \   / |   |
|   |  \ /     \ /%%%%%\ /     \ /  |   |
|   |   o       o%%%%%%%o       o   |   |
|   |   |\     / \%%%%%/ \     /|   |   |
|   |   | \   /   \%%%/   \   / |   |   |
|   | u |  \ /     \%/     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 44-a.  Difference Map of J (Areal)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         /%\         \   |
                                                  |  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                 /|  o         o%%%o         o  |
                                                / |   \         \%/         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /%%%%%%%\ /       \     |
|          o---------o   o---------o          |   |    /%%%%%%%%%o         \    |
|         /           \ /           \         |   |   /%%%%%%%%%/ \         \   |
|        /             o             \        |   |  o%%%%%%%%%o   o         o  |
|       /             /`\      @------\-----------@  |%% du %%%|   |    dv   |  |
|      /             /```\             \      |   |  o%%%%%%%%%o   o         o  |
|     /             /`````\             \     |   |   \%%%%%%%%%\ /         /   |
|    /             /```````\             \    |   |    \%%%%%%%%%o         /    |
|   o             o`````````o             o   |   |     \%%%%%%%/ \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /%%%%%%%\     |
|    \             \```````/   \         /    |   |    /         o%%%%%%%%%\    |
|     \             \`````/     \       /     |   |   /         / \%%%%%%%%%\   |
|      \             \```/       \     /      |   |  o         o   o%%%%%%%%%o  |
|       \      @------\-/---------\---------------@  |   du    |   |%%% dv %%|  |
|        \             o           \ /        |   |  o         o   o%%%%%%%%%o  |
|         \           / \           /         |   |   \         \ /%%%%%%%%%/   |
|          o---------o   o---------o \        |   |    \         o%%%%%%%%%/    |
|                                     \       |   |     \       / \%%%%%%%/     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |                             |
                                             \    |      o-----o   o-----o      |
                                              \   |     /%%%%%%%\ /%%%%%%%\     |
                                               \  |    /%%%%%%%%%o%%%%%%%%%\    |
                                                \ |   /%%%%%%%%%/%\%%%%%%%%%\   |
                                                 \|  o%%%%%%%%%o%%%o%%%%%%%%%o  |
                                                  @  |%% du %%%|%%%|%%% dv %%|  |
                                                  |  o%%%%%%%%%o%%%o%%%%%%%%%o  |
                                                  |   \%%%%%%%%%\%/%%%%%%%%%/   |
                                                  |    \%%%%%%%%%o%%%%%%%%%/    |
                                                  |     \%%%%%%%/ \%%%%%%%/     |
                                                  |      o-----o   o-----o      |
                                                  |                             |
                                                  o-----------------------------o
Figure 44-b.  Difference Map of J (Bundle)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |              dv .(du) |           | du .(dv)              |    |
|    |     u     @<--------------->@<--------------->@     v     |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  v                                  |
|                                  @                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 44-c.  Difference Map of J (Compact)

o-----------------------------------------------------------o
|                                                           |
|                            u v                            |
|                                                           |
|                             @                             |
|                            ^^^                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|               (du) dv /     |     \ du (dv)               |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  v          |          v                  |
|                 @           |           @                 |
|               u (v)         |         (u) v               |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du | dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             v                             |
|                             @                             |
|                                                           |
|                          (u) (v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 44-d.  Difference Map of J (Digraph)

Diff Log D 2004 • Note 57


Differential of Conjunction

| By deploying discourse throughout a calendar,
| and by giving a date to each of its elements,
| one does not obtain a definitive hierarchy of
| precessions and originalities;  this hierarchy
| is never more than relative to the systems of
| discourse that it sets out to evaluate.
|
| Michel Foucault, 'The Archaeology of Knowledge', [Fou, 143]

Finally, at long last, the differential proposition dJ can be gleaned from the
difference proposition DJ by ranging over the cells of [u, v] and picking out
the linear proposition of [du, dv] that is "closest" to the portion of DJ that
touches on each point.  The idea of distance that would give this definition
unequivocal sense has been referred to in cautionary quotes, the kind we use
to distance ourselves from taking a final position.  There are obvious notions
of approximation that suggest themselves, but finding one that can be justified
as ultimately correct is not as straightforward as it seems.

| He had drifted into the very heart of the world.
| From him to the distant beloved was as far as to
| the next tree.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 144]

Let us venture a guess about where these developments might be heading.
From the present vantage point, it appears that the ultimate answer to
the quandary of distances and the question of a fitting measure may be
that, rather than having the constitution of an analytic series depend
on our familiar notions of approach, proximity, and approximation, it
will be found preferable, and perhaps unavoidable, to turn the tables
and let the orders of approximation be defined in terms of our favored
and operative notions of formal analysis.  Only the aftermath of this
conversion, if it does converge, could be hoped to prove whether this
hortatory form of analysis and the cohort idea of an analytic form --
the limitary concept of a self-corrective process and the coefficient
concept of a completable product -- are truly (in practical reality)
the more inceptive and persistent of principles and really (for all
practical purposes) the more effective and regulative of ideas.

Awaiting that determination, I proceed with what seems like
the obvious course, and compute dJ according to the pattern
in Table 45.

Table 45.  Computation of dJ
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  u v ((du)(dv))  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du dv  |
|                                                                               |
| =>                                                                            |
|                                                                               |
| dj  =  u v  (du, dv)   +   u (v) dv       +  (u) v  du       +  (u)(v) . 0    |
|                                                                               |
o-------------------------------------------------------------------------------o

Diff Log D 2004 • Note 58


Differential of Conjunction (concl.)

Figures 46-a through 46-d illustrate the proposition dJ, rounded out
in our usual array of prospects.  This proposition of EU% is what we
refer to as the (first order) differential of J, and normally regard
as 'the' differential proposition corresponding to J.

o---------------------------------------o
|                                       |
|                   o                   |
|                  / \                  |
|                 /   \                 |
|                /     \                |
|               o       o               |
|              /%\     /%\              |
|             /%%%\   /%%%\             |
|            /%%%%%\ /%%%%%\            |
|           o%%%%%%%o%%%%%%%o           |
|          /%\%%%%%/ \%%%%%/%\          |
|         /%%%\%%%/   \%%%/%%%\         |
|        /%%%%%\%/     \%/%%%%%\        |
|       o%%%%%%%o       o%%%%%%%o       |
|      / \%%%%%/%\     /%\%%%%%/ \      |
|     /   \%%%/%%%\   /%%%\%%%/   \     |
|    /     \%/%%%%%\ /%%%%%\%/     \    |
|   o       o%%%%%%%o%%%%%%%o       o   |
|   |\     / \%%%%%/ \%%%%%/ \     /|   |
|   | \   /   \%%%/   \%%%/   \   / |   |
|   |  \ /     \%/     \%/     \ /  |   |
|   |   o       o       o       o   |   |
|   |   |\     / \     / \     /|   |   |
|   |   | \   /   \   /   \   / |   |   |
|   | u |  \ /     \ /     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 46-a.  Differential of J (Areal)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         / \         \   |
                                                  |  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                 /|  o         o   o         o  |
                                                / |   \         \ /         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /%%%%%%%\ /       \     |
|          o---------o   o---------o          |   |    /%%%%%%%%%o         \    |
|         /           \ /           \         |   |   /%%%%%%%%%/%\         \   |
|        /             o             \        |   |  o%%%%%%%%%o%%%o         o  |
|       /             /`\      @------\-----------@  |%% du %%%|%%%|    dv   |  |
|      /             /```\             \      |   |  o%%%%%%%%%o%%%o         o  |
|     /             /`````\             \     |   |   \%%%%%%%%%\%/         /   |
|    /             /```````\             \    |   |    \%%%%%%%%%o         /    |
|   o             o`````````o             o   |   |     \%%%%%%%/ \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /%%%%%%%\     |
|    \             \```````/   \         /    |   |    /         o%%%%%%%%%\    |
|     \             \`````/     \       /     |   |   /         /%\%%%%%%%%%\   |
|      \             \```/       \     /      |   |  o         o%%%o%%%%%%%%%o  |
|       \      @------\-/---------\---------------@  |   du    |%%%|%%% dv %%|  |
|        \             o           \ /        |   |  o         o%%%o%%%%%%%%%o  |
|         \           / \           /         |   |   \         \%/%%%%%%%%%/   |
|          o---------o   o---------o \        |   |    \         o%%%%%%%%%/    |
|                                     \       |   |     \       / \%%%%%%%/     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |                             |
                                             \    |      o-----o   o-----o      |
                                              \   |     /%%%%%%%\ /%%%%%%%\     |
                                               \  |    /%%%%%%%%%o%%%%%%%%%\    |
                                                \ |   /%%%%%%%%%/ \%%%%%%%%%\   |
                                                 \|  o%%%%%%%%%o   o%%%%%%%%%o  |
                                                  @  |%% du %%%|   |%%% dv %%|  |
                                                  |  o%%%%%%%%%o   o%%%%%%%%%o  |
                                                  |   \%%%%%%%%%\ /%%%%%%%%%/   |
                                                  |    \%%%%%%%%%o%%%%%%%%%/    |
                                                  |     \%%%%%%%/ \%%%%%%%/     |
                                                  |      o-----o   o-----o      |
                                                  |                             |
                                                  o-----------------------------o
Figure 46-b.  Differential of J (Bundle)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /   @   \                       \      |
|     /                       /   ^ ^   \                       \     |
|    o                       o   /   \   o                       o    |
|    |                       |  /     \  |                       |    |
|    |                       | /       \ |                       |    |
|    |                       |/         \|                       |    |
|    |         u         (du)/ dv     du \(dv)         v         |    |
|    |                      /|           |\                      |    |
|    |                     / |           | \                     |    |
|    |                    /  |           |  \                    |    |
|    o                   /   o           o   \                   o    |
|     \                 /     \         /     \                 /     |
|      \               v       \ du dv /       v               /      |
|       \             @<----------------------->@             /       |
|        \                       \   /                       /        |
|         \                       \ /                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 46-c.  Differential of J (Compact)

o-----------------------------------------------------------o
|                                                           |
|                            u v                            |
|                             @                             |
|                            ^ ^                            |
|                           /   \                           |
|                          /     \                          |
|                         /       \                         |
|                        /         \                        |
|               (du) dv /           \ du (dv)               |
|                      /             \                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  v                     v                  |
|           u (v) @<--------------------->@ (u) v           |
|                           du dv                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                             @                             |
|                          (u) (v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 46-d.  Differential of J (Digraph)

Diff Log D 2004 • Note 59


Remainder of Conjunction

| I bequeath myself to the dirt to grow from the grass I love,
| If you want me again look for me under your bootsoles.
|
| You will hardly know who I am or what I mean,
| But I shall be good health to you nevertheless,
| And filter and fibre your blood.
|
| Failing to fetch me at first keep encouraged,
| Missing me one place search another,
| I stop some where waiting for you
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]

Let us now recapitulate the story so far.  In effect, we have
been carrying out a decomposition of the enlarged proposition
EJ in a series of stages.  First, we considered the equation
EJ = !e!J + DJ, which was involved in the definition of DJ as
the difference EJ - !e!J.  Next, we contemplated the equation
DJ = dJ + rJ, which expresses DJ in terms of two components,
the differential dJ that was just extracted and the residual
component rJ = DJ - dJ.  This remaining proposition rJ can
be computed as shown in Table 47.

Table 47.  Computation of rJ
o-------------------------------------------------------------------------------o
|                                                                               |
| rJ  =        DJ        +        dJ                                            |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  u v ((du)(dv))  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du dv  |
|                                                                               |
| dJ  =  u v  (du, dv)   +   u (v) dv       +  (u) v  du       +  (u)(v) . 0    |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| rJ  =  u v   du  dv    +   u (v) du  dv   +  (u) v  du  dv   +  (u)(v) du dv  |
|                                                                               |
o-------------------------------------------------------------------------------o

As it happens, the remainder rJ falls under the description
of a second order differential rJ = d^2.J.  This means that
the expansion of EJ in the form:

   EJ   =   !e!J    +   DJ

        =   !e!J    +   dJ      +   rJ

        =   d^0.J   +   d^1.J   +   d^2.J

which is nothing other than the propositional analogue
of a Taylor series, is a decomposition that terminates
in a finite number of steps.

Diff Log D 2004 • Note 60


Remainder of Conjunction (concl.)

Figures 48-a through 48-d illustrate the proposition rJ = d^2.J,
which forms the remainder map of J and also, in this instance,
the second order differential of J.

o---------------------------------------o
|                                       |
|                   o                   |
|                  / \                  |
|                 /   \                 |
|                /     \                |
|               o       o               |
|              / \     / \              |
|             /   \   /   \             |
|            /     \ /     \            |
|           o       o       o           |
|          / \     /%\     / \          |
|         /   \   /%%%\   /   \         |
|        /     \ /%%%%%\ /     \        |
|       o       o%%%%%%%o       o       |
|      / \     /%\%%%%%/%\     / \      |
|     /   \   /%%%\%%%/%%%\   /   \     |
|    /     \ /%%%%%\%/%%%%%\ /     \    |
|   o       o%%%%%%%o%%%%%%%o       o   |
|   |\     / \%%%%%/%\%%%%%/ \     /|   |
|   | \   /   \%%%/%%%\%%%/   \   / |   |
|   |  \ /     \%/%%%%%\%/     \ /  |   |
|   |   o       o%%%%%%%o       o   |   |
|   |   |\     / \%%%%%/ \     /|   |   |
|   |   | \   /   \%%%/   \   / |   |   |
|   | u |  \ /     \%/     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 48-a.  Remainder of J (Areal)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         /%\         \   |
                                                  |  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                 /|  o         o%%%o         o  |
                                                / |   \         \%/         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /       \ /       \     |
|          o---------o   o---------o          |   |    /         o         \    |
|         /           \ /           \         |   |   /         /%\         \   |
|        /             o             \        |   |  o         o%%%o         o  |
|       /             /`\      @------\-----------@  |   du    |%%%|    dv   |  |
|      /             /```\             \      |   |  o         o%%%o         o  |
|     /             /`````\             \     |   |   \         \%/         /   |
|    /             /```````\             \    |   |    \         o         /    |
|   o             o`````````o             o   |   |     \       / \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /       \     |
|    \             \```````/   \         /    |   |    /         o         \    |
|     \             \`````/     \       /     |   |   /         /%\         \   |
|      \             \```/       \     /      |   |  o         o%%%o         o  |
|       \      @------\-/---------\---------------@  |   du    |%%%|    dv   |  |
|        \             o           \ /        |   |  o         o%%%o         o  |
|         \           / \           /         |   |   \         \%/         /   |
|          o---------o   o---------o \        |   |    \         o         /    |
|                                     \       |   |     \       / \       /     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |                             |
                                             \    |      o-----o   o-----o      |
                                              \   |     /       \ /       \     |
                                               \  |    /         o         \    |
                                                \ |   /         /%\         \   |
                                                 \|  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                  |  o         o%%%o         o  |
                                                  |   \         \%/         /   |
                                                  |    \         o         /    |
                                                  |     \       / \       /     |
                                                  |      o-----o   o-----o      |
                                                  |                             |
                                                  o-----------------------------o
Figure 48-b.  Remainder of J (Bundle)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |   du dv   |                       |    |
|    |       u       @<------------------------->@       v       |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    o                       o     @     o                       o    |
|     \                       \    ^    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                    du | dv                    /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                                  |                                  |
|                                  v                                  |
|                                  @                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 48-c.  Remainder of J (Compact)

o-----------------------------------------------------------o
|                                                           |
|                            u v                            |
|                             @                             |
|                             ^                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du | dv                          |
|           u (v) @<----------|---------->@ (u) v           |
|                          du | dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             v                             |
|                             @                             |
|                          (u) (v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 48-d.  Remainder of J (Digraph)

Diff Log D 2004 • Note 61


Summary of Conjunction

To establish a convenient reference point for further discussion,
Table 49 summarizes the operator actions that have been computed
for the form of conjunction, as exemplified by the proposition J.

Table 49.  Computation Summary for J
o-------------------------------------------------------------------------------o
|                                                                               |
| !e!J  =  uv .     1       + u(v) .    0    + (u)v .   0     + (u)(v) .   0    |
|                                                                               |
|   EJ  =  uv .  (du)(dv)   + u(v) . (du)dv  + (u)v . du(dv)  + (u)(v) . du dv  |
|                                                                               |
|   DJ  =  uv . ((du)(dv))  + u(v) . (du)dv  + (u)v . du(dv)  + (u)(v) . du dv  |
|                                                                               |
|   dJ  =  uv .  (du, dv)   + u(v) .     dv  + (u)v . du      + (u)(v) .   0    |
|                                                                               |
|   rJ  =  uv .   du  dv    + u(v) .  du dv  + (u)v . du dv   + (u)(v) . du dv  |
|                                                                               |
o-------------------------------------------------------------------------------o

Diff Log D 2004 • Note 62


Analytic Series:  Coordinate Method

| And if he is told that something 'is' the way it is, then he thinks:
| Well, it could probably just as easily be some other way.  So the
| sense of possibility might be defined outright as the capacity to
| think how everything could "just as easily" be, and to attach no
| more importance to what is than to what is not.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 12]

Table 50 exhibits a truth table method for computing the analytic series
(or the differential expansion) of a proposition in terms of coordinates.

Table 50.  Computation of an Analytic Series in Terms of Coordinates
o-----------o-------------o-------------oo-------------o---------o-------------o
|  u     v  |  du     dv  |  u'     v'  || !e!J    EJ  |   DJ    |  dJ   d^2.J |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  0     0  |  0      0   |  0      0   ||  0      0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  0      1   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  1      0   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  1      1   ||         1   |    1    |  0      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  0     1  |  0      0   |  0      1   ||  0      0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  0      0   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  1      1   ||         1   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  1      0   ||         0   |    0    |  1      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  1     0  |  0      0   |  1      0   ||  0      0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  1      1   ||         1   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  0      0   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  0      1   ||         0   |    0    |  1      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  1     1  |  0      0   |  1      1   ||  1      1   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  1      0   ||         0   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  0      1   ||         0   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  0      0   ||         0   |    1    |  0      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o

The first six columns of the Table, taken as a whole, represent the variables of
a construct that I describe as the "contingent universe" [u, v, du, dv, u', v'],
or the bundle of "contingency spaces" [du, dv, u', v'] over the universe [u, v].
Their placement to the left of the double bar indicates that all of them amount
to independent variables, but there is a co-dependency among them, as described
by the following equations:

o-------------------------------------------------o
|                                                 |
|         u'   =   u + du   =   (u, du)           |
|                                                 |
|         v'   =   v + du   =   (v, dv)           |
|                                                 |
o-------------------------------------------------o

These relations correspond to the formal substitutions that are made in
defining EJ and DJ.  For now, the whole rigamarole of contingency spaces
can be regarded as a technical device for achieving the effect of these
substitutions, adapted to a setting where functional compositions and
other symbolic manipulations are difficult to contemplate and execute.

The five columns to the right of the double bar in Table 50 contain the
values of the dependent variables {!e!J, EJ, DJ, dJ, d^2.J}.  These are
normally interpreted as values of functions WJ : EU -> B or as values of
propositions in the extended universe [u, v, du, dv], but the dependencies
prevailing in the contingent universe make it possible to regard these same
final values as arising via functions on alternative lists of arguments, say,
<u, v, u', v'>.

The column for !e!J is computed as J<u, v> = uv.  This, along with the
columns for u and v, illustrates the Table's "structure-sharing" scheme,
listing only the initial entries of each constant block.

The column for EJ is computed by means of the following chain of
identities, where the contingent variables u' and v' are defined
as u' = u + du and v' = v + dv.

o--------------------------------------------------------------o
|                                                              |
|   EJ<u, v, du, dv>   =   J<u + du, v + dv>   =   J<u', v'>   |
|                                                              |
o--------------------------------------------------------------o

This makes it easy to determine EJ by inspection, computing the
conjunction J<u', v'> = u' v' from the columns headed u' and v'.
Since all of these forms express the same proposition EJ in EU%,
the dependence on du and dv is still present but merely left
implicit in the final variant J<u', v'>.

NB.  On occasion, it is tempting to use the further notation J'<u, v> = J<u', v'>,
especially to suggest a transformation that acts on whole propositions, for example,
taking the proposition J into the proposition J' = EJ.  The prime ['] then signifies
an action that is mediated by a field of choices, namely, the values that are picked
out for the contingent variables in sweeping through the initial universe.  But this
heaps an unwieldy lot of construed intentions on a rather slight character, and puts
too high a premium on the constant correctness of its interpretation.  In practice,
therefore, it is best to avoid this usage.

Given the values of !e!J and EJ, the columns for the remaining functions
can be filled in quickly.  The difference map is computed according to the
relation DJ = !e!J + EJ.  The first order differential dJ is found by looking
in each block of constant <u, v> and choosing the linear function of <du, dv>
that best approximates DJ in that block.  Finally, the remainder is computed
as rJ = DJ + dJ, in this case yielding the second order differential d^2.J.

Diff Log D 2004 • Note 63


Analytic Series:  Recap

Let us now summarize the results of Table 50 by writing down for
each column, and for each block of constant <u, v>, a reasonably
canonical symbolic expression for the function of <du, dv> that
appears there.  The synopsis formed in this way is presented
in Table 51.  As one has a right to expect, it confirms the
results that were obtained previously by operating solely
in terms of the formal calculus.

Table 51.  Computation of an Analytic Series in Symbolic Terms
o-----------o---------o------------o------------o------------o-----------o
|  u     v  |    J    |     EJ     |     DJ     |     dJ     |   d^2.J   |
o-----------o---------o------------o------------o------------o-----------o
|           |         |            |            |            |           |
|  0     0  |    0    |   du  dv   |   du  dv   |     ()     |   du dv   |
|           |         |            |            |            |           |
|  0     1  |    0    |   du (dv)  |   du (dv)  |     du     |   du dv   |
|           |         |            |            |            |           |
|  1     0  |    0    |  (du) dv   |  (du) dv   |     dv     |   du dv   |
|           |         |            |            |            |           |
|  1     1  |    1    |  (du)(dv)  | ((du)(dv)) |  (du, dv)  |   du dv   |
|           |         |            |            |            |           |
o-----------o---------o------------o------------o------------o-----------o

Figures 52 and 53 provide a quick overview of the analysis performed so far,
giving the successive decompositions of EJ = J + DJ and DJ = dJ + rJ in two
different styles of diagram.

            o                           o                           o
           /%\                         /%\                         / \
          /%%%\                       /%%%\                       /   \
         o%%%%%o                     o%%%%%o                     o     o
        / \%%%/ \                   /%\%%%/%\                   /%\   /%\
       /   \%/   \                 /%%%\%/%%%\                 /%%%\ /%%%\
      o     o     o               o%%%%%o%%%%%o               o%%%%%o%%%%%o
     /%\   / \   /%\             / \%%%/%\%%%/ \             /%\%%%/%\%%%/%\
    /%%%\ /   \ /%%%\           /   \%/%%%\%/   \           /%%%\%/%%%\%/%%%\
   o%%%%%o     o%%%%%o         o     o%%%%%o     o         o%%%%%o%%%%%o%%%%%o
  / \%%%/ \   / \%%%/ \       / \   / \%%%/ \   / \       / \%%%/ \%%%/ \%%%/ \
 /   \%/   \ /   \%/   \     /   \ /   \%/   \ /   \     /   \%/   \%/   \%/   \
o     o     o     o     o   o     o     o     o     o   o     o     o     o     o
|\   / \   /%\   / \   /|   |\   / \   / \   / \   /|   |\   / \   /%\   / \   /|
| \ /   \ /%%%\ /   \ / |   | \ /   \ /   \ /   \ / |   | \ /   \ /%%%\ /   \ / |
|  o     o%%%%%o     o  |   |  o     o     o     o  |   |  o     o%%%%%o     o  |
|  |\   / \%%%/ \   /|  |   |  |\   / \   / \   /|  |   |  |\   / \%%%/ \   /|  |
|u | \ /   \%/   \ / | v|   |u | \ /   \ /   \ / | v|   |u | \ /   \%/   \ / | v|
o--+--o     o     o--+--o   o--+--o     o     o--+--o   o--+--o     o     o--+--o
   |   \   / \   /   |         |   \   / \   /   |         |   \   / \   /   |
   | du \ /   \ / dv |         | du \ /   \ / dv |         | du \ /   \ / dv |
   o-----o     o-----o         o-----o     o-----o         o-----o     o-----o
          \   /                       \   /                       \   /
           \ /                         \ /                         \ /
            o                           o                           o

           EJ             =             J             +            DJ

o-----------------------o   o-----------------------o   o-----------------------o
|                       |   |                       |   |                       |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|     /    \ /    \     |   |     /    \ /    \     |   |     /    \ /    \     |
|    /      o      \    |   |    /      o      \    |   |    /      o      \    |
|   /  u   / \   v  \   |   |   /  u   / \   v  \   |   |   /  u   / \   v  \   |
|  o      /->-\      o  |   |  o      /->-\      o  |   |  o      /   \      o  |
|  |     o \ / o     |  |   |  |     o \ / o     |  |   |  |     o     o     |  |
|  |  @--|->@<-|--@  |  |   |  |  @<-|--@--|->@  |  |   |  |  @<-|->@<-|->@  |  |
|  |     o  ^  o     |  |   |  |     o  |  o     |  |   |  |     o  ^  o     |  |
|  o      \ | /      o  |   |  o      \ | /      o  |   |  o      \ | /      o  |
|   \      \|/      /   |   |   \      \|/      /   |   |   \      \|/      /   |
|    \      |      /    |   |    \      |      /    |   |    \      |      /    |
|     \    /|\    /     |   |     \    /|\    /     |   |     \    /|\    /     |
|      o--o | o--o      |   |      o--o v o--o      |   |      o--o v o--o      |
|           @           |   |           @           |   |           @           |
o-----------------------o   o-----------------------o   o-----------------------o
Figure 52.  Decomposition of the Enlarged Conjunction EJ = (J, DJ)

            o                           o                           o
           / \                         / \                         / \
          /   \                       /   \                       /   \
         o     o                     o     o                     o     o
        /%\   /%\                   /%\   /%\                   / \   / \
       /%%%\ /%%%\                 /%%%\%/%%%\                 /   \ /   \
      o%%%%%o%%%%%o               o%%%%%o%%%%%o               o     o     o
     /%\%%%/%\%%%/%\             /%\%%%/ \%%%/%\             / \   /%\   / \
    /%%%\%/%%%\%/%%%\           /%%%\%/   \%/%%%\           /   \ /%%%\ /   \
   o%%%%%o%%%%%o%%%%%o         o%%%%%o     o%%%%%o         o     o%%%%%o     o
  / \%%%/ \%%%/ \%%%/ \       / \%%%/%\   /%\%%%/ \       / \   /%\%%%/%\   / \
 /   \%/   \%/   \%/   \     /   \%/%%%\ /%%%\%/   \     /   \ /%%%\%/%%%\ /   \
o     o     o     o     o   o     o%%%%%o%%%%%o     o   o     o%%%%%o%%%%%o     o
|\   / \   /%\   / \   /|   |\   / \%%%/ \%%%/ \   /|   |\   / \%%%/%\%%%/ \   /|
| \ /   \ /%%%\ /   \ / |   | \ /   \%/   \%/   \ / |   | \ /   \%/%%%\%/   \ / |
|  o     o%%%%%o     o  |   |  o     o     o     o  |   |  o     o%%%%%o     o  |
|  |\   / \%%%/ \   /|  |   |  |\   / \   / \   /|  |   |  |\   / \%%%/ \   /|  |
|u | \ /   \%/   \ / | v|   |u | \ /   \ /   \ / | v|   |u | \ /   \%/   \ / | v|
o--+--o     o     o--+--o   o--+--o     o     o--+--o   o--+--o     o     o--+--o
   |   \   / \   /   |         |   \   / \   /   |         |   \   / \   /   |
   | du \ /   \ / dv |         | du \ /   \ / dv |         | du \ /   \ / dv |
   o-----o     o-----o         o-----o     o-----o         o-----o     o-----o
          \   /                       \   /                       \   /
           \ /                         \ /                         \ /
            o                           o                           o

           DJ             =            dJ             +            ddJ

o-----------------------o   o-----------------------o   o-----------------------o
|                       |   |                       |   |                       |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|     /    \ /    \     |   |     /    \ /    \     |   |     /    \ /    \     |
|    /      o      \    |   |    /      o      \    |   |    /      o      \    |
|   /  u   / \   v  \   |   |   /  u   / \   v  \   |   |   /  u   / \   v  \   |
|  o      /   \      o  |   |  o      /   \      o  |   |  o      /   \      o  |
|  |     o     o     |  |   |  |     o     o     |  |   |  |     o     o     |  |
|  |  @<-|->@<-|->@  |  |   |  |  @<-|->@<-|->@  |  |   |  |  @<-|-----|->@  |  |
|  |     o  ^  o     |  |   |  |   ^ o     o ^   |  |   |  |     o  @  o     |  |
|  o      \ | /      o  |   |  o    \ \   / /    o  |   |  o      \ ^ /      o  |
|   \      \|/      /   |   |   \    --\-/--    /   |   |   \      \|/      /   |
|    \      |      /    |   |    \      o      /    |   |    \      |      /    |
|     \    /|\    /     |   |     \    / \    /     |   |     \    /|\    /     |
|      o--o v o--o      |   |      o--o   o--o      |   |      o--o v o--o      |
|           @           |   |           @           |   |           @           |
o-----------------------o   o-----------------------o   o-----------------------o
Figure 53.  Decomposition of the Differed Conjunction DJ = (dJ, ddJ)

Diff Log D 2004 • Note 64


Terminological Interlude

| Lastly, my attention was especially attracted, not so much to the scene,
| as to the mirrors that produced it.  These mirrors were broken in parts.
| Yes, they were marked and scratched;  they had been "starred", in spite
| of their solidity ...
|
| Gaston Leroux, 'The Phantom of the Opera', [Ler, 230]

At this point several issues of terminology have accrued enough substance
to intrude on our discussion.  The remarks of this Section are intended to
accomplish two goals.  First, I call attention to important aspects of the
previous series of Figures, translating into literal terms what they depict
in iconic forms, and I restress the most important structural elements that
they indicate.  Next, I prepare the way for taking on more complex examples
of transformations, whose target universes have more than a single dimension.

In talking about the actions of operators it is important to keep in mind the
distinctions between the operators per se, their operands, and their results.
Furthermore, in working with composite forms of operators W = <W_1, ..., W_n>,
transformations F = <F_1, ..., F_n>, and target domains X% = [x_1, ..., x_n],
we need to preserve a clear distinction between the compound entity of each
given type and any one of its separate components.  It is curious, given the
usefulness of the concepts "operator" and "operand", that we seem to lack
a generic term, formed on the same root, for the corresponding result of
an operation.  Following the obvious paradigm would lead on to words like
"opus", "opera", and "operant", but these words are too affected with clang
associations to work well at present, though they might be adapted in time.
One current usage gets around this problem by using the substantive "map"
as a systematic epithet to express the result of each operator's action.
I am following this practice as far as possible, for example, using the
phrase "tangent map" to denote the end product of the tangent functor
acting on its operand map.

Scholium.  See [JGH, 6-9] for a good account of tangent functors
and tangent maps in ordinary analysis, and for examples of their
use in mechanics.  This work as a whole is a model of clarity in
applying functorial principles to problems in physical dynamics.

Whenever we focus on isolated propositions, on single components of composite
operators, or on the portions of transformations that have 1-dimensional ranges,
we are free to shift between the native form of a proposition J : U -> B and the
thematized form of a mapping J : U% -> [x] without much trouble.  In these cases
we are able to tolerate a higher degree of ambiguity about the precise nature of
the input and output domains of an operator than we otherwise might.  For example,
in the preceding treatment of the example J, and for each operator W in the set
{!e!, !h!, E, D, d, r}, both the operand J and the result WJ could be viewed in
either one of two ways.  On the one hand, we could regard them as propositions
J : U -> B and WJ : EU -> B, ignoring the qualitative distinction between the
range [x] ~=~ B of !e!J and the range [dx] ~=~ D of the other WJ's.  This is
what we usually do when we content ourselves simply with coloring in regions
of venn diagrams.  On the other hand, we could view these entities as maps
J : U% -> [x] = X% and !e!J : EU% -> [x] c EX% or WJ : EU% -> [dx] c EX%,
in which case the qualitative characters of the output features are not
allowed to go without saying, nor thus at the risk of being forgotten.

Diff Log D 2004 • Note 65


Terminological Interlude (cont.)

At the beginning of this Division I recast the natural form of a proposition
J : U -> B into the thematic role of a transformation J : U% -> [x], where x
was a variable recruited to express the newly independent ¢(J).  However, in
my computations and representations of operator actions I immediately lapsed
back to viewing the results as native elements of the extended universe EU%,
in other words, as propositions WJ : EU -> B, where W ranged over the set
{!e!, E, D, d, r}.  That is as it should be.  In fact, I have worked hard
to devise a language that gives us all of these competing advantages, the
flexibility to exchange terms and types that bear equal information value,
and the capacity to reflect as quickly and as wittingly as a controlled
reflex on the fibers of our propositions, independently of whether they
express amusements, beliefs, or conjectures.

As we take on target spaces of increasing dimension, however, these
types of confusions (and confusions of types) become less and less
permissible.  For this reason, Tables 54 and 55 present a rather
detailed summary of the notation and the terminology that I am
using here, as applied to the case of J = uv.  The rationale
of these Tables is not so much to train more elephant guns
on this poor drosophila of an example, but to establish
the general paradigm with enough solidity to bear the
weight of abstraction that is coming on down the road.

Table 54 provides basic notation and descriptive information for the
objects and operators that are used used in this Example, giving the
generic type (or broadest defined type) for each entity.  Here, the
sans serif operators, $W$ in {$e$, $E$, $D$, $d$, $r$}, and their
components W in {!e!, !h!, E, D, d, r} both have the same broad
type $W$, W : (U% -> X%) -> (EU% -> EX%), as would be expected
of operators that map transformations J : U% -> X% to extended
transformations $W$J, WJ : EU% -> EX%.

Table 54.  Cast of Characters:  Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation                | Description      | Type                       |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| U%   | = [u, v]                | Source Universe  | [B^2]                      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| X%   | = [x]                   | Target Universe  | [B^1]                      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EU%  | = [u, v, du, dv]        | Extended         | [B^2 x D^2]                |
|      |                         | Source Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EX%  | = [x, dx]               | Extended         | [B^1 x D^1]                |
|      |                         | Target Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| J    | J : U -> B              | Proposition      | (B^2 -> B) c [B^2]         |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| J    | J : U% -> X%            | Transformation,  | [B^2] -> [B^1]             |
|      |                         | or Mapping       |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| W    | W :                     | Operator         |                            |
|      | U% -> EU%,              |                  | [B^2] -> [B^2 x D^2],      |
|      | X% -> EX%,              |                  | [B^1] -> [B^1 x D^1],      |
|      | (U%->X%)->(EU%->EX%),   |                  | ([B^2] -> [B^1])           |
|      | for each W among:       |                  | ->                         |
|      | e!, !h!, E, D, d        |                  | ([B^2 x D^2]->[B^1 x D^1]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| !e!  |                         | Tacit Extension Operator   !e!                |
| !h!  |                         | Trope Extension Operator   !h!                |
|  E   |                         | Enlargement Operator        E                 |
|  D   |                         | Difference Operator         D                 |
|  d   |                         | Differential Operator       d                 |
|      |                         |                                               |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| $W$  | $W$ :                   | Operator         |                            |
|      | U% -> $T$U% = EU%,      |                  | [B^2] -> [B^2 x D^2],      |
|      | X% -> $T$X% = EX%,      |                  | [B^1] -> [B^1 x D^1],      |
|      | (U%->X%)->($T$U%->$T$X%)|                  | ([B^2] -> [B^1])           |
|      | for each $W$ among:     |                  | ->                         |
|      | $e$, $E$, $D$, $T$      |                  | ([B^2 x D^2]->[B^1 x D^1]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| $e$  |                         | Radius Operator            $e$ = <!e!, !h!>   |
| $E$  |                         | Secant Operator            $E$ = <!e!,  E >   |
| $D$  |                         | Chord Operator             $D$ = <!e!,  D >   |
| $T$  |                         | Tangent Functor            $T$ = <!e!,  d >   |
|      |                         |                                               |
o------o-------------------------o-----------------------------------------------o

Diff Log D 2004 • Note 66


Terminological Interlude (concl.)

Table 55 supplies a more detailed outline of terminology for operators and
their results.  Here, I list the restrictive subtype (or narrowest defined
subtype) that applies to each entity, and I indicate across the span of the
Table the whole spectrum of alternative types that color the interpretation
of each symbol.  Accordingly, each of the component operator maps WJ, since
their ranges are 1-dimensional (of type B^1 or D^1), can be regarded either
as propositions WJ : EU -> B or as logical transformations WJ : EU% -> X%.
As a rule, the plan of the Table allows us to name each entry by detaching
the adjective at the left of its row and prefixing it to the generic noun
at the top of its column.  In one case, however, it is customary to depart
from this scheme.  Because the phrase "differential proposition", applied
to the result dJ : EU -> D, does not distinguish it from the general run
of differential propositions G : EU -> B, it is usual to single out dJ
as the "tangent proposition" of J.

Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
|              | Operator             | Proposition        | Map                  |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Tacit        | !e! :                | !e!J :             | !e!J :               |
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x]     |
|              | (U%->X%)->(EU%->X%)  | B^2 x D^2 -> B     | [B^2 x D^2]->[B^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Trope        | !h! :                | !h!J :             | !h!J :               |
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Enlargement  | E :                  | EJ :               | EJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Difference   | D :                  | DJ :               | DJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Differential | d :                  | dJ :               | dJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Remainder    | r :                  | rJ :               | rJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Radius       | $e$ = <!e!, !h!> :   |                    | $e$J :               |
| Operator     | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Secant       | $E$ = <!e!, E> :     |                    | $E$J :               |
| Operator     | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Chord        | $D$ = <!e!, D> :     |                    | $D$J :               |
| Operator     | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Tangent      | $T$ = <!e!, d> :     | dJ :               | $T$J :               |
| Functor      | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o

Diff Log D 2004 • Note 67


End of Perfunctory Chatter:  Time to Roll the Clip!

Two steps remain to finish the analysis of J that I began so long ago.
First, I need to paste the accumulated heap of flat pictures into the
frames of transformations, filling out the shapes of the operator maps
$W$J : EU% -> EX%.  This scheme is executed in two styles, using the
"areal views" in Figures 56-a and the "box views" in Figures 56-b.
Finally, in Figures 57-1 to 57-4 I put all the pieces together to
construct the full operator diagrams for $W$ : J -> $W$J.  There
is a large amount of redundancy in these three series of figures.
At this early stage of exposition I thought that it would be better
not to tax the reader's imagination, and to guarantee that the author,
at least, has worked through the relevant exercises.  I hope the reader
will excuse the flagrant use of space and try to view these snapshots as
successive frames in the animation of logic that they are meant to become.

Diff Log D 2004 • Note 68


Operator Maps:  Areal Views

                              o
                             /X\
                            /XXX\
                           oXXXXXo
                          /X\XXX/X\
                         /XXX\X/XXX\
                        oXXXXXoXXXXXo
                       / \XXX/X\XXX/ \
                      /   \X/XXX\X/   \
                     o     oXXXXXo     o
                    / \   / \XXX/ \   / \
                   /   \ /   \X/   \ /   \
                  o     o     o     o     o
                 =|\   / \   / \   / \   /|=
                = | \ /   \ /   \ /   \ / | =
               =  |  o     o     o     o  |  =
              =   |  |\   / \   / \   /|  |   =
             =    |u | \ /   \ /   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       /\\
          ////\      | du \ /   \ / dv |      /\\\\
         o/////o     o-----o     o-----o     o\\\\\o
        //\/////\           \   /           /\\\\\/\\
       ////\/////\           \ /           /\\\\\/\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         / \\\\/\\\\\/ \
    /   \/////\//   \       =   =       /   \\/\\\\\/   \
   o     o/////o     o     =     =     o     o\\\\\o     o
  / \   / \//// \   / \   =       =   / \   / \\\\/ \   / \
 /   \ /   \//   \ /   \ =         = /   \ /   \\/   \ /   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   / \   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /   \ /   \ / |
|  o     o     o     o  |           |  o     o     o     o  |
|  |\   / \   / \   /|  |           |  |\   / \   / \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \ /   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\          !h!J
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       /
      x = uv               \     /             dx = uv
                            \   /
                             \ /
                              o
Figure 56-a1.  Radius Map of the Conjunction J = uv

                              o
                             /X\
                            /XXX\
                           oXXXXXo
                          //\XXX//\
                         ////\X////\
                        o/////o/////o
                       /\\/////\////\\
                      /\\\\/////\//\\\\
                     o\\\\\o/////o\\\\\o
                    / \\\\/ \//// \\\\/ \
                   /   \\/   \//   \\/   \
                  o     o     o     o     o
                 =|\   / \   /\\   / \   /|=
                = | \ /   \ /\\\\ /   \ / | =
               =  |  o     o\\\\\o     o  |  =
              =   |  |\   / \\\\/ \   /|  |   =
             =    |u | \ /   \\/   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       /\\
          ////\      | du \ /   \ / dv |      /\\\\
         o/////o     o-----o     o-----o     o\\\\\o
        //\/////\           \   /           / \\\\/ \
       ////\/////\           \ /           /   \\/   \
      o/////o/////o           o           o     o     o
     / \/////\//// \         = =         /\\   / \   /\\
    /   \/////\//   \       =   =       /\\\\ /   \ /\\\\
   o     o/////o     o     =     =     o\\\\\o     o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/ \   / \\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/   \ /   \\/   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   /\\   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /\\\\ /   \ / |
|  o     o     o     o  |           |  o     o\\\\\o     o  |
|  |\   / \   / \   /|  |           |  |\   / \\\\/ \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \\/   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           EJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |. 
                  o-----o           o-----o 
                         \         /
                          \       / dx = (u, du)(v, dv)
      x = uv               \     /
                            \   /   dx = uv + u dv + v du + du dv
                             \ /
                              o
Figure 56-a2.  Secant Map of the Conjunction J = uv

                              o
                             //\
                            ////\
                           o/////o
                          /X\////X\
                         /XXX\//XXX\
                        oXXXXXoXXXXXo
                       /\\XXX/X\XXX/\\
                      /\\\\X/XXX\X/\\\\
                     o\\\\\oXXXXXo\\\\\o
                    / \\\\/ \XXX/ \\\\/ \
                   /   \\/   \X/   \\/   \
                  o     o     o     o     o
                 =|\   / \   /\\   / \   /|=
                = | \ /   \ /\\\\ /   \ / | =
               =  |  o     o\\\\\o     o  |  =
              =   |  |\   / \\\\/ \   /|  |   =
             =    |u | \ /   \\/   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       / \
          ////\      | du \ /   \ / dv |      /   \
         o/////o     o-----o     o-----o     o     o
        //\/////\           \   /           /\\   /\\
       ////\/////\           \ /           /\\\\ /\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         /\\\\\/\\\\\/\\
    /   \/////\//   \       =   =       /\\\\\/\\\\\/\\\\
   o     o/////o     o     =     =     o\\\\\o\\\\\o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/ \\\\/ \\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/   \\/   \\/   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   /\\   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /\\\\ /   \ / |
|  o     o     o     o  |           |  o     o\\\\\o     o  |
|  |\   / \   / \   /|  |           |  |\   / \\\\/ \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \\/   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           DJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       / dx = (u, du)(v, dv) - uv
      x = uv               \     /
                            \   /   dx = u dv + v du + du dv
                             \ /
                              o
Figure 56-a3.  Chord Map of the Conjunction J = uv

                              o
                             //\
                            ////\
                           o/////o
                          /X\////X\
                         /XXX\//XXX\
                        oXXXXXoXXXXXo
                       /\\XXX//\XXX/\\
                      /\\\\X////\X/\\\\
                     o\\\\\o/////o\\\\\o
                    / \\\\/\\////\\\\\/ \
                   /   \\/\\\\//\\\\\/   \
                  o     o\\\\\o\\\\\o     o
                 =|\   / \\\\/ \\\\/ \   /|=
                = | \ /   \\/   \\/   \ / | =
               =  |  o     o     o     o  |  =
              =   |  |\   / \   / \   /|  |   =
             =    |u | \ /   \ /   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       / \
          ////\      | du \ /   \ / dv |      /   \
         o/////o     o-----o     o-----o     o     o
        //\/////\           \   /           /\\   /\\
       ////\/////\           \ /           /\\\\ /\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         /\\\\\/ \\\\/\\
    /   \/////\//   \       =   =       /\\\\\/   \\/\\\\
   o     o/////o     o     =     =     o\\\\\o     o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/\\   /\\\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/\\\\ /\\\\\/   \
o     o     o     o     o           o     o\\\\\o\\\\\o     o
|\   / \   / \   / \   /|           |\   / \\\\/ \\\\/ \   /|
| \ /   \ /   \ /   \ / |           | \ /   \\/   \\/   \ / |
|  o     o     o     o  |           |  o     o     o     o  |
|  |\   / \   / \   /|  |           |  |\   / \   / \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \ /   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           dJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       /
      x = uv               \     /  dx = u dv + v du
                            \   /
                             \ /
                              o
Figure 56-a4.  Tangent Map of the Conjunction J = uv

Diff Log D 2004 • Note 69


Operator Maps:  Box Views

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  / \  dv  \   |
|  o      /   \      o  |
|  |     o     o     |  |
|  |     |     |     |  |
|  |     o     o     |  |
|  o      \   /      o  |
|   \      \ /      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /    \ /    \     |      \
|    /      o      \    |       \
|   /  du  / \  dv  \   |        \
|  o      /   \      o  |         \
|  |     o     o     |  @          \
|  |     |     |     |  |\          \
|  |     o     o     |  | \          \
|  o      \   /      o  |  \          \
|   \      \ /      /   |   \          \
|    \      o      /    |    \          \
|     \    / \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |```````````````````````|
|                       | \ |          \          @ |   |```````````````````````|
|                       |  \|           \           |   |```````````````````````|
|      o--o   o--o      |   \      o--o  \o--o      |   |``````o--o```o--o``````|
|     /    \ /    \     |   |\    /    \ /\   \     |   |`````/````\`/````\`````|
|    /      o      \    |   | \  /      o  @   \    |   |````/``````o``````\````|
|   /  du  / \  dv  \   |   |  \/  du  /`\  dv  \   |   |```/``du``/`\``dv``\```|
|  o      /   \      o  |   |  o\     /```\      o  |   |``o``````/```\``````o``|
|  |     o     o     |  |   |  | \   o`````o     |  |   |``|`````o`````o`````|``|
|  |     |     |     |  |   |  |  @  |``@--|-----|------@``|`````|`````|`````|``|
|  |     o     o     |  |   |  |     o`````o     |  |   |``|`````o`````o`````|``|
|  o      \   /      o  |   |  o      \```/      o  |   |``o``````\```/``````o``|
|   \      \ /      /   |   |   \      \`/      /   |   |```\``````\`/``````/```|
|    \      o      /    |   |    \      o      /    |   |````\``````o``````/````|
|     \    / \    /     |   |     \    / \    /     |   |`````\````/`\````/`````|
|      o--o   o--o      |   |      o--o   o--o      |   |``````o--o```o--o``````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \       !h!J        /        \        J        /        \       !h!J        /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o
Figure 56-b1.  Radius Map of the Conjunction J = uv

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  /`\  dv  \   |
|  o      /```\      o  |
|  |     o`````o     |  |
|  |     |`````|     |  |
|  |     o`````o     |  |
|  o      \```/      o  |
|   \      \`/      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /````\ /    \     |      \
|    /``````o      \    |       \
|   /``du``/ \  dv  \   |        \
|  o``````/   \      o  |         \
|  |`````o     o     |  @          \
|  |`````|     |     |  |\          \
|  |`````o     o     |  | \          \
|  o``````\   /      o  |  \          \
|   \``````\ /      /   |   \          \
|    \``````o      /    |    \          \
|     \````/ \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |```````````````````````|
|                       | \ |          \          @ |   |```````````````````````|
|                       |  \|           \           |   |```````````````````````|
|      o--o   o--o      |   \      o--o  \o--o      |   |``````o--o```o--o``````|
|     /    \ /````\     |   |\    /    \ /\   \     |   |`````/    \`/    \`````|
|    /      o``````\    |   | \  /      o  @   \    |   |````/      o      \````|
|   /  du  / \``dv``\   |   |  \/  du  /`\  dv  \   |   |```/  du  / \  dv  \```|
|  o      /   \``````o  |   |  o\     /```\      o  |   |``o      /   \      o``|
|  |     o     o`````|  |   |  | \   o`````o     |  |   |``|     o     o     |``|
|  |     |     |`````|  |   |  |  @  |``@--|-----|------@``|     |     |     |``|
|  |     o     o`````|  |   |  |     o`````o     |  |   |``|     o     o     |``|
|  o      \   /``````o  |   |  o      \```/      o  |   |``o      \   /      o``|
|   \      \ /``````/   |   |   \      \`/      /   |   |```\      \ /      /```|
|    \      o``````/    |   |    \      o      /    |   |````\      o      /````|
|     \    / \````/     |   |     \    / \    /     |   |`````\    /`\    /`````|
|      o--o   o--o      |   |      o--o   o--o      |   |``````o--o```o--o``````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \        EJ         /        \        J        /        \        EJ         /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o
Figure 56-b2.  Secant Map of the Conjunction J = uv

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  /`\  dv  \   |
|  o      /```\      o  |
|  |     o`````o     |  |
|  |     |`````|     |  |
|  |     o`````o     |  |
|  o      \```/      o  |
|   \      \`/      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /````\ /    \     |      \
|    /``````o      \    |       \
|   /``du``/ \  dv  \   |        \
|  o``````/   \      o  |         \
|  |`````o     o     |  @          \
|  |`````|     |     |  |\          \
|  |`````o     o     |  | \          \
|  o``````\   /      o  |  \          \
|   \``````\ /      /   |   \          \
|    \``````o      /    |    \          \
|     \````/ \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |                       |
|                       | \ |          \          @ |   |                       |
|                       |  \|           \           |   |                       |
|      o--o   o--o      |   \      o--o  \o--o      |   |      o--o   o--o      |
|     /    \ /````\     |   |\    /    \ /\   \     |   |     /````\ /````\     |
|    /      o``````\    |   | \  /      o  @   \    |   |    /``````o``````\    |
|   /  du  / \``dv``\   |   |  \/  du  /`\  dv  \   |   |   /``du``/`\``dv``\   |
|  o      /   \``````o  |   |  o\     /```\      o  |   |  o``````/```\``````o  |
|  |     o     o`````|  |   |  | \   o`````o     |  |   |  |`````o`````o`````|  |
|  |     |     |`````|  |   |  |  @  |``@--|-----|------@  |`````|`````|`````|  |
|  |     o     o`````|  |   |  |     o`````o     |  |   |  |`````o`````o`````|  |
|  o      \   /``````o  |   |  o      \```/      o  |   |  o``````\```/``````o  |
|   \      \ /``````/   |   |   \      \`/      /   |   |   \``````\`/``````/   |
|    \      o``````/    |   |    \      o      /    |   |    \``````o``````/    |
|     \    / \````/     |   |     \    / \    /     |   |     \````/ \````/     |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \        DJ         /        \        J        /        \        DJ         /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o
Figure 56-b3.  Chord Map of the Conjunction J = uv

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  / \  dv  \   |
|  o      /   \      o  |
|  |     o     o     |  |
|  |     |     |     |  |
|  |     o     o     |  |
|  o      \   /      o  |
|   \      \ /      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /````\ /    \     |      \
|    /``````o      \    |       \
|   /``du``/`\  dv  \   |        \
|  o``````/```\      o  |         \
|  |`````o`````o     |  @          \
|  |`````|`````|     |  |\          \
|  |`````o`````o     |  | \          \
|  o``````\```/      o  |  \          \
|   \``````\`/      /   |   \          \
|    \``````o      /    |    \          \
|     \````/ \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |                       |
|                       | \ |          \          @ |   |                       |
|                       |  \|           \           |   |                       |
|      o--o   o--o      |   \      o--o  \o--o      |   |      o--o   o--o      |
|     /    \ /````\     |   |\    /    \ /\   \     |   |     /````\ /````\     |
|    /      o``````\    |   | \  /      o  @   \    |   |    /``````o``````\    |
|   /  du  /`\``dv``\   |   |  \/  du  /`\  dv  \   |   |   /``du``/ \``dv``\   |
|  o      /```\``````o  |   |  o\     /```\      o  |   |  o``````/   \``````o  |
|  |     o`````o`````|  |   |  | \   o`````o     |  |   |  |`````o     o`````|  |
|  |     |`````|`````|  |   |  |  @  |``@--|-----|------@  |`````|     |`````|  |
|  |     o`````o`````|  |   |  |     o`````o     |  |   |  |`````o     o`````|  |
|  o      \```/``````o  |   |  o      \```/      o  |   |  o``````\   /``````o  |
|   \      \`/``````/   |   |   \      \`/      /   |   |   \``````\ /``````/   |
|    \      o``````/    |   |    \      o      /    |   |    \``````o``````/    |
|     \    / \````/     |   |     \    / \    /     |   |     \````/ \````/     |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \        dJ         /        \        J        /        \        dJ         /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o
Figure 56-b4.  Tangent Map of the Conjunction J = uv

Diff Log D 2004 • Note 70


Operator Diagrams for the Conjunction J = uv

            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             oXXXXXo
        ////////\                           /X\XXX/X\
       //////////\                         /XXX\X/XXX\
      o///////////o                       oXXXXXoXXXXXo
     / \////////// \                     / \XXX/X\XXX/ \
    /   \////////   \                   /   \X/XXX\X/   \
   /     \//////     \                 o     oXXXXXo     o
  /       \////       \               / \   / \XXX/ \   / \
 /         \//         \             /   \ /   \X/   \ /   \
o           o           o           o     o     o     o     o
|\         / \         /|           |\   / \   / \   / \   /|
| \       /   \       / |           | \ /   \ /   \ /   \ / |
|  \     /     \     /  |           |  o     o     o     o  |
|   \   /       \   /   |           |  |\   / \   / \   /|  |
| u  \ /         \ /  v |           |u | \ /   \ /   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $e$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $e$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $e$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o
Figure 57-1.  Radius Operator Diagram for the Conjunction J = uv

            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             oXXXXXo
        ////////\                           //\XXX//\
       //////////\                         ////\X////\
      o///////////o                       o/////o/////o
     / \////////// \                     /\\/////\////\\
    /   \////////   \                   /\\\\/////\//\\\\
   /     \//////     \                 o\\\\\o/////o\\\\\o
  /       \////       \               / \\\\/ \//// \\\\/ \
 /         \//         \             /   \\/   \//   \\/   \
o           o           o           o     o     o     o     o
|\         / \         /|           |\   / \   /\\   / \   /|
| \       /   \       / |           | \ /   \ /\\\\ /   \ / |
|  \     /     \     /  |           |  o     o\\\\\o     o  |
|   \   /       \   /   |           |  |\   / \\\\/ \   /|  |
| u  \ /         \ /  v |           |u | \ /   \\/   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $E$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $E$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $E$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o
Figure 57-2.  Secant Operator Diagram for the Conjunction J = uv

            o                                   o
           //\                                 //\
          ////\                               ////\
         //////\                             o/////o
        ////////\                           /X\////X\
       //////////\                         /XXX\//XXX\
      o///////////o                       oXXXXXoXXXXXo
     / \////////// \                     /\\XXX/X\XXX/\\
    /   \////////   \                   /\\\\X/XXX\X/\\\\
   /     \//////     \                 o\\\\\oXXXXXo\\\\\o
  /       \////       \               / \\\\/ \XXX/ \\\\/ \
 /         \//         \             /   \\/   \X/   \\/   \
o           o           o           o     o     o     o     o
|\         / \         /|           |\   / \   /\\   / \   /|
| \       /   \       / |           | \ /   \ /\\\\ /   \ / |
|  \     /     \     /  |           |  o     o\\\\\o     o  |
|   \   /       \   /   |           |  |\   / \\\\/ \   /|  |
| u  \ /         \ /  v |           |u | \ /   \\/   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $D$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $D$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $D$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o
Figure 57-3.  Chord Operator Diagram for the Conjunction J = uv

            o                                   o
           //\                                 //\
          ////\                               ////\
         //////\                             o/////o
        ////////\                           /X\////X\
       //////////\                         /XXX\//XXX\
      o///////////o                       oXXXXXoXXXXXo
     / \////////// \                     /\\XXX//\XXX/\\
    /   \////////   \                   /\\\\X////\X/\\\\
   /     \//////     \                 o\\\\\o/////o\\\\\o
  /       \////       \               / \\\\/\\////\\\\\/ \
 /         \//         \             /   \\/\\\\//\\\\\/   \
o           o           o           o     o\\\\\o\\\\\o     o
|\         / \         /|           |\   / \\\\/ \\\\/ \   /|
| \       /   \       / |           | \ /   \\/   \\/   \ / |
|  \     /     \     /  |           |  o     o     o     o  |
|   \   /       \   /   |           |  |\   / \   / \   /|  |
| u  \ /         \ /  v |           |u | \ /   \ /   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $T$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $T$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $T$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o
Figure 57-4.  Tangent Functor Diagram for the Conjunction J = uv

Diff Log D 2004 • Note 71


| The past and present wilt . . . . I have filled them and
|    emptied them,
| And proceed to fill my next fold of the future.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 87]

Taking Aim at Higher Dimensional Targets

In the next Subdivision I consider a logical transformation F that has the
concrete type F : [u, v] -> [x, y] and the abstract type F : [B^2] -> [B^2].
From the standpoint of propositional calculus, the task of understanding such
a transformation is naturally approached by parsing it into component maps with
1-dimensional ranges, as follows:

o-----------------------------------------------------------o
|                                                           |
|   F   =   <f, g>  =  <F_1, F_2>  :  [u, v]  ->  [x, y]    |
|                                                           |
|   where      f    =      F_1     :  [u, v]  ->  [x]       |
|                                                           |
|   and        g    =      F_2     :  [u, v]  ->  [y]       |
|                                                           |
o-----------------------------------------------------------o

Then one tackles the separate components, now viewed as propositions
F_i : U -> B, one at a time.  At the completion of this analytic phase,
one returns to the task of synthesizing all of these partial and transient
impressions into an agile form of integrity, a solidly coordinated and deeply
integrated comprehension of the ongoing transformation.  (Very often, of course,
in tangling with refractory cases, one never gets as far as the beginning again.)

Diff Log D 2004 • Note 72


Taking Aim at Higher Dimensional Targets (concl.)

Let us now refer to the dimension of the target space or codomain as
the "toll" (or "tole") of a transformation, as distinguished from the
dimension of the range or image that is customarily called the "rank".
When we keep to transformations with a toll of 1, as J : [u, v] -> [x],
we tend to get lazy about distinguishing a logical transformation from
its component propositions.  However, if we deal with transformations
of a higher toll, this form of indolence can no longer be tolerated.

Well, perhaps we can carry it a little further.  After all,
the operator result WJ : EU% -> EX% is a map of toll 2, and
cannot be unfolded in one piece as a proposition.  But when a
map has rank 1, like !e!J : EU -> X c EX or dJ : EU -> dX c EX,
we naturally choose to concentrate on the 1-dimensional range of
the operator result WJ, ignoring the final difference in quality
between the spaces X and dX, and view WJ as a proposition about EU.

In this way, an initial ambivalence about the role of the operand J
conveys a double duty to the result WJ.  The pivot that is formed by
our focus of attention is essential to the linkage that transfers this
double moment, as the whole process takes its bearing and wheels around
the precise measure of a narrow bead that we can draw on the range of WJ.
This is the escapement that it takes to get away with what may otherwise
seem to be a simple duplicity, and this is the tolerance that is needed
to counterbalance a certain arrogance of equivocation, by all of which
machinations we make ourselves free to indicate the operator results
WJ as propositions or as transformations, indifferently.

But that's it, and no further.  Neglect of these distinctions in range and
target universes of higher dimensions is bound to cause a hopeless confusion.
To guard against these adverse prospects, Tables 58 and 59 lay the groundwork
for discussing a typical map F : [B^2] -> [B^2], and begin to pave the way, to
some extent, for discussing any transformation of the form F : [B^n] -> [B^k].

Table 58.  Cast of Characters:  Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation                | Description      | Type                       |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| U%   | = [u, v]                | Source Universe  | [B^n]                      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| X%   | = [x, y]                | Target Universe  | [B^k]                      |
|      | = [f, g]                |                  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EU%  | = [u, v, du, dv]        | Extended         | [B^n x D^n]                |
|      |                         | Source Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EX%  | = [x, y, dx, dy]        | Extended         | [B^k x D^k]                |
|      | = [f, g, df, dg]        | Target Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| F    | F = <f, g> : U% -> X%   | Transformation,  | [B^n] -> [B^k]             |
|      |                         | or Mapping       |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
|      | f, g : U -> B           | Proposition,     | B^n -> B                   |
|      |                         |   special case   |                            |
| f    | f : U -> [x] c X%       |   of a mapping,  | c (B^n, B^n -> B)          |
|      |                         |   or component   |                            |
| g    | g : U -> [y] c X%       |   of a mapping.  | = (B^n +-> B) = [B^n]      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| W    | W :                     | Operator         |                            |
|      | U% -> EU%,              |                  | [B^n] -> [B^n x D^n],      |
|      | X% -> EX%,              |                  | [B^k] -> [B^k x D^k],      |
|      | (U%->X%)->(EU%->EX%),   |                  | ([B^n] -> [B^k])           |
|      | for each W among:       |                  | ->                         |
|      | !e!, !h!, E, D, d       |                  | ([B^n x D^n]->[B^k x D^k]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| !e!  |                         | Tacit Extension Operator   !e!                |
| !h!  |                         | Trope Extension Operator   !h!                |
|  E   |                         | Enlargement Operator        E                 |
|  D   |                         | Difference Operator         D                 |
|  d   |                         | Differential Operator       d                 |
|      |                         |                                               |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| $W$  | $W$ :                   | Operator         |                            |
|      | U% -> $T$U% = EU%,      |                  | [B^n] -> [B^n x D^n],      |
|      | X% -> $T$X% = EX%,      |                  | [B^k] -> [B^k x D^k],      |
|      | (U%->X%)->($T$U%->$T$X%)|                  | ([B^n] -> [B^k])           |
|      | for each $W$ among:     |                  | ->                         |
|      | $e$, $E$, $D$, $T$      |                  | ([B^n x D^n]->[B^k x D^k]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| $e$  |                         | Radius Operator        $e$  =  <!e!, !h!>     |
| $E$  |                         | Secant Operator        $E$  =  <!e!,  E >     |
| $D$  |                         | Chord Operator         $D$  =  <!e!,  D >     |
| $T$  |                         | Tangent Functor        $T$  =  <!e!,  d >     |
|      |                         |                                               |
o------o-------------------------o-----------------------------------------------o

Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
|              | Operator             | Proposition        | Transformation       |
|              |    or                |    or              |    or                |
|              | Operand              | Component          | Mapping              |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Operand      | F = <F_1, F_2>       | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
|              |                      |                    |                      |
|              | F = <f, g> : U -> X  | F_i : B^n -> B     | F : B^n -> B^k       |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Tacit        | !e! :                | !e!F_i :           | !e!F :               |
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y]  |
|              | (U%->X%)->(EU%->X%)  | B^n x D^n -> B     | [B^n x D^n]->[B^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Trope        | !h! :                | !h!F_i :           | !h!F :               |
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Enlargement  | E :                  | EF_i :             | EF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Difference   | D :                  | DF_i :             | DF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Differential | d :                  | dF_i :             | dF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Remainder    | r :                  | rF_i :             | rF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Radius       | $e$ = <!e!, !h!> :   |                    | $e$F :               |
| Operator     |                      |                    |                      |
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
|              |                      |                    |                      |
|              |                      |                    | [B^n x D^n] ->       |
|              |                      |                    | [B^k x D^k]          |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Secant       | $E$ = <!e!, E> :     |                    | $E$F :               |
| Operator     |                      |                    |                      |
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
|              |                      |                    |                      |
|              |                      |                    | [B^n x D^n] ->       |
|              |                      |                    | [B^k x D^k]          |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Chord        | $D$ = <!e!, D> :     |                    | $D$F :               |
| Operator     |                      |                    |                      |
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
|              |                      |                    |                      |
|              |                      |                    | [B^n x D^n] ->       |
|              |                      |                    | [B^k x D^k]          |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Tangent      | $T$ = <!e!, d> :     | dF_i :             | $T$F :               |
| Functor      |                      |                    |                      |
|              | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u, v, du, dv] ->    |
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
|              |                      |                    |                      |
|              |                      | B^n x D^n -> D     | [B^n x D^n] ->       |
|              |                      |                    | [B^k x D^k]          |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o

Diff Log D 2004 • Note 73


Transformations of Type B^2 -> B^2

To take up a slightly more complex example, but one that remains
simple enough to pursue through a complete series of developments,
consider the transformation from U% = [u, v] to X% = [x, y] that is
defined by the following system of equations:

o-----------------------------------------------------------o
|                                                           |
|         x   =   f(u, v)   =   ((u)(v))                    |
|                                                           |
|         y   =   g(u, v)   =   ((u, v))                    |
|                                                           |
o-----------------------------------------------------------o

The component notation F = <F_1, F_2> = <f, g> : U% -> X% allows
us to give a name and a type to this transformation, and permits
us to define it by means of the compact description that follows:

o-----------------------------------------------------------o
|                                                           |
|    <x, y>   =   F<u, v>   =   <((u)(v)), ((u, v))>        |
|                                                           |
o-----------------------------------------------------------o

The information that defines the logical transformation F can be represented
in the form of a truth table, as in Table 60.  To cut down on subscripts in
this example I continue to use plain letter equivalents for all components
of spaces and maps.

Table 60.  Propositional Transformation
o-------------o-------------o-------------o-------------o
|      u      |      v      |      f      |      g      |
o-------------o-------------o-------------o-------------o
|             |             |             |             |
|      0      |      0      |      0      |      1      |
|             |             |             |             |
|      0      |      1      |      1      |      0      |
|             |             |             |             |
|      1      |      0      |      1      |      0      |
|             |             |             |             |
|      1      |      1      |      1      |      1      |
|             |             |             |             |
o-------------o-------------o-------------o-------------o
|             |             |  ((u)(v))   |  ((u, v))   |
o-------------o-------------o-------------o-------------o

Diff Log D 2004 • Note 74


Transformations of Type B^2 -> B^2 (cont.)

Figure 61 shows how one might paint a picture of
the logical transformation F on the canvass that
was earlier primed for this purpose (way back in
Figure 30).

             o-----------------------------------------------------o
             | U                                                   |
             |                                                     |
             |            o-----------o   o-----------o            |
             |           /             \ /             \           |
             |          /               o               \          |
             |         /               / \               \         |
             |        /               /   \               \        |
             |       o               o     o               o       |
             |       |               |     |               |       |
             |       |       u       |     |       v       |       |
             |       |               |     |               |       |
             |       o               o     o               o       |
             |        \               \   /               /        |
             |         \               \ /               /         |
             |          \               o               /          |
             |           \             / \             /           |
             |            o-----------o   o-----------o            |
             |                                                     |
             |                                                     |
             o-----------------------------------------------------o
            / \                                                   / \
           /   \                                                 /   \
          /     \                                               /     \
         /       \                                             /       \
        /         \                                           /         \
       /           \                                         /           \
      /             \                                       /             \
     /               \                                     /               \
    /                 \                                   /                 \
   /                   \                                 /                   \
  /                     \                               /                     \
 /                       \                             /                       \
o-------------------------o                           o-------------------------o
| U                       |                           |\U \\\\\\\\\\\\\\\\\\\\\\|
|      o---o   o---o      |                           |\\\\\\o---o\\\o---o\\\\\\|
|     //////\ //////\     |                           |\\\\\/     \\/     \\\\\\|
|    ////////o///////\    |                           |\\\\/       o       \\\\\|
|   //////////\///////\   |                           |\\\/       /\\       \\\\|
|  o///////o///o///////o  |                           |\\o       o\\\o       o\\|
|  |// u //|///|// v //|  |                           |\\|   u   |\\\|   v   |\\|
|  o///////o///o///////o  |                           |\\o       o\\\o       o\\|
|   \///////\//////////   |                           |\\\\       \\/       /\\\|
|    \///////o////////    |                           |\\\\\       o       /\\\\|
|     \////// \//////     |                           |\\\\\\     /\\     /\\\\\|
|      o---o   o---o      |                           |\\\\\\o---o\\\o---o\\\\\\|
|                         |                           |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o                           o-------------------------o
 \                        |                           |                        /
   \                      |                           |                      /
     \                    |                           |                    /
       \        f         |                           |         g        /
         \                |                           |                /
           \              |                           |              /
             \            |                           |            /
               \          |                           |          /
                 \        |                           |        /
                   \      |                           |      /
             o-------\----|---------------------------|----/-------o
             | X       \  |                           |  /         |
             |           \|                           |/           |
             |            o-----------o   o-----------o            |
             |           //////////////\ /\\\\\\\\\\\\\\           |
             |          ////////////////o\\\\\\\\\\\\\\\\          |
             |         /////////////////X\\\\\\\\\\\\\\\\\         |
             |        /////////////////XXX\\\\\\\\\\\\\\\\\        |
             |       o///////////////oXXXXXo\\\\\\\\\\\\\\\o       |
             |       |///////////////|XXXXX|\\\\\\\\\\\\\\\|       |
             |       |////// x //////|XXXXX|\\\\\\ y \\\\\\|       |
             |       |///////////////|XXXXX|\\\\\\\\\\\\\\\|       |
             |       o///////////////oXXXXXo\\\\\\\\\\\\\\\o       |
             |        \///////////////\XXX/\\\\\\\\\\\\\\\/        |
             |         \///////////////\X/\\\\\\\\\\\\\\\/         |
             |          \///////////////o\\\\\\\\\\\\\\\/          |
             |           \////////////// \\\\\\\\\\\\\\/           |
             |            o-----------o   o-----------o            |
             |                                                     |
             |                                                     |
             o-----------------------------------------------------o
Figure 61.  Propositional Transformation

Diff Log D 2004 • Note 75


Transformations of Type B^2 -> B^2 (cont.)

Figure 62 extracts the gist of Figure 61,
exemplifying a style of diagram that is
adequate for most purposes.

o-------------------------o o-------------------------o
| U                       | |\U \\\\\\\\\\\\\\\\\\\\\\|
|      o---o   o---o      | |\\\\\\o---o\\\o---o\\\\\\|
|     //////\ //////\     | |\\\\\/     \\/     \\\\\\|
|    ////////o///////\    | |\\\\/       o       \\\\\|
|   //////////\///////\   | |\\\/       /\\       \\\\|
|  o///////o///o///////o  | |\\o       o\\\o       o\\|
|  |// u //|///|// v //|  | |\\|   u   |\\\|   v   |\\|
|  o///////o///o///////o  | |\\o       o\\\o       o\\|
|   \///////\//////////   | |\\\\       \\/       /\\\|
|    \///////o////////    | |\\\\\       o       /\\\\|
|     \////// \//////     | |\\\\\\     /\\     /\\\\\|
|      o---o   o---o      | |\\\\\\o---o\\\o---o\\\\\\|
|                         | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
 \                       /   \                       /
  \                     /     \                     /
   \                   /       \                   /
    \        f        /         \        g        /
     \               /           \               /
      \             /             \             /
       \           /               \           /
        \         /                 \         /
         \       /                   \       /
o---------\-----/---------------------\-----/---------o
| X        \   /                       \   /          |
|           \ /                         \ /           |
|            o-----------o   o-----------o            |
|           //////////////\ /\\\\\\\\\\\\\\           |
|          ////////////////o\\\\\\\\\\\\\\\\          |
|         /////////////////X\\\\\\\\\\\\\\\\\         |
|        /////////////////XXX\\\\\\\\\\\\\\\\\        |
|       o///////////////oXXXXXo\\\\\\\\\\\\\\\o       |
|       |///////////////|XXXXX|\\\\\\\\\\\\\\\|       |
|       |////// x //////|XXXXX|\\\\\\ y \\\\\\|       |
|       |///////////////|XXXXX|\\\\\\\\\\\\\\\|       |
|       o///////////////oXXXXXo\\\\\\\\\\\\\\\o       |
|        \///////////////\XXX/\\\\\\\\\\\\\\\/        |
|         \///////////////\X/\\\\\\\\\\\\\\\/         |
|          \///////////////o\\\\\\\\\\\\\\\/          |
|           \////////////// \\\\\\\\\\\\\\/           |
|            o-----------o   o-----------o            |
|                                                     |
|                                                     |
o-----------------------------------------------------o
Figure 62.  Propositional Transformation (Short Form)

Diff Log D 2004 • Note 76


Transformations of Type B^2 -> B^2 (cont.)

Figure 63 give a more complete picture of the transformation F,
showing how the points of U% are transformed into points of X%.
The lines that cross from one universe to the other trace the
action that F induces on points, in other words, they depict
the aspect of the transformation that acts as a mapping from
points to points, and chart its effects on the elements that
are variously called cells, points, positions, or singular
propositions.

             o-----------------------------------------------------o
             |`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
             |` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
             |` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
             |` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
             |` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
             |` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
             |` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
             |` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
             |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
             |` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
             |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
             |` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
             |` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
             |` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
             |` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
             |` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
             |` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
             |` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
             |` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
             o-----------\----|---------|---------|----------------o
            " "           \   |         |         |               " "
         "       "         \  |         |         |            "       "
      "             "       \ |         |         |         "             "
   "                   "     \|         |         |      "                   "
o-------------------------o   \         |         |   o-------------------------o
| U                       |   |\        |         |   |`U```````````````````````|
|      o---o   o---o      |   | \       |         |   |``````o---o```o---o``````|
|     /'''''\ /'''''\     |   |  \      |         |   |`````/     \`/     \`````|
|    /'''''''o'''''''\    |   |   \     |         |   |````/       o       \````|
|   /'''''''/'\'''''''\   |   |    \    |         |   |```/       /`\       \```|
|  o'''''''o'''o'''''''o  |   |     \   |         |   |``o       o```o       o``|
|  |'''u'''|'''|'''v'''|  |   |      \  |         |   |``|   u   |```|   v   |``|
|  o'''''''o'''o'''''''o  |   |       \ |         |   |``o       o```o       o``|
|   \'''''''\'/'''''''/   |   |        \|         |   |```\       \`/       /```|
|    \'''''''o'''''''/    |   |         \         |   |````\       o       /````|
|     \'''''/ \'''''/     |   |         |\        |   |`````\     /`\     /`````|
|      o---o   o---o      |   |         | \       |   |``````o---o```o---o``````|
|                         |   |         |  \      *   |`````````````````````````|
o-------------------------o   |         |   \    /    o-------------------------o
 \                        |   |         |    \  /     |                        /
   \      ((u)(v))        |   |         |     \/      |        ((u, v))      /
     \                    |   |         |     /\      |                    /
       \                  |   |         |    /  \     |                  /
         \                |   |         |   /    \    |                /
           \              |   |         |  /      *   |              /
             \            |   |         | /       |   |            /
               \          |   |         |/        |   |          /
                 \        |   |         /         |   |        /
                   \      |   |        /|         |   |      /
             o-------\----|---|-------/-|---------|---|----/-------o
             | X       \  |   |      /  |         |   |  /         |
             |           \|   |     /   |         |   |/           |
             |            o---|----/--o | o-------|---o            |
             |           /' ' | ' / ' '\|/` ` ` ` | ` `\           |
             |          / ' ' | '/' ' ' | ` ` ` ` | ` ` \          |
             |         /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\         |
             |        / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \        |
             |   @   o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o       |
             |       |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `|       |
             |       |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `|       |
             |       |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `|       |
             |       o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o       |
             |        \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` /        |
             |         \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/         |
             |          \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` /          |
             |           \' ' ' ' ' ' '/ \` ` ` ` ` ` `/           |
             |            o-----------o   o-----------o            |
             |                                                     |
             |                                                     |
             o-----------------------------------------------------o
Figure 63.  Transformation of Positions

Diff Log D 2004 • Note 77


Transformations of Type B^2 -> B^2 (cont.)

Table 64 shows how the action of the transformation F
on cells or points is computed in terms of coordinates.

Table 64.  Transformation of Positions
o-----o----------o----------o-------o-------o--------o--------o-------------o
| u v |    x     |    y     |  x y  |  x(y) | (x)y   | (x)(y) | X% = [x, y] |
o-----o----------o----------o-------o-------o--------o--------o-------------o
|     |          |          |       |       |        |        |      ^      |
| 0 0 |    0     |    1     |   0   |   0   |   1    |   0    |      |      |
|     |          |          |       |       |        |        |             |
| 0 1 |    1     |    0     |   0   |   1   |   0    |   0    |      F      |
|     |          |          |       |       |        |        |      =      |
| 1 0 |    1     |    0     |   0   |   1   |   0    |   0    |   <f , g>   |
|     |          |          |       |       |        |        |             |
| 1 1 |    1     |    1     |   1   |   0   |   0    |   0    |      ^      |
|     |          |          |       |       |        |        |      |      |
o-----o----------o----------o-------o-------o--------o--------o-------------o
|     | ((u)(v)) | ((u, v)) |  u v  | (u,v) | (u)(v) |   0    | U% = [u, v] |
o-----o----------o----------o-------o-------o--------o--------o-------------o

Table 65 extends this scheme from single cells to arbitrary regions of
the source and target universes, and illustrates a form of computation
that can be used to determine how a logical transformation acts on all
of the propositions in the universe.  The way that a transformation of
positions affects the propositions, or any other structure that can be
built on top of the positions, is normally called the "induced action"
of the given transformation on the system of structures in question.

Table 65.  Induced Transformation on Propositions
o------------o---------------------------------o------------o
|     X%     |   <---   F  =  <f , g>   <---   |     U%     |
o------------o----------o-----------o----------o------------o
|            |      u = |  1 1 0 0  | = u      |            |
|            |      v = |  1 0 1 0  | = v      |            |
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
|            |      x = |  1 1 1 0  | = f<u,v> |            |
|            |      y = |  1 0 0 1  | = g<u,v> |            |
o------------o----------o-----------o----------o------------o
|            |          |           |          |            |
|    f_0     |    ()    |  0 0 0 0  |    ()    |    f_0     |
|            |          |           |          |            |
|    f_1     |  (x)(y)  |  0 0 0 1  |    ()    |    f_0     |
|            |          |           |          |            |
|    f_2     |  (x) y   |  0 0 1 0  |  (u)(v)  |    f_1     |
|            |          |           |          |            |
|    f_3     |  (x)     |  0 0 1 1  |  (u)(v)  |    f_1     |
|            |          |           |          |            |
|    f_4     |   x (y)  |  0 1 0 0  |  (u, v)  |    f_6     |
|            |          |           |          |            |
|    f_5     |     (y)  |  0 1 0 1  |  (u, v)  |    f_6     |
|            |          |           |          |            |
|    f_6     |  (x, y)  |  0 1 1 0  |  (u  v)  |    f_7     |
|            |          |           |          |            |
|    f_7     |  (x  y)  |  0 1 1 1  |  (u  v)  |    f_7     |
|            |          |           |          |            |
o------------o----------o-----------o----------o------------o
|            |          |           |          |            |
|    f_8     |   x  y   |  1 0 0 0  |   u  v   |    f_8     |
|            |          |           |          |            |
|    f_9     | ((x, y)) |  1 0 0 1  |   u  v   |    f_8     |
|            |          |           |          |            |
|    f_10    |      y   |  1 0 1 0  | ((u, v)) |    f_9     |
|            |          |           |          |            |
|    f_11    |  (x (y)) |  1 0 1 1  | ((u, v)) |    f_9     |
|            |          |           |          |            |
|    f_12    |   x      |  1 1 0 0  | ((u)(v)) |    f_14    |
|            |          |           |          |            |
|    f_13    | ((x) y)  |  1 1 0 1  | ((u)(v)) |    f_14    |
|            |          |           |          |            |
|    f_14    | ((x)(y)) |  1 1 1 0  |   (())   |    f_15    |
|            |          |           |          |            |
|    f_15    |   (())   |  1 1 1 1  |   (())   |    f_15    |
|            |          |           |          |            |
o------------o----------o-----------o----------o------------o

Diff Log D 2004 • Note 78


Transformations of Type B^2 -> B^2 (cont.)

Given the alphabets !U! = {u, v} and !X! = {x, y},
along with the corresponding universes of discourse
U% and X% ~=~ [B^2], how many logical transformations
of the general form G = <G_1, G_2> : U% -> X% are there?

Since G_1 and G_2 can be any propositions of the type B^2 -> B,
there are 2^4 = 16 choices for each of the maps G_1 and G_2, and
thus there are 2^4 * 2^4 = 2^8 = 256 different mappings altogether
of the form G : U% -> X%.  The set of all functions of a given type
is customarily denoted by placing its type indicator in parentheses,
in the present instance writing (U% -> X%) = {G : U% -> X%}, and so
the cardinality of this "function space" can be most conveniently
summed up by writing |(U% -> X%)| = |(B^2 -> B^2)| = 4^4 = 256.

Given any transformation G = <G_1, G_2> : U% -> X% of this type,
one can define a couple of further transformations, related to G,
that operate between the extended universes, EU% and EX%, of its
source and target domains.

First, the enlargement map (or the secant transformation)
EG = <EG_1, EG_2> : EU% -> EX% is defined by the following
set of component equations:

o-------------------------------------------------o
|                                                 |
|   EG_i  =  G_i <u + du, v + dv>                 |
|                                                 |
o-------------------------------------------------o

Second, the difference map (or the chordal transformation)
DG = <DG_1, DG_2> : EU% -> EX% is defined in component-wise
fashion as the boolean sum of the initial proposition G_i
and the enlarged proposition EG_i, for i = 1, 2, according
to the following set of equations:

o-------------------------------------------------o
|                                                 |
|   DG_i  =  G_i <u, v>  +  EG_i <u, v, du, dv>   |
|                                                 |
|         =  G_i <u, v>  +  G_i <u + du, v + dv>  |
|                                                 |
o-------------------------------------------------o

Maintaining a strict analogy with ordinary difference calculus
would perhaps have us write DG_i = EG_i - G_i, but the sum and
difference operations are the same thing in boolean arithmetic.
It is more often natural in the logical context to consider an
initial proposition q, then to compute the enlargement Eq, and
finally to determine the difference Dq = q + Eq, so we let the
variant order of terms reflect this sequence of considerations.

Viewed in this light the difference operator D is imagined to be a function
of very wide scope and polymorphic application, one that is able to realize
the association between each transformation G and its difference map DG, for
instance, taking the function space (U% -> X%) into (EU% -> EX%).  Given the
interpretive flexibility of contexts in which we are allowing a proposition
to appear, it should be clear that an operator of this scope is not at all
a trivial matter to define properly, and may take some trouble to work out.
For the moment, let's content ourselves with returning to particular cases.

Diff Log D 2004 • Note 79


Transformations of Type B^2 -> B^2 (cont.)

In their application to the present example, namely, the
logical transformation F = <f, g> = <((u)(v)), ((u, v))>,
the operators E and D respectively produce the enlarged
map EF = <Ef, Eg> and the difference map DF = <Df, Dg>,
whose components can be given as follows, if the reader,
in lieu of a special font for the logical parentheses,
can forgive a syntactically bilingual formulation:

o-------------------------------------------------o
|                                                 |
|   Ef  =  ((u + du)(v + dv))                     |
|                                                 |
|   Eg  =  ((u + du, v + dv))                     |
|                                                 |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|   Df  =  ((u)(v))  +  ((u + du)(v + dv))        |
|                                                 |
|   Dg  =  ((u, v))  +  ((u + du, v + dv))        |
|                                                 |
o-------------------------------------------------o

But these initial formulas are purely definitional, and help us little
in understanding either the purpose of the operators or the meaning of
their results.  Working symbolically, let us apply the same method to
the separate components f and g that we earlier used on J.  This work
is recorded in Appendix 1 and a summary of the results is presented
in Tables 66-i and 66-ii.

Table 66-i.  Computation Summary for f<u, v> = ((u)(v))
o--------------------------------------------------------------------------------o
|                                                                                |
| !e!f  =  uv.    1      + u(v).    1      + (u)v.    1      + (u)(v).    0      |
|                                                                                |
|   Ef  =  uv. (du  dv)  + u(v). (du (dv)) + (u)v.((du) dv)  + (u)(v).((du)(dv)) |
|                                                                                |
|   Df  =  uv.  du  dv   + u(v).  du (dv)  + (u)v. (du) dv   + (u)(v).((du)(dv)) |
|                                                                                |
|   df  =  uv.    0      + u(v).  du       + (u)v.      dv   + (u)(v). (du, dv)  |
|                                                                                |
|   rf  =  uv.  du  dv   + u(v).  du  dv   + (u)v.  du  dv   + (u)(v).  du  dv   |
|                                                                                |
o--------------------------------------------------------------------------------o

Table 66-ii.  Computation Summary for g<u, v> = ((u, v))
o--------------------------------------------------------------------------------o
|                                                                                |
| !e!g  =  uv.    1      + u(v).    0      + (u)v.    0      + (u)(v).    1      |
|                                                                                |
|   Eg  =  uv.((du, dv)) + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v).((du, dv)) |
|                                                                                |
|   Dg  =  uv. (du, dv)  + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v). (du, dv)  |
|                                                                                |
|   dg  =  uv. (du, dv)  + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v). (du, dv)  |
|                                                                                |
|   rg  =  uv.    0      + u(v).    0      + (u)v.    0      + (u)(v).    0      |
|                                                                                |
o--------------------------------------------------------------------------------o

Diff Log D 2004 • Note 80


Transformations of Type B^2 -> B^2 (cont.)

Table 67 shows how to compute the analytic series
for F = <f, g> = <((u)(v)), ((u, v))> in terms of
coordinates, and Table 68 recaps these results in
symbolic terms, agreeing with earlier derivations.

Table 67.  Computation of an Analytic Series in Terms of Coordinates
o--------o-------o-------o--------o-------o-------o-------o-------o
|  u  v  | du dv | u' v' |  f  g  | Ef Eg | Df Dg | df dg | rf rg |
o--------o-------o-------o--------o-------o-------o-------o-------o
|        |       |       |        |       |       |       |       |
|  0  0  | 0  0  | 0  0  |  0  1  | 0  1  | 0  0  | 0  0  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 0  1  | 0  1  |        | 1  0  | 1  1  | 1  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  0  | 1  0  |        | 1  0  | 1  1  | 1  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  1  | 1  1  |        | 1  1  | 1  0  | 0  0  | 1  0  |
|        |       |       |        |       |       |       |       |
o--------o-------o-------o--------o-------o-------o-------o-------o
|        |       |       |        |       |       |       |       |
|  0  1  | 0  0  | 0  1  |  1  0  | 1  0  | 0  0  | 0  0  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 0  1  | 0  0  |        | 0  1  | 1  1  | 1  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  0  | 1  1  |        | 1  1  | 0  1  | 0  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  1  | 1  0  |        | 1  0  | 0  0  | 1  0  | 1  0  |
|        |       |       |        |       |       |       |       |
o--------o-------o-------o--------o-------o-------o-------o-------o
|        |       |       |        |       |       |       |       |
|  1  0  | 0  0  | 1  0  |  1  0  | 1  0  | 0  0  | 0  0  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 0  1  | 1  1  |        | 1  1  | 0  1  | 0  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  0  | 0  0  |        | 0  1  | 1  1  | 1  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  1  | 0  1  |        | 1  0  | 0  0  | 1  0  | 1  0  |
|        |       |       |        |       |       |       |       |
o--------o-------o-------o--------o-------o-------o-------o-------o
|        |       |       |        |       |       |       |       |
|  1  1  | 0  0  | 1  1  |  1  1  | 1  1  | 0  0  | 0  0  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 0  1  | 1  0  |        | 1  0  | 0  1  | 0  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  0  | 0  1  |        | 1  0  | 0  1  | 0  1  | 0  0  |
|        |       |       |        |       |       |       |       |
|        | 1  1  | 0  0  |        | 0  1  | 1  0  | 0  0  | 1  0  |
|        |       |       |        |       |       |       |       |
o--------o-------o-------o--------o-------o-------o-------o-------o

Table 68.  Computation of an Analytic Series in Symbolic Terms
o-----o-----o------------o----------o----------o----------o----------o----------o
| u v | f g |     Df     |    Dg    |    df    |    dg    |    rf    |    rf    |
o-----o-----o------------o----------o----------o----------o----------o----------o
|     |     |            |          |          |          |          |          |
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
| 0 1 | 1 0 |  (du) dv   | (du, dv) |    dv    | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
| 1 0 | 1 0 |   du (dv)  | (du, dv) |    du    | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
| 1 1 | 1 1 |   du  dv   | (du, dv) |    ()    | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
o-----o-----o------------o----------o----------o----------o----------o----------o

Diff Log D 2004 • Note 81


Transformations of Type B^2 -> B^2 (cont.)

Figure 69 gives a graphical picture of the difference map DF = <Df, Dg>
for the transformation F = <f, g> = <((u)(v)), ((u, v))>.  This depicts
the same information about Df and Dg that was given in the corresponding
rows of the computation summary in Tables 66-i and 66-ii, excerpted here:

o-------------------------------------------------------------------------o
|                                                                         |
|  Df  =  uv. du  dv  + u(v). du (dv) + (u)v.(du) dv  + (u)(v).((du)(dv)) |
|                                                                         |
|  Dg  =  uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv)  |
|                                                                         |
o-------------------------------------------------------------------------o

o-----------------------------------o o-----------------------------------o
| U                                 | |`U`````````````````````````````````|
|                                   | |```````````````````````````````````|
|                 ^                 | |```````````````````````````````````|
|                 |                 | |```````````````````````````````````|
|       o-------o | o-------o       | |```````o-------o```o-------o```````|
| ^    /`````````\|/`````````\    ^ | | ^ ```/      ^  \`/  ^      \``` ^ |
|  \  /```````````|```````````\  /  | |``\``/        \  o  /        \``/``|
|   \/`````u`````/|\`````v`````\/   | |```\/     u    \/`\/    v     \/```|
|   /\``````````/`|`\``````````/\   | |```/\          /\`/\          /\```|
|  o``\````````o``@``o````````/``o  | |``o  \        o``@``o        /  o``|
|  |```\```````|`````|```````/```|  | |``|   \       |`````|       /   |``|
|  |````@``````|`````|``````@````|  | |``|    @-------->`<--------@    |``|
|  |```````````|`````|```````````|  | |``|           |`````|           |``|
|  o```````````o` ^ `o```````````o  | |``o           o`````o           o``|
|   \```````````\`|`/```````````/   | |```\           \```/           /```|
|    \```` ^ ````\|/```` ^ ````/    | |````\     ^     \`/     ^     /````|
|     \`````\`````|`````/`````/     | |`````\     \     o     /     /`````|
|      \`````\```/|\```/`````/      | |``````\     \   /`\   /     /``````|
|       o-----\-o | o-/-----o       | |```````o-----\-o```o-/-----o```````|
|              \  |  /              | |``````````````\`````/``````````````|
|               \ | /               | |```````````````\```/```````````````|
|                \|/                | |````````````````\`/````````````````|
|                 @                 | |`````````````````@`````````````````|
o-----------------------------------o o-----------------------------------o
 \                                 /   \                                 /
   \                             /       \                             /
     \         ((u)(v))        /           \        ((u, v))         /
       \                     /               \                     /
         \                 /                   \                 /
o----------\-------------/-----------------------\-------------/----------o
| X          \         /                           \         /            |
|              \     /                               \     /              |
|                \ /                                   \ /                |
|                 o----------------o   o----------------o                 |
|                /                  \ /                  \                |
|               /                    o                    \               |
|              /                    / \                    \              |
|             /                    /   \                    \             |
|            /                    /     \                    \            |
|           /                    /       \                    \           |
|          /                    /         \                    \          |
|         o                    o           o                    o         |
|         |                    |           |                    |         |
|         |                    |           |                    |         |
|         |         f          |           |          g         |         |
|         |                    |           |                    |         |
|         |                    |           |                    |         |
|         o                    o           o                    o         |
|          \                    \         /                    /          |
|           \                    \       /                    /           |
|            \                    \     /                    /            |
|             \                    \   /                    /             |
|              \                    \ /                    /              |
|               \                    o                    /               |
|                \                  / \                  /                |
|                 o----------------o   o----------------o                 |
|                                                                         |
|                                                                         |
|                                                                         |
o-------------------------------------------------------------------------o
Figure 69.  Difference Map of F = <f, g> = <((u)(v)), ((u, v))>

Diff Log D 2004 • Note 82


Transformations of Type B^2 -> B^2 (cont.)

Figure 70-a shows a graphical way of picturing the tangent functor map
dF = <df, dg> for the transformation F = <f, g> = <((u)(v)), ((u, v))>.
This amounts to the same information about df and dg that was given in
the computation summary of Tables 66-i and 66-ii, the relevant rows of
which are repeated here:

o-------------------------------------------------------------------------------o
|                                                                               |
|  df  =  uv.   0      +  u(v). du       +  (u)v.     dv   +  (u)(v).(du, dv)   |
|                                                                               |
|  dg  =  uv.(du, dv)  +  u(v).(du, dv)  +  (u)v.(du, dv)  +  (u)(v).(du, dv)   |
|                                                                               |
o-------------------------------------------------------------------------------o

                o                                   o
               / \                                 / \
              /   \                               /   \
             /     \                             /  O  \
            /       \                           o  /@\  o
           /         \                         / \     / \
          /           \                       /   \   /   \
         /      O      \                     /  O  \ /  O  \
        o      /@\      o                   o  /@\  o  /@\  o
       / \             / \                 / \   \ / \   \ / \
      /   \           /   \               /   \   /   \   /   \
     /     \         /     \             /  O  \ /  O  \ /  O  \
    /       \       /       \           o  /@   o  /@\  o  /@   o
   /         \     /         \         / \   \ / \     / \   \ / \
  /           \   /           \       /   \   /   \   /   \   /   \
 /      O      \ /      O      \     /  O  \ /  O  \ /  O  \ /  O  \
o      /@       o      /@       o   o  /@   o  /@   o  /@   o  /@   o
|\             / \             /|   |\     / \ /   / \ /   / \     /|
| \           /   \           / |   | \   /   \   /   \   /   \   / |
|  \         /     \         /  |   |  \ /  O  \ /  O  \ /  O  \ /  |
|   \       /       \       /   |   |   o  /@   o   @\  o  /@   o   |
|    \     /         \     /    |   |   |\ / \ / \     / \ / \ /|   |
|     \   /           \   /     |   |   | \   /   \   /   \   / |   |
| u    \ /      O      \ /    v |   | u |  \ /  O  \ /  O  \ /  | v |
o-------o       @\      o-------o   o---+---o   @\  o   @\  o---+---o
         \             /                |    \ / \ / \ / \ /    |
          \           /                 |     \   /   \   /     |
           \         /                  | du   \ /  O  \ /   dv |
            \       /                   o-------o   @\  o-------o
             \     /                             \     /
              \   /                               \   /
               \ /                                 \ /
                o                                   o
                     U%          $T$          $E$U%
                        o------------------>o
                        |                   |
                        |                   |
                        |                   |
                        |                   |
                     F  |                   | $T$F
                        |                   |
                        |                   |
                        |                   |
                        v                   v
                        o------------------>o
                     X%          $T$          $E$X%
                o                                   o
               / \                                 / \
              /   \                               /   \
             /     \                             /  O  \
            /       \                           o  /@\  o
           /         \                         / \     / \
          /           \                       /   \   /   \
         /      O      \                     /  O  \ /  O  \
        o      /@\      o                   o  /@\  o  /@\  o
       / \             / \                 / \   \ / \ /   / \
      /   \           /   \               /   \   /   \   /   \
     /     \         /     \             /  O  \ /  O  \ /  O  \
    /       \       /       \           o  /@   o  /@\  o   @\  o
   /         \     /         \         / \   \ / \ / \ / \ /   / \
  /           \   /           \       /   \   /   \   /   \   /   \
 /      O      \ /      O      \     /  O  \ /  O  \ /  O  \ /  O  \
o      /@       o       @\      o   o  /@   o  /@   o   @\  o   @\  o
|\             / \             /|   |\     / \ / \ / \ / \ / \     /|
| \           /   \           / |   | \   /   \   /   \   /   \   / |
|  \         /     \         /  |   |  \ /  O  \ /  O  \ /  O  \ /  |
|   \       /       \       /   |   |   o  /@   o   @   o   @\  o   |
|    \     /         \     /    |   |   |\ /   / \ / \ / \   \ /|   |
|     \   /           \   /     |   |   | \   /   \   /   \   / |   |
| x    \ /      O      \ /    y |   | x |  \ /  O  \ /  O  \ /  | y |
o-------o       @       o-------o   o---+---o   @   o   @   o---+---o
         \             /                |    \ /   / \   \ /    |
          \           /                 |     \   /   \   /     |
           \         /                  | dx   \ /  O  \ /   dy |
            \       /                   o-------o   @   o-------o
             \     /                             \     /
              \   /                               \   /
               \ /                                 \ /
                o                                   o
Figure 70-a.  Tangent Functor Diagram for F<u, v> = <((u)(v)), ((u, v))>

Diff Log D 2004 • Note 83


Transformations of Type B^2 -> B^2 (concl.)

Figure 70-b shows another way to picture the action of the tangent functor
on the logical transformation F<u, v> = <((u)(v)), ((u, v))>, roughly in
the style of the "bundle of universes" type of diagram.  [NB.  I can't
really do justice to the original Figure in ascii graphics, but this
collection of pictures may serve as a construction kit, with some
assembly required, to convey the general idea.]

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /////\ /////\     |  |     /XXXX\ /XXXX\     |  |     /\\\\\ /\\\\\     |
|    ///////o//////\    |  |    /XXXXXXoXXXXXX\    |  |    /\\\\\\o\\\\\\\    |
|   //////// \//////\   |  |   /XXXXXX/ \XXXXXX\   |  |   /\\\\\\/ \\\\\\\\   |
|  o///////   \//////o  |  |  oXXXXXX/   \XXXXXXo  |  |  o\\\\\\/   \\\\\\\o  |
|  |/////o     o/////|  |  |  |XXXXXo     oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  |/du//|     |//dv/|  |  |  |XXXXX|     |XXXXX|  |  |  |\du\\|     |\\dv\|  |
|  |/////o     o/////|  |  |  |XXXXXo     oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  o//////\   ///////o  |  |  oXXXXXX\   /XXXXXXo  |  |  o\\\\\\\   /\\\\\\o  |
|   \//////\ ////////   |  |   \XXXXXX\ /XXXXXX/   |  |   \\\\\\\\ /\\\\\\/   |
|    \//////o///////    |  |    \XXXXXXoXXXXXX/    |  |    \\\\\\\o\\\\\\/    |
|     \///// \/////     |  |     \XXXX/ \XXXX/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @ (u)(v)       o-----------------------o          dv' @ (u)(v)   =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /    \ /////\     |  |     /\\\\\ /XXXX\     |  |     /\\\\\ /\\\\\     |
|    /      o//////\    |  |    /\\\\\\oXXXXXX\    |  |    /\\\\\\o\\\\\\\    |
|   /      //\//////\   |  |   /\\\\\\//\XXXXXX\   |  |   /\\\\\\/ \\\\\\\\   |
|  o      ////\//////o  |  |  o\\\\\\////\XXXXXXo  |  |  o\\\\\\/   \\\\\\\o  |
|  |     o/////o/////|  |  |  |\\\\\o/////oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  | du  |/////|//dv/|  |  |  |\\\\\|/////|XXXXX|  |  |  |\du\\|     |\\dv\|  |
|  |     o/////o/////|  |  |  |\\\\\o/////oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  o      \//////////o  |  |  o\\\\\\\////XXXXXXo  |  |  o\\\\\\\   /\\\\\\o  |
|   \      \/////////   |  |   \\\\\\\\//XXXXXX/   |  |   \\\\\\\\ /\\\\\\/   |
|    \      o///////    |  |    \\\\\\\oXXXXXX/    |  |    \\\\\\\o\\\\\\/    |
|     \    / \/////     |  |     \\\\\/ \XXXX/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @ (u) v        o-----------------------o          dv' @ (u) v    =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /////\ /    \     |  |     /XXXX\ /\\\\\     |  |     /\\\\\ /\\\\\     |
|    ///////o      \    |  |    /XXXXXXo\\\\\\\    |  |    /\\\\\\o\\\\\\\    |
|   /////////\      \   |  |   /XXXXXX//\\\\\\\\   |  |   /\\\\\\/ \\\\\\\\   |
|  o//////////\      o  |  |  oXXXXXX////\\\\\\\o  |  |  o\\\\\\/   \\\\\\\o  |
|  |/////o/////o     |  |  |  |XXXXXo/////o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  |/du//|/////|  dv |  |  |  |XXXXX|/////|\\\\\|  |  |  |\du\\|     |\\dv\|  |
|  |/////o/////o     |  |  |  |XXXXXo/////o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  o//////\////      o  |  |  oXXXXXX\////\\\\\\o  |  |  o\\\\\\\   /\\\\\\o  |
|   \//////\//      /   |  |   \XXXXXX\//\\\\\\/   |  |   \\\\\\\\ /\\\\\\/   |
|    \//////o      /    |  |    \XXXXXXo\\\\\\/    |  |    \\\\\\\o\\\\\\/    |
|     \///// \    /     |  |     \XXXX/ \\\\\/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @  u (v)       o-----------------------o          dv' @  u (v)   =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /    \ /    \     |  |     /\\\\\ /\\\\\     |  |     /\\\\\ /\\\\\     |
|    /      o      \    |  |    /\\\\\\o\\\\\\\    |  |    /\\\\\\o\\\\\\\    |
|   /      / \      \   |  |   /\\\\\\/ \\\\\\\\   |  |   /\\\\\\/ \\\\\\\\   |
|  o      /   \      o  |  |  o\\\\\\/   \\\\\\\o  |  |  o\\\\\\/   \\\\\\\o  |
|  |     o     o     |  |  |  |\\\\\o     o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  | du  |     |  dv |  |  |  |\\\\\|     |\\\\\|  |  |  |\du\\|     |\\dv\|  |
|  |     o     o     |  |  |  |\\\\\o     o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  o      \   /      o  |  |  o\\\\\\\   /\\\\\\o  |  |  o\\\\\\\   /\\\\\\o  |
|   \      \ /      /   |  |   \\\\\\\\ /\\\\\\/   |  |   \\\\\\\\ /\\\\\\/   |
|    \      o      /    |  |    \\\\\\\o\\\\\\/    |  |    \\\\\\\o\\\\\\/    |
|     \    / \    /     |  |     \\\\\/ \\\\\/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @  u v         o-----------------------o          dv' @  u v     =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| U                     |  |\U\\\\\\\\\\\\\\\\\\\\\|  |\U\\\\\\\\\\\\\\\\\\\\\|
|      o--o   o--o      |  |\\\\\\o--o\\\o--o\\\\\\|  |\\\\\\o--o\\\o--o\\\\\\|
|     /////\ /////\     |  |\\\\\/////\\/////\\\\\\|  |\\\\\/    \\/    \\\\\\|
|    ///////o//////\    |  |\\\\///////o//////\\\\\|  |\\\\/      o      \\\\\|
|   /////////\//////\   |  |\\\////////X\//////\\\\|  |\\\/      /\\      \\\\|
|  o//////////\//////o  |  |\\o///////XXX\//////o\\|  |\\o      /\\\\      o\\|
|  |/////o/////o/////|  |  |\\|/////oXXXXXo/////|\\|  |\\|     o\\\\\o     |\\|
|  |//u//|/////|//v//|  |  |\\|//u//|XXXXX|//v//|\\|  |\\|  u  |\\\\\|  v  |\\|
|  |/////o/////o/////|  |  |\\|/////oXXXXXo/////|\\|  |\\|     o\\\\\o     |\\|
|  o//////\//////////o  |  |\\o//////\XXX///////o\\|  |\\o      \\\\/      o\\|
|   \//////\/////////   |  |\\\\//////\X////////\\\|  |\\\\      \\/      /\\\|
|    \//////o///////    |  |\\\\\//////o///////\\\\|  |\\\\\      o      /\\\\|
|     \///// \/////     |  |\\\\\\/////\\/////\\\\\|  |\\\\\\    /\\    /\\\\\|
|      o--o   o--o      |  |\\\\\\o--o\\\o--o\\\\\\|  |\\\\\\o--o\\\o--o\\\\\\|
|                       |  |\\\\\\\\\\\\\\\\\\\\\\\|  |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o  o-----------------------o  o-----------------------o
 =          u'             o-----------------------o              v'         =
  =                        | U'                    |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o
Figure 70-b.  Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>

Diff Log D 2004 • Note 84


Epilogue, Enchoiry, Exodus

| It is time to explain myself . . . . let us stand up.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 79]

Diff Log D 2004 • Appendices

Appendix 1-A. Operator Maps for the Disjunction f
Table A1. Computation of εf
Table A2. Computation of Ef
Table A3. Computation of Df (Method 1)
Table A4. Computation of Df (Method 2)
Table A5. Computation of df
Table A6. Computation of rf
Table A7. Computation Summary for f
Appendix 1-B. Operator Maps for the Equality g
Table B1. Computation of εg
Table B2. Computation of Eg
Table B3. Computation of Dg (Method 1)
Table B4. Computation of Dg (Method 2)
Table B6. Computation of rg
Table B7. Computation Summary for g
Appendix 2
EF Arranged by Differential Features
DF Arranged by Differential Features
EF Arranged by Original Features
DF Arranged by Original Features
Differential Forms : Expanded on a Logical Basis {(dA)(dB), dA(dB), (dA)dB, dA∙dB}
Differential Forms : Expanded on an Algebraic Basis {1, dA, dB, dA∙dB}
Pointwise Differential dF = Pointwise Linear Approximation to the Difference DF
Taylor Series Expansion
Partial & Relative Differentials
Detail of Calculation for DF = EF + F
Appendix 3

[Sophus Lie, 1880]

Problem.  To determine the most general function f of x and r parameters a1, a2, ... , ar satisfying an equation of the form

  f ( f (x, a1, ... , ar), b1, ... , br)   =   f (x, c1, ... , cr)

in which it is assumed that the ci depend only on the a's and the b's.

 This problem can perhaps be more clearly formulated by using the concept of a transformation group, which we now define.

Definition.  A family of transformations

  x? = f (x, a1, ... , ar),

where x? denotes the original variable, x the new one, and the ai parameters, forms a transformation group if the composition of two transformations of the family is a transformation of the family, i.e., when from the equations

  x? ? f (x, a1, ... , ar),

  x? ? f (x?, b1, ... , br),

there follows

  x? ? f (x, c1, ... , cr),

where the ci are functions of the a's and b's alone.  [p. 117]



 If in the equations

  xi? ? fi (x1, ... , xn,   a1, ... , ar), (i = 1, 2, ... , n),

one considers x1?, ... , xn? as original variables and x1, ... , xn as new variables and a1, ... , ar as parameters, then these equations define 8r transformations.  I say that such a family of transformations forms a group if the composition of two transformations of the family is again a transformation of the family, i.e., when from the equations

  xi? ? fi (x1, ... , xn,   a1, ... , ar)   =   fi (a),

  xi? ? fi (x1?, ... , xn?, b1, ... , br),

follows

  xi? ? fi (x1, ... , xn,   c1, ... , cr),

where c1, ... , cr depend only on the a's and the b's, and neither on x or the index i.  In other words, we require equations

  fi ( f1 (a), ... , fn (a),  b1, ... , br)   =   fi (x1, ... , xn,  c1, ... , cr).

      [p. 171]

[Sophus Lie, 1880]

 A transformation is said to be infinitesimal if it can be put in the form

  xi? = xi  +  Xi (x1, ... , xn).?t

where ?t is an infinitesimal.  Generally, we shall write such equations as

  ?xi = Xi (x1, ... , xn).?t .

 If one replaces x1, ... , xn by new variables, say y1, ... , yn, then our infinitesimal transformation assumes the form

  ?yi = ?t.?k ?yi/?xk X .

 On the other hand, if we make the same change of variables in the expression

  A(F) = X1.?F/?x1 + ... + Xn.?F/?xn ,

we get

  A(F) = ?F/?y1 ?k ?y1/?xk Xk + ... + ?F/?yn ?k ?yn/?xk Xk .

Thus we see that the equations of the infinitesimal transformation and the expression A(F) transform in the same way.  Therefore, it is analytically permissible to consider A(F) as the symbol of our infinitesimal transformation.

      [Lie, p. 177]

Appendix 4

Various Definitions of the Tangent Vector

Sources to be classified:

Refs:  [Sp65], [Sp79]

1st Approach.  Tangent vector as differential operator.

Refs:  [Che46, 76], [dCa, 6-7], [Hic, 5-6, 18], [KoA, 13]

2nd Approach.  Tangent vector as path classifier.

Refs:  [DoM]

3rd Approach.  Tangent vector via tangent functor.

Refs:  [JGH]

4th Approach.  Tangent vector as equivalence class of triples (U, ?, u), using chart maps ?, ?.

(U, ?, u) ˜ (V, ?, v) if "derivative" (differential) of transition map ? ° ?-1 at ?(x) takes u to v.

(? ° ?-1)? (?x) u  =  v

Helpful hint:  Read "(? ° ?-1)(x)" as:  "my name for what you call x",

or  "the new name for what I used to call x".

Refs:  [La83, 489], [La85, 26], [La93, 533], [Hir, 16-17]

References

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[KoA] Kosinski, A.A.  Differential Manifolds.  Academic Press, San Diego, CA, 1993.

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[La93] Lang, S.  Real and Functional Analysis.  3rd edition.  Springer-Verlag, New York, NY, 1993.

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[Lie84] Lie, S.  "Sophus Lie's 1884 Differential Invariant Paper".  In Lie Groups:  History, Frontiers, and Applications, Volume 3.  Translated by M. Ackerman.  Comments by R. Hermann.  Math Sci Press, Brookline, MA, 1976.  Original paper 1884.

[LoS] Loomis, L.H. & Sternberg, S.  Advanced Calculus.  Addison-Wesley, Reading, MA, 1968.

[Mel] Melzak, Z.A.  Mathematical Ideas, Modeling and Applications.  Volume 2 of Companion to Concrete Mathematics.  John Wiley & Sons, New York, NY, 1976.

[Men] Menabrea, L.F.  "Sketch of the Analytical Engine Invented by Charles Babbage"  With Notes by the Translator:  Ada Augusta (Byron), Countess of Lovelace.  In [M&M, 225-297].  Originally published 1842.

[M&M] Morrison, P. & Morrison, E. (eds.)  Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others.  With an Introduction by the Editors.  Dover, Mineola, NY, 1961.

[P1] Peirce, C.S.  Collected Papers of Charles Sanders Peirce.  Volumes 1-8.  Edited by:

C. Hartshorne, P. Weiss, & A.W. Burks.  Harvard University Press, Cambridge, MA, 1931-1960.  References are given as:  "CP [vol.].[paragraph]".

[P2] -.  "Qualitative Logic".  In The New Elements of Mathematics, Volume 4.  Edited by:  C. Eisele.  Mouton, The Hague, 1976.  References are given as:  "NE [vol.],[page]".

[Rob] Roberts, D.D.  The Existential Graphs of Charles S. Peirce.  Mouton, The Hague, 1973.

[SeH] Seldin, J.P. & Hindley, J.R. (eds.)  To H.B. Curry:  Essays on Combinatory Logic, Lambda Calculus, and Formalism.  Academic Press, London, UK, 1980.

[SpB] Spencer-Brown, G.  Laws of Form.  George Allen & Unwin, London, UK, 1969.

[Sp65] Spivak, M.  Calculus on Manifolds:  A Modern Approach to Classical Theorems of Advanced Calculus.  W.A. Benjamin, New York, NY, 1965.

[Sp79] Spivak, M.  A Comprehensive Introduction to Differential Geometry.  Volumes 1 & 2.  1st edition 1970.  2nd edition:  Publish or Perish Inc., Houston, TX, 1979.

[Sty] Styazhkin, N.I.  History of Mathematical Logic from Leibniz to Peano.  1st published in Russian:  Nauka, Moscow, 1964.  MIT Press, Cambridge, MA, 1969.

[Wie] Wiener, N.  Cybernetics:  or Control and Communication in the Animal and the Machine.  1st edition 1948.  2nd edition:  MIT Press, Cambridge, MA, 1961.

Works Consulted

[Ami] Amit, D.J.  Modeling Brain Function:  The World of Attractor Neural Networks.  Cambridge University Press, Cambridge, UK, 1989.

[Ed87] Edelman, G.M.  Neural Darwinism:  The Theory of Neuronal Group Selection.  Basic Books, New York, NY, 1987.

[Ed88] Edelman, G.M.  Topobiology:  An Introduction to Molecular Embryology.  Basic Books, New York, NY, 1988.

[Fla] Flanders, H.  Differential Forms with Applications to the Physical Sciences.  Dover, Mineola, NY, 1989.  First published:  Academic Press, 1963.

[Has] Hassoun, M.H. (ed.)  Associative Neural Memories:  Theory and Implementation.  Oxford University Press, New York, NY, 1993.

[KoB] Kosko, B.  Neural Networks and Fuzzy Systems:  A Dynamical Systems Approach to Machine Intelligence.  Prentice-Hall, Englewood Cliffs, NJ, 1992.

[MaB] Mac Lane, S. & Birkhoff, G.  Algebra.  3rd edition.  Chelsea, New York, NY, 1993.

[Mac] Mac Lane, S.  Categories for the Working Mathematician.  Springer-Verlag, New York, NY, 1971.

[McC] McCulloch, W.S.  Embodiments of Mind.  MIT Press, Cambridge, MA, 1965.

[Mc1] -.  "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, Vol. 7 (1945), pp. 89-93.  Reprinted in [McC].

[MiP] Minsky, M.L. & Papert, S.A.  Perceptrons:  An Introduction to Computational Geometry.  MIT Press, Cambridge, MA, 1969.  2nd printing 1972.  Expanded edition 1988.

[Rum] Rumelhart, D.E., Hinton, G.E., & McClelland, J.L.  "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, & the PDP Research Group, Parallel Distributed Processing;  Explorations in the Microstructure of Cognition, Volume 1:  Foundations.  MIT Press, Cambridge, MA, 1986.

Incidental Works

[Dew] Dewey, John.  How We Think.  Prometheus Books, Buffalo, NY, 1991.  Originally published:  D.C. Heath, Lexington, MA, 1910.

[Fou] Foucault, Michel.  The Archaeology of Knowledge & The Discourse on Language.  Translated by A.M. Sheridan-Smith & Rupert Swyer.  Pantheon, New York, NY, 1972.  Originally published as L'Archéologie du Savoir & L'ordre du discours, Editions Gallimard, 1969 & 1971.

[Hom] Homer.  The Odyssey.  With an English translation by A.T. Murray.  Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980.  First printed 1919.

[Jam] James, William.  Pragmatism:  A New Name for Some Old Ways of Thinking.  Longmans, Green, & Co., New York, NY, 1907.

[Ler] Leroux, Gaston.  The Phantom of the Opera.  Foreword by P. Haining.  Dorset Press, New York, NY, 1988.  Originally published in French, 1911.

[Mus] Musil, Robert.  The Man Without Qualities.  3 volumes.  Translated with a Foreword by Eithne Wilkins and Ernst Kaiser.  Pan Books, London, UK, 1979.  English edition first published by Secker & Warburg, 1954.  Originally published in German:  Der Mann ohne Eigenschaften, 1930 & 1932.

[PlaR] Plato.  The Republic.  With an English translation by Paul Shorey.  Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980.  First printed 1930 & 1935.

[PlaS] Plato.  The Sophist.  Loeb Classical Library, William Heinemann, London, 1921, 1987.

[Qui] Quine, W.V.  Mathematical Logic.  1st edition, 1940.  Revised edition, 1951.  Harvard University Press, Cambridge, MA, 1981.

[SaD] Santillana, Giorgio de, & Dechend, Hertha von.  Hamlet's Mill:  An Essay on Myth and the Frame of Time.  David R. Godine, Publisher, Boston, MA, 1977.  1st published 1969.

[Sha] Shakespeare, Wm.  William Shakespeare:  The Complete Works.  Compact Edition.  General editors:  S. Wells & G. Taylor.  Oxford University Press, Oxford, UK, 1988.

[Sh1] -.  A Midsummer Night's Dream.  Washington Square Press, New York, NY, 1958.

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[Web] Webster's Ninth New Collegiate Dictionary.  Merriam-Webster, Springfield, MA, 1983.

[Whi] Whitman, Walt.  Leaves of Grass.  Vintage Books / The Library of America, New York, NY, 1992.  Originally published in numerous editions, 1855-1892.

[Wil] Wilhelm, R. (trans.)  The I Ching, or Book of Changes.  Translated by R. Wilhelm & C.F. Baynes.  Foreword by C.G. Jung.  Preface by H. Wilhelm.  3rd edition:  Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.

Differential Logic 2004 • Work Area

Polymorphous Example (Edelman)


Table 4.  A Polymorphous Function
 _____________________________
|                     |       |
|                     |   T   |
|_____________________|_______|
|                     |       |
|    A     B     C    |   t   |
|    A     B   ( C )  |   t   |
|    A   ( B )   C    |   t   |
|    A   ( B ) ( C )  |   f   |
|  ( A )   B     C    |   t   |
|  ( A )   B   ( C )  |   f   |
|  ( A ) ( B )   C    |   f   |
|  ( A ) ( B ) ( C )  |   f   |
|_____________________|_______|

With these preliminaries we are ready to see how the
form of expression and means of evaluation used in our
version of ExG (Existential Graphs) may be related to
the usual pictures.  Looking at the initial columns of
the truth table, and having seen that the relation between
the heading and the rows is one of interpretation, we can
let the rows suffice to interpret themselves by replacing
their 1 and 0 values with their "A" and "not A" positions
regarding each proposition A in the column heads.  In our
current example, this gives the arrangement in Table 4.
Here we have used the one-slot operator to express "not A"
by "( A )".  Also, because the parser that we use in Study
will later require it, we have adopted the habit of terminating
each logical term (even those of one symbol) with a blank space.

| Note for later use:
|
| The parser for log files will accept <space> and
| also <end of line> or <carriage return> characters to
| terminate a word, but not <end of file> or <control-z>
| characters.  Extra spaces and lines, not within a word,
| may be added freely.

Figure 3.  A Polymorphous Function

T
  :
    \ ____ A ___ B ___ C     *
     |     |     |
     |     |     |__ ( C )   *
     |     |
     |     |__ ( B ) _ C     *
     |           |
     |           |__ ( C )   -
     |
     |__ ( A ) _ B ___ C     *
           |     |
           |     |__ ( C )   -
           |
           |__ ( B ) _ C     -
                 |
                 |__ ( C )   -

In our next transformation of the Truth table we take the rows
in their present order as though they were the word sequences
in some language, treating strings of the form "A" and "( A )"
as distinct words, and then we unify common initial segments
of these Strands to form a tree.  As the leaves of this tree
we take the truth functional values of the proposition, here
signifying "true" by "*" and "false" by "-".  This gives us
the tree form shown in Figure 3.

Figure 4.  A Polymorphous Function
 _____________________________
|                             |
| A                           |
|  B *                        |
|  (B )                       |
|   C *                       |
|   (C ) -                    |
| (A )                        |
|  B                          |
|   C *                       |
|   (C ) -                    |
|  (B ) -                     |
|_____________________________|


Finally we compress this tree into an outline form, as shown in Figure 4.
This is the actual output of the Model function in the Study section when
put to work on a log file containing the proposition:

(( A B )( B C )( C A )),

that is,

{ A and B } or { B and C } or { C and A }.

The process of making up log files will be taken up shortly,
but first we need to discuss some unexplained aspects of the
outline form just given.  You have probably noticed that the
outline produced for our current example did not cover all of
the branches of the preceding tree in equal detail or list all
of its possible paths.  Namely, it summarizes only the following:

 A    B         *
 A   (B )  C    *
 A   (B ) (C )  -
(A )  B    C    *
(A )  B   (C )  -
(A ) (B )       -

Thus, we can see that the tree actualized by Model
has only six leaves in comparison to the eight that
were possible.  This is because the Model function
has taken each path only as far as necessary to be
certain of its value under the truth function given.
In this case, the first and last paths did not have
to be taken through their final branch because their
values were already detectable.

This is exactly the kind of advantage in efficiency that we want
to make the maximum possible use of, since the work of exploring
2^N rows of a table or paths of a tree becomes quickly prohibitive.
But there are limits to how well we can do.  Some expressions are
just so complex, and there is a minimum complexity associated with
all equivalent ways of expressing them.  For many of the natural
examples however, the universes of discourse we actually find
ourselves needing to model, things are usually not so bad.

In these situations we can treat Propositional Calculus as a very
simple type of declarative programming language, not too rich in
what it can express but useful at any rate for certain jobs, with
the Model function acting as the evaluator or interpreter of the
language.  Seen in this light it is not so surprising that there
can be arbitrarily inefficient programs for the same task, the
question is whether we can find better ones.  For many of the
problems that arise spontaneously in prctice, and this can only
be an empirically defined class for now, it often happens that
more efficient descriptions can be found literally by adding more
variables, so long as these are used to supply more constraining
information on the space of models.

There is one other way that we can improve the procedure for evaluating
interpretations, that is, for finding the value of a proposition at the
end of a table row, a tree path, or a sequence of indented topic headings
in an outline.  In the original truth table, each row must list the variables
in the same order, but now that we have given each interpretation its own head
we can let them ride off in all directions at once.  In other words, once two
roads have diverged they need not visit the remaining places in the same order,
not when some advantage can be seen to doing otherwise.  These remarks will
become clearer when we see some larger examples where the Model function
has room to exploit this strategy.

Finally, before we leave our polymorphous example, there is one last aspect
of the Model function that we need to discuss, the subject of Normal Forms.
We have seen the concept of a logical model appear in several guises:  the
true interpretations, the rows of a truth table or the paths of a tree that
lead to a true value or that satisfy a proposition, the cells of an indicated
region in a Venn diagram.

Notice that we can use the set of models of a proposition to obtain a logically
equivalent proposition whose meaning across cases is manifestly clear, being
expressed by independent pieces.  We do this by transforming each model into
the conjunction of its features and then forming the disjunction of the whole
set of such conjuncts.  The result is called a Disjunctive Normal Form (DNF).
In our present example we get:

o-----------------------o             o--------------------------------o
|                       |             |                                |
|   ((   A    B         |   which     |   either {  A and  B  }        |
|   )(   A   (B )  C    |    means:   |     or   {  A and -B and C }   |
|   )(  (A )  B    C    |             |     or   { -A and  B and C }   |
|   ))                  |             |   <end>                        |
|                       |             |                                |
o-----------------------o             o--------------------------------o

Here we illustrate how the log file parser lets us arrange the disjuncts on
separate lines, and use extra spaces for ease of reading.  The DNF that Model
gave us here is probably not too impressive, since the initial expression was
already in such a form, and a shorter one at that -- still, it does the job.

There are a number of other functions in the Study section, all of which start
with the normal form produced by Model and derive various abstracts of it which,
depending on the proposition, may be complete enough and even more clear for most
purposes.  Rather than think up new qualifiers (like pseudo-, quasi-, semi-) for
these new forms, we will simply refer to them all as "normal forms" for now.

With this we arrive at our ultimate description of the type of logical
modeling process implemented here.  In sum, Study is a set of functions
for computing normal forms of propositions.  Their essential character
is that of clarifying expression while preserving meaning.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| References
|
| Angluin, Dana,
|"Learning with Hints", in:
|'Proceedings of the 1988 Workshop on Computational Learning Theory',
| edited by D. Haussler & L. Pitt, Morgan Kaufmann, San Mateo, CA, 1989.
|
| Peirce, C.S.,
|'Collected Papers of Charles Sanders Peirce',
| edited by C. Hartshorne, P. Weiss, & A.W. Burks, 8 Volumes,
| Harvard University Press, Cambridge, MA, 1931-1960.
| Cited in the form:  CP Volume.Paragraph.
|
| Spencer Brown, George, (1969),
|'Laws of Form', George Allen & Unwin, London, UK, 1969.
|
| Edelman, Gerald M., (1988),
|'Topobiology:  An Introduction to Molecular Embryology',
| Basic Books, New York, NY, 1988.
|
| McClelland, J.L. & Rumelhart, D.E., (1988),
|'Explorations in Parallel Distributed Processing:
| A Handbook of Models, Programs, and Exercises',
| MIT Press, Cambridge, MA, 1988.
|
| Maier, D. & Warren, D.S.,
|'Computing with Logic:  Logic Programming with Prolog',
| Benjamin/Cummings, Menlo Park, CA, 1988.
|
| Denning, P.J., Dennis, J.B., & Qualitz, J.E.,
|'Machines, Languages, & Computation',
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
|
| Lloyd, J.W.,
|'Foundations of Logic Programming',
| Springer-Verlag, Berlin, Germany, 1984.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

One way to do this is as follows.  Look at the logical features
that are mentioned in a proposition or in a set of propositions,
and take down their names as one's initial or minimal "alphabet"
of logical variable names, for example:

| For the single proposition,
|
| q = (( u v )( u w )( v w )),
|
| adopt the initial alphabet:
|
| #X# = {"u", "v", "w"}.

Given an "alphabet" #X# = {"x<1>", ..., "x<k>"}, whose terms may
be interpreted as names of logical features a<j>, for j = 1 to k,
we can use this as a basis to construct a concrete logical space,
notated as X = <#X#>, that can be defined in the following steps:

1.  Define the "i^th coordinate dimension" as X<i> = {"(x<i>)", "x<i>"}.
    For the moment, this is just a set that consists of two expressions.
    As interpreted, these expressions are particular ways of referring
    to values in B, where "(x<i>)" denotes the negation of the value
    of x<i>, and where "x<i>" denotes the value of x<i>, respectively.
    

2.  ...

Diff Log A 2004 • Omitted Material


Continuing to draw on the reduced example of group representations,
I would like to draw out a few of the finer points and problems of
regarding the maxim of pragmatism as a principle of representation.

Let us revisit the example of an abstract group that we had befour:

Table 1.  Klein Four-Group V_4
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    .    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o=========o=========o=========o=========o=========o
|         %         |         |         |         |
|    e    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    f    %    f    |    e    |    h    |    g    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    g    %    g    |    h    |    e    |    f    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    h    %    h    |    g    |    f    |    e    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o

I presented the regular post-representation
of the four-group V_4 in the following form:

Reading "+" as a logical disjunction:

   G  =  e  +  f  +  g  + h

And so, by expanding effects, we get:

   G  =  e:e  +  f:f  +  g:g  +  h:h

      +  e:f  +  f:e  +  g:h  +  h:g

      +  e:g  +  f:h  +  g:e  +  h:f

      +  e:h  +  f:g  +  g:f  +  h:e

This presents the group in one big bunch,
and there are occasions when one regards
it this way, but that is not the typical
form of presentation that we'd encounter.
More likely, the story would go a little
bit like this:

I cannot remember any of my math teachers
ever invoking the pragmatic maxim by name,
but it would be a very regular occurrence
for such mentors and tutors to set up the
subject in this wise:  Suppose you forget
what a given abstract group element means,
that is, in effect, 'what it is'.  Then a
sure way to jog your sense of 'what it is'
is to build a regular representation from
the formal materials that are necessarily
left lying about on that abstraction site.

Working through the construction for each
one of the four group elements, we arrive
at the following exegeses of their senses,
giving their regular post-representations:

   e  =  e:e  +  f:f  +  g:g  +  h:h

   f  =  e:f  +  f:e  +  g:h  +  h:g

   g  =  e:g  +  f:h  +  g:e  +  h:f

   h  =  e:h  +  f:g  +  g:f  +  h:e

So if somebody asks you, say, "What is g?",
you can say, "I don't know for certain but
in practice its effects go a bit like this:
Converting e to g, f to h, g to e, h to f".

I will have to check this out later on, but my impression is
that Peirce tended to lean toward the other brand of regular,
the "second", the "left", or the "ante-representation" of the
groups that he treated in his earliest manuscripts and papers.
I believe that this was because he thought of the actions on
the pattern of dyadic relative terms like the "aftermath of".

Working through this alternative for each
one of the four group elements, we arrive
at the following exegeses of their senses,
giving their regular ante-representations:

   e  =  e:e  +  f:f  +  g:g  +  h:h

   f  =  f:e  +  e:f  +  h:g  +  g:h

   g  =  g:e  +  h:f  +  e:g  +  f:h

   h  =  h:e  +  g:f  +  f:g  +  e:h

Your paraphrastic interpretation of what this all
means would come out precisely the same as before.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Erratum

Oops!  I think that I have just confounded two entirely different issues:

1.  The substantial difference between right and left regular representations.

2.  The inessential difference between two conventions of presenting matrices.

I will sort this out and correct it later, as need be.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

I have been planning for quite some time now to make my return to Peirce's
skyshaking "Description of a Notation for the Logic of Relatives" (1870),
and I can see that it's just about time to get down tuit, so let this
current bit of rambling inquiry function as the preamble to that.
All we need at the present, though, is a modus vivendi/operandi
for telling what is substantial from what is inessential in
the brook between symbolic conceits and dramatic actions
that we find afforded by means of the pragmatic maxim.

Back to our "subinstance", the example in support of our first example.
I will now reconstruct it in a way that may prove to be less confusing.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| Bein' on the twenty-third of June,
|      As I sat weaving all at my loom,
| Bein' on the twenty-third of June,
|      As I sat weaving all at my loom,
| I heard a thrush, singing on yon bush,
|      And the song she sang was The Jug of Punch.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| In the beginning was the three-pointed star,
| One smile of light across the empty face;
| One bough of bone across the rooting air,
| The substance forked that marrowed the first sun;
| And, burning ciphers on the round of space,
| Heaven and hell mixed as they spun.
|
| Dylan Thomas, "In The Beginning", Verse 1

I'm afrayed that this thread is just bound to keep
encountering its manifold of tensuous distractions,
but I'd like to try and return now to the topic of
inquiry, espectrally viewed in differential aspect.

Here's one picture of how it begins,
one angle on the point of departure:

o-----------------------------------------------------------o
|                                                           |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |                         |                |
|                |                         |                |
|                |       Observation       |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \          \ /          /    \            |
|           /      \   d_I ^  o  ^ d_E   /      \           |
|          /        \       \/ \/       /        \          |
|         /          \      /\ /\      /          \         |
|        /            \    /  @  \    /            \        |
|       o              o--o---|---o--o              o       |
|       |                 |   |   |                 |       |
|       |                 |   v   |                 |       |
|       |   Expectation   |  d_O  |    Intention    |       |
|       |                 |       |                 |       |
|       |                 |       |                 |       |
|       o                 o       o                 o       |
|        \                 \     /                 /        |
|         \                 \   /                 /         |
|          \                 \ /                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o

From what we must assume was a state of "Unconscious Nirvana" (UN),
since we do not acutely become conscious until after we are exiled
from that garden of our blissful innocence, where our Expectations,
our Intentions, our Observations all subsist in a state of perfect
harmony, one with every barely perceived other, something intrudes
on that scene of paradise to knock us out of that blessed isle and
to trouble our countenance forever after at the retrospect thereof.

The least disturbance, it being provident and prudent both to take
that first up, will arise in just one of three ways, in accord with
the mode of discord that importunes on our equanimity, whether it is
Expectation, Intention, Observation that incipiently incites the riot,
departing as it will from congruence with the other two modes of being.

In short, we cross just one of the three lines that border on the center,
or perhaps it is better to say that the objective situation transits one
of the chordal bounds of harmony, for the moment marked as d_E, d_I, d_O
to note the fact one's Expectation, Intention, Observation, respectively,
is the mode that we duly indite as the one that's sounding the sour note.

A difference between Expectation and Observation is experienced
as a "Surprise", a phenomenon that cries out for an Explanation.

A discrepancy between Intention and Observation is experienced
as a "Problem", of the species that calls for a Plan of Action.

I can remember that I once thought up what I thought up an apt
name for a gap between Expectation and Intention, but I cannot
recall what it was, nor yet find the notes where I recorded it.

At any rate, the modes of experiencing a surprising phenomenon
or a problematic situation, as described just now, are already
complex modalities, and will need to be analyzed further if we
want to relate them to the minimal changes d_E, d_I, d_O.  Let
me think about that for a little while and see what transpires.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| In the beginning was the pale signature,
| Three-syllabled and starry as the smile;
| And after came the imprints on the water,
| Stamp of the minted face upon the moon;
| The blood that touched the crosstree and the grail
| Touched the first cloud and left a sign.
|
| Dylan Thomas, "In The Beginning", Verse 2

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| In the beginning was the mounting fire
| That set alight the weathers from a spark,
| A three-eyed, red-eyed spark, blunt as a flower;
| Life rose and spouted from the rolling seas,
| Burst in the roots, pumped from the earth and rock
| The secret oils that drive the grass.
|
| Dylan Thomas, "In The Beginning", Verse 3

Diff Log B 2004 • Omitted Material

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| Cast of Characters
|
| @A@  =  attic A = greek alpha,
| #A#  =  bold  A,
| $A$  =  curly A = script A,
| !A!  =  singly underscored A,
| %A%  =  doubly underscored A,
| x^j  =  x superscript j,
| x_j  =  x subscript j.
|
| The part of @a@, the lower case greek,
| especially in the mathematical scenes,
| is often played by the underscore !a!.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Speaking of initial conditions --

Though, to speak in truth, if come the day,
when do such as we, or all the likes of us,
ever truly speak of our initial conditions,
sensitive toward them how ever may we grow?
So let me then speak of initial conditions
in just the way I wit then I ultimately do,
in medias res, in thick midsts of the plot.
For my part there is much that begins here:

[Ashby Quote]

I will now make yet another attempt to re-introduce this subject
to which I have alluded before under some of the following names:
"Analytic Differential Ontology" (ADO), when we imagine it to be
a question of being itself, or the "Differential Extension" (DE)
of logic, for now, just the propositional or the sentential type
of logic, specifically, for my part, as expressed in what I call
a "Reflective Extension" (RE) of Peirce's Logical Graphs, in sum,
named "DiffLog" and "RefLog", respectively.  I will apologize at
this point for all of the "rude mnemonic mechanics", but I think
that you can probably guess by now that this is utterly the only
way that I have found to keep track of all this stuff in my head.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

To make the present discussion more self-contained, I will need to insert
my discussion of an example from the psych research literature of the times,
what was known in the cognitive science, immune system models, and neural net
circles as a "polymorphous set".

In some parts of this old discussion, I find myself occasionally guilty
of engaging in the rather sloppy practice of using the same symbol for
any of the correlated notions of a concept, a set, or a truth function.
Please be forgiving of this less sad and less wise person who did this.
I have tried to go back through the text, marking the concept symbols
in bold letters, as #Q#, the proposition names or truth-function names
in plain lower case, as q, and the set names in plain capitals, as Q,
but I may have missed a few, and I sometimes omit the extra markings
in figures and tables.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Notation

In the process of editing this uncolorized version,
I have tried to fix several problems with notation
that are begining to become a bother in this plain
courier context, like the excessive "busy-ness" of
the letter "B" being used not only for the boolean
domain B = {0, 1}, but also for a logical variable.

I am also switching to the more usual mathematical
convention of designating the more global space of
one's immediate interest with the letter "X", thus
leaving the letters "U", "V", "W", and so on, free
to designate the more local and transitory patches
of spaces that will come to occupy one's attention
in the moment to moment concern with the situation
that is given as the site of one's manifold senses.

To save capital letters for sets and spaces, in so
far as it may be convenient, I will default to the
use of lower case letters for variables or indices
whenever possible, especially in abstract examples.

Just as an experiment, as I am not too sure of how
it will work out if pursued in a persistent manner,
whenever a sign like "u" serves as a logical index,
by which I mean the index of a logical proposition,
in which case you may notice that it is very usual
to treat "u" as being subject to the extra reading
by which it can also denote a function, that is to
say, a proposition of the form u : X -> B = {0, 1},
then I will invest the associated capital with the
meaning of the antecedent pre-image of 1 under the
function, what some call the "fiber of truth" in u,
in summary, accordingly, writing "U = (u^(-1))(1)".

Finally, notice that we have two ways of referring
to particular points of a logical space X, what we
may on divers occasions elect to treat as cells in
a venn diagram, as boolean vectors in a coordinate
space B^k, or as rows of values in a truth tableau,
namely, we can indicate such a particular point as
the vector space sum, x = <x1, x2, ..., xk>, or as
the conjunctive product x = x1 x2  ...  xk, and so
I will quite freely switch between these two forms.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

The next thing that we typically do is consider the effects of
various operators on our target proposition -- and, by the way,
not too coincidentally, I will need to shift our paradigm from
the psychologist's to what is more akin to the mathematician's
way of using the terms "source" and "target" -- and so I shall
begin to speak about the "operand" or the "source" proposition,
instead of the "target" proposition as I have been up till now.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Subj:  Differential Logic
Date:  Fri, 06 Jun 2003 09:24:06 -0400
From:  Jon Awbrey <jawbrey@oakland.edu>
  To:  SUO <standard-upper-ontology@ieee.org>
  CC:  Inquiry <inquiry@stderr.org>, SemioCom <gdsemiocom@univ-perp.fr>

One way to think about mappings between different ontologies, and also about
ontologies that develop through time -- the two problems are intimately related --
is in terms of transformations from universe of discourse to universe of discourse,
the sort of thing that one is naturally tempted to call a "transformation of discourse".

Re-starting from the ground up, as experience constantly teaches that we must,
we may contemplate simple propositions and unanalyzed predications of the sort
that one finds pictured in euler-venn diagrams and that one computes in terms
of bits and boolean functions.

So, working at the basement level, a mapping between two different
universes of discourse, which may of course be only two different
ways of describing the same universe of discourse, could be written
as F : U -> V, and a transformation that describes the changes that
occur in a single universe of discourse, which may of course come in
the corresponding varieties of "alias" and "alibi" flavors, could be
written as F : U -> U.

Transformations like these can be very complex things to think about --
for instance, one may be thinking of a neuroid system of formal neurons
that carry one bit each, and so one's universe is a state space U that is
isomorphic to B^n, where B = {0, 1} and n is roughly 10^10, give or take --
so we usually end up having to approach such creatures, the transformations
F : U -> U and F : X -> Y, in a series of increasing orders of approximation.

That is what differential calculus is all about.  A "derivative" or a "differential"
of a transformation F is a "locally linear approximation" to F.  In many ways, one
can think of differentiation as an operation that takes the global description of
a transformation and distributes the information into locally relevant forms.

These days, differential calculus and differential geometry are carried out in terms
of a thing called the "tangent functor", which is the category theoretic expression
of what we do when we take derivatives.  A "functor" W is a "mapping of maps" or
a "transformation of transformations", so W takes a map F : X -> Y into another
map WF : WX -> WY.  Roughly speaking then, the particular sort of functor that
we will soon know and love as the "tangent functor" T is one that takes the
map F : X -> Y and gives back the locally relevant version TF : TX -> TY.

Cranking the analogy for logic produces the subject of "differential logic",
which has been my pursuit for a decade or two.  I am almost done serializing
one of my more coherent, but also more detailed, papers on the subject, and
I have appended the outline of links so far.

Speaking of initial conditions --

Though, to speak in truth, if come the day,
When do such as we, or all the likes of us,
Ever truly speak of our initial conditions,
Sensitive toward them how ever we may grow?
So let me then speak of initial conditions
In just the way I wit then I ultimately do,
In medias res, in thick midsts of the plot,
For my part there is much that begins here:

| The most fundamental concept in cybernetics is that of "difference",
| either that two things are recognisably different or that one thing
| has changed with time.
|
| Ashby, W. Ross,
|'An Introduction to Cybernetics',
| Chapman & Hall, London, UK, 1956,
| Methuen & Company, London, UK, 1964,
| Page 9.

The quickest way for me to open up this topic is just to jump in,
and so I will cite here my very first outline of it, that formed
an appendix to my Master's (in Psych) Thesis Substitute Document.
I am hoping that the very simplicity of this will make it useful
as an initial invitation, aided by the circumstance that I wrote
it in such a way as to form a gentle bridge between the ordinary
difference calculus, executed over the reals R or the integers Z,
and the logical difference calculus, working over the booleans B.
Accordingly, all of the definitions, equations, expressions, and
formulas in the following presentation can be read independently
of whether you interpret the ostensible names as denoting values
in R, or Z, or B.

Diff Log B 2004 • Work Area 1


"vague object impressions" (VOI's) or you might
say "hints of objects" (HOO's)

"vague impressions of objects" (VIOO's)
"impressions of vague objects" (IOVO's)

ambit, a unit of ambiguous information
that's not a bit till it's interpreted.

SPONS

ACOYA ("a couple of years ago")

pro
epi

note
rest

rock
roll

bound
blank

spike
stone

active
stable

mobile
static

moving
steady

charged
resting

working
playing
resting

changing
enduring

creation
devotion

creating
reciting

producing
receiving

evanescence
equilibrium

Diff Log B 2004 • Work Area 2


Put Off 2.  

What happened to the fabled distinction
between individuals and predicates?

Differential Logic 2004 • Discussion

Differential Logic 2004 • Series A • Discussion

Diff Log A 2004 • Discussion Note 1


GR = Gary Richmond
JA = Jon Awbrey
JS = John Sowa

Re: DLOG A10.  http://suo.ieee.org/ontology/msg05373.html

In Texas they tell a variety of joke that goes a bit like this:

| Q.  What do you do when your 100-dollar 10-gallon hat
|     blows off in a dust-storm?
|
| A.  Reach up in the air and pull down another one.

The story that I called a "genealogy" not a "history"
has to do with the sorts of ideas that are always in
the air and only occasionally get seized on in novel
fashions.  Or call it a Hegelian history if you like.

It is customary to give Bentham first billing for
the "paraphrasis" idea, so I went along with that.
Quine's comment in Van Heijenoort gives additional
snippets about Schonfinkel's "Bausteine".  A very
enjoyable way to learn about combinator calculus
is provided by the second half of Ray Smullyan's
'To Mock a Mockingbird'.  Folks usually give Curry
and Church credit for independently rediscovering
what are more or less computationally equivalent
ideas in the various lambda calculi.  I tend to be
suspicious of how independent anybody can be from
their collective unconscious background/culture,
but that's just me.

B. and/or C. Peirce still get credit, I haven't been
able to sort out which deserves the lion's share yet,
for a very general form of algebraic representation
principle that's been a blowin' in the same wind
more or less since about the days of Galois.

Somewhere in the mid 1970's I figured out the relationship between
the pragmatic maxim and these representation principles, and that
has been the important thing to me since then.  Mathematics and
physics are still just about the only places where something
like the prag-maxim gets applied on a routine basis, most
often by people who never heard of it under that name.

I see John's note covers most of the other questions.

CSP: | Consider what effects that might 'conceivably'
     | have practical bearings you 'conceive' the
     | objects of your 'conception' to have.  Then,
     | your 'conception' of those effects is the
     | whole of your 'conception' of the object.
     |
     | Peirce, "Maxim of Pragmaticism",
     | 'Collected Papers', CP 5.438.

JA: The genealogy of this conception of pragmatic representation is very intricate.
    I will delineate some details that I presently fancy I remember clearly enough,
    subject to later correction.  Without checking historical accounts, I will not
    be able to pin down anything like a real chronology, but most of these notions
    were standard furnishings of the 19th Century mathematical study, and only the
    last few items date as late as the 1920's.

JA: The idea about the regular representations of a group is universally known
    as "Cayley's Theorem", usually in the form:  "Every group is isomorphic to
    a subgroup of Aut(X), the group of automorphisms of an appropriate set X".
    There is a considerable generalization of these regular representations to
    a broad class of relational algebraic systems in Peirce's earliest papers.
    The crux of the whole idea is this:

JA: Contemplate the effects of the symbol
    whose meaning you wish to investigate
    as they play out on all the stages of
    conduct on which you have the ability
    to imagine that symbol playing a role.

JA: This idea of contextual definition is basically the same as Jeremy Bentham's
    notion of "paraphrasis", a "method of accounting for fictions by explaining
    various purported terms away" (Quine, in Van Heijenoort, page 216).  Today
    we'd call these constructions "term models".  This, again, is the big idea
    behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus,
    and I reckon you know where that leads.

GR: I managed to follow this discussion pretty well,
    except for your last comment.

JA: This, again, is the big idea behind Schönfinkel's combinators {S, K, I},
    and hence of lambda calculus, and I reckon you know where that leads.

GR: Confronted with *lambda calculus*, I think first of what John Sowa has
    written (see, especially, 'Conceptual Structures' [1984], 162-3, 373-4).
    A few matters I'm quite unclear about:

GR: 1. Sowa describes the lambda calculus in this way (I'm starting
       in media res, with comments involving a shipping example):

GR, quoting JS:

JS: | The denotations operation [delta] would search the database to find
    | some part x, date y, and shipment z that would make the predicates
    | in the body of the formula true.  If the denotation is true, then
    | the answer to the question is yes. If the denotation is false,
    | then the answer is no.
    |
    | For a wh-type type question *What suppliers shipped parts to Dept. 85?*,
    | the answer is a set of suppliers.  In symbolic logic, a yes-no question
    | corresponds to a proposition where every variable is *bound* by a quantifier.
    | A wh-question, however, is mapped to a *lambda expressions* with one or more
    | parameters ... 162 [the text diagrams this situation through symbolic logic
    | and a conceptual graph] 162-3

GR: The upshot of this is:

JS: | The denotation of the lambda expression is not a truth value like
    | true or false, but rather the set of all instances of SUPPLIERs that
    | could be substituted for w to make the body of the expression true.
    | Lambda calculus combined with symbolic logic makes a powerful database
    | query language. 163

GR: What do you think of this description, which really constitutes a kind
    of definition?  Is it yours?  Is it pointing to what your last, perhaps
    rhetorical, question is meant to point to?  (Collateral knowledge needed,
    Jon.  Less literary device, more information please  ;-)

GR: 2. In the second passage discussing the lambda calculus there 
       is no mention made of Schönfinkel in Sowa's discussion of
       the history of the lambda calculus.

GR, quoting JS:

JS: | A definition by extension is only possible when the domain A is finite.
    | In all other cases, the function must be defined by a rule, which is
    | called the *intension* of f.  (One could, of course, define the
    | extension of a function as an infinite set, but the set itself
    | would have to be defined by some rule or intension.)
    |
    | Defining a function by a rule is more natural  or intuitive than defining
    | it as a set of ordered pairs.  But a problem can occur when two or more
    | different rules or intensions lead to the same sets of ordered pairs or
    | extensions.  Are two functions considered the same if they have the same
    | sets of ordered pairs, but different mapping rules?  To distinguish the
    | intension and extension of functions and to formalize the rules for
    | defining them, Alonzo Church (1941) developed a system called the
    | *lambda calculus* which uses the Greek letter [lambda] to indicate
    | the parameters of a function 373]

GR: So what's the exact intellectual history here?  I it connected to Peirce
    in any way?  John's earlier discussion hints at this being a triadic logic
    in the Peircean sense (yes/no being insufficient for the particular purpose
    at hand -- see the shipping example above) though I don't believe he explicitly
    states it as such.  Later he comments:

JS: | An important advantage of the lambda notation is that it defines
    | a mapping independently of the act of naming it.  As a result, an
    | unnamed lambda expression can be used anywhere that a function name
    | could be used.  This feature is especially useful for applications
    | that create new functions dynamically and then immediately pass them
    | as arguments to another function. [e.g., some database systems]

GR: John concludes this analysis thus:

JS: An important result of the lambda calculus is the Church-Rosser theorem:
    when more than one function in an expression is expandable, the order of
    expansion is irrelevant because the same canonical form would be obtained
    with any sequence of expansion.  [He then points to Sect. 3.6 of his text
    relating this to graphs] 374

GR: What do you think about all this (perhaps even beyond databases)?
    What are the pragmatic import of the lambda calculus?  [Btw, I have
    not yet read much of John's most recent textbook, so perhaps he's
    commented on some of these matters there.]

GR: If my questions appear naive, I hope you will realize that I am a mere
    beginning student of logic hoping to get some light thrown on matters
    that look to me to be of potential (pragmatic) value.

GR: Can Peirce's Gamma graphs be related to any of this, btw?

Diff Log A 2004 • Discussion Note 2


RM = Richard Martin

Re: DLOG A10.  http://suo.ieee.org/ontology/msg05373.html

CSP: | Consider what effects that might conceivably
     | have practical bearings you conceive the
     | objects of your conception to have.  Then,
     | your conception of those effects is the
     | whole of your conception of the object.
     |
     | Peirce, "Maxim of Pragmatism",
     | 'Collected Papers', CP 5.438.

RM: I've been pondering this maxim again.  If I paraphrase it thus,

    "My concept of an object's effects is my concept of the object",

    am I even close to capturing CSP's intent?

    Even simpler,

    "My concept of an object is my comprehension of its effects".

    Am I missing something more profound?  The first sentence of
    the maxim seems to ask no more than to cast "effects" in the
    broadest of terms.

It seems that both of the list servers/archivers that I'm using for this
are having problems at the moment, and I have to run out in a second and
do some errands, so I will just send out this quick echo now, as a test,
and try to get back to your questions later today.

Diff Log A 2004 • Discussion Note 3


RM = Richard Martin

Re: DLOG A10.  http://suo.ieee.org/ontology/msg05373.html

CSP: | Consider what effects that might conceivably
     | have practical bearings you conceive the
     | objects of your conception to have.  Then,
     | your conception of those effects is the
     | whole of your conception of the object.
     |
     | Peirce, "Maxim of Pragmatism",
     | 'Collected Papers', CP 5.438.

RM: I've been pondering this maxim again.  If I paraphrase it thus,

    "My concept of an object's effects is my concept of the object",

    am I even close to capturing CSP's intent?

    Even simpler,

    "My concept of an object is my comprehension of its effects".

    Am I missing something more profound?  The first sentence of
    the maxim seems to ask no more than to cast "effects" in the
    broadest of terms.

Let me sum up my own rough sense of what I think Peirce is trying to do here.
He is trying to identify the critical difference between methods of inquiry
that work and those that do not, between those that get us somewhere down
the road to greater knowledge and those that just seem to go in circles
forever.  Experience and history provide lots of examples of both types,
always provisionally classified, of course, but there has always been
a problem about telling the effective ingredients from the excipients.
The maxim expresses one of Peirce's best guesses about what makes the
difference, stating it in the form of a practical heuristic that is
meant to serve as a guide under all the perplexities of realistic
field conditions.

Part of the utility of a practical heuristic, a rule of thumb,
depends on it implementing a proportionate type of continuity,
where using an approximation to the exact rule on a sample of
the conceivable data will be proportionately satisficing and
not just utterly useless or totally misleading.  Robustness,
I think they'd call it today.

I think what you said above makes a good first approximation,
one that I frequently use myself, and one that will serve as
a sufficient guideline in the great majority of applications.

The way I read it, the Pragmatic Maxim describes and recommends
a certain technique for reflecting on, critically examining, and
thereby facilitating the clarification and continuous development
of one's concepts.

At first strike, I see a rule that reminds me of a closure principle.
Ordinarily, a closure operator C satisfies a law like C(C(X)) = C(X).
But here we have a principle that has the rough shape C(E(X)) = C(X).

In words:  My conception of the effects of X is my conception of X.

Said so succinctly, though, it leaves out a lot of important details,
and it's almost impossible to figure out what good such a purported
platitude would be unless you look at concrete examples of its use.

I started one, but it got more involved
than I can track this time of the night,
so I will leave it to the morning light.

P.S.  I'm posting this series to the Inquiry List also,
but there's some kind of echo in the Ontology channel,
which forced me to isolate it from my other CC lists.
Also, the Ontology server is taking about 12 hours
to distribute posts, so I will copy you directly.

Diff Log A 2004 • Discussion Note 4


In trying to work through this version of the Pragmatic Maxim one more time,
I begin to understand another one of those problems that most of us have in
reading Peirce.  As a person who read Kant's 'Critique of Pure Reason' up,
down, sideways, and, yes, even forward at the age of 13, Peirce sometimes
writes as if everybody else has too.  Whereas a person like me, who has
made two stabs at the buch a decade apart, is still only two thirds of
the way through it.  Still, I can dimly grasp that many of the things
Peirce doesn't say in his statement he doesn't say because he thinks
that Kant already painted the background in illustratively enough.

One of the bits of fading scenery that I just now noticed is a distinctive
ambiguity of force or mood in the use of any maxim for reflective critique,
that is, with the aim to bring deliberate reflection to train on a present
condition but also to promote a beneficial change of that condition in the
future, and thus the effective value of a maxim compounds both descriptive
and normative elements.

In the present case, the Pragmatic Maxim says something about how to recognize
our present concept of an object when we see it, but also how to improve on it.

Putting those generalities back in the background,
though not entirely wishing to forget them, I'll
try once again to get down to details next time.

Diff Log A 2004 • Discussion Note 5


| Consider what effects that might conceivably
| have practical bearings you conceive the
| objects of your conception to have.  Then,
| your conception of those effects is the
| whole of your conception of the object.
|
| Peirce, "Maxim of Pragmatism",
| 'Collected Papers', CP 5.438.

Let me see what happens if I try to follow Peirce's instructions
in the sort of context where they were meant to be used, that is,
to clarify the meaning of a concept.

Interpreter J picks a concept y whose meaning J wants to clarify.

For the sake of the exercise, let us say that y has some object x.
We can safely suppose this without making any real commitments of
the ontological sort, since we can always abandon the supposition
if it leads to absurd consequences, whether logical or practical.

As far as what a concept is, Peirce follows a classical tradition
that regards a concept as a mental sign.  More exactly, a concept
is a mental symbol.  Now, a symbol is a type of sign that denotes
whatever objects it does just because some interpreter interprets
it as doing so.  Let's put off saying what that means until later,
but it is bound to involve Peirce's definition of a sign relation,
and so knowing that much allows us to draw a picture of this sort:

                  y
                 o
                /
               /
   x o--------@
               \
                \
                 o
                  z

This places the 3-ple <x, y, z> of the form <object, sign, sign'>
in the context of a suitable sign relation L c !O! x !S! x !I!,
where !O!, !S!, and !I! are the relational domains of objects,
signs, and interpretant signs available to the interpreter J.

Regarding y as a sign, J can ask whether y
has objects, and what its objects might be.

Regarding y as a symbol, what objects it has
depends on its interpreters, for instance, J.
That is, it depends on its interpreters in a
way that is "essential" and not eliminatable,
not without loss of generativity as a symbol.

For any interpreter K, let C_K (x) be K's concept of x.
For instance, in the present case, we have C_J (x) = y.

Relative to the object x of the concept y, the maxim advises J to consider
the set of effects, that might conceivably have practical bearings, that J
conceives the object x to have.  Let E_J (x) be this collection of effects,
that might conceivably have practical bearings, that J conceives x to have.

The entities and relationships that we've seen so far
may be sketched in the form of the following diagram:

                  o y = C_J (x)
                 /
                /
    x o--------@
      .         \
      .          \
      .           o z
      .
      .
      .
      .           o C_J (E_J (x))
      .          /
      .         /
      o--------@
   E_J (x)      \
                 \
                  o

For the sake of a first approximation to the maxim,
I am overlooking the subtleties that may be lurking
in the proviso "conceivably have practical bearing".

Apart from the catches of that one remaining wrinkle,
the maxim apparently suggests some sort of descriptive
or normative equation between C_J (x) and C_J (E_J (x)).

Now, what possible use could such a formula have,
when it comes to revealing the meaning of C_J (x)?

One thing that comes to mind right off is the similarity of the
diagram that I drew above to the kinds that I commonly draw in
cases of functions defined by recursion.  What I have in mind
here is the type of function whose value on "complicated"
arguments is arrived at by way of a specified relation
to its values on "simpler" arguments.

If this form of analogy is apt, then C_J would be analogous to
the recursive function in question, and E_J would be analogous
to the recursion relation between complex and simple arguments.
Of course, it has to be kept in mind that we are talking about
structures more general than functions, namely, sign relations.

But I think that I will sleep on it now
and explore this idea further tomorrow.

Diff Log A 2004 • Discussion Note 6


| Consider what effects that might conceivably
| have practical bearings you conceive the
| objects of your conception to have.  Then,
| your conception of those effects is the
| whole of your conception of the object.
|
| Peirce, "Maxim of Pragmatism",
| 'Collected Papers', CP 5.438.

We left off last time with this picture:

                  o y = C_J (x)
                 /
                /
    x o--------@
      .         \
      .          \
      .           o z
      .
      .
      .
      .           o C_J (E_J (x))
      .          /
      .         /
      o--------@
   E_J (x)      \
                 \
                  o

We have the following legend for the labels:

   C_J (x)  =  J's concept of the object x.

   E_J (x)  =  the [set of] effects, that might conceivably have
               practical bearings, that J conceives x to have.

   C_J (E_J (x)))  =  J's concept of the [set of] effects,
                      that might conceivably have practical
                      bearings, that J conceives x to have.

I have flagged my intrusion of set-theoretic concepts into Peirce's statement
of the Pragmatic Maxim, because there is a potential distortion at this point
that we may have to track back to at a later stage if we find that something
has gone seriously awry the attempted interpretation.

And now that I've stopped to look more carefully at the possible road-blocks,
there is what may be a related reservation about Peirce's use of the operator
"whole of", as paraphrased in the form:  C_J (E_J (x)) is the whole of C_J (x).
Is this really meant to be an exact equation, as I've been reading it so far,
or is there a more significant operation of integration or synthesis that is
being invoked by the 2-adic relative term "whole of"?

Let's leave those worries, duly flagged, aside for the moment,
and proceed with our attempt at the simplest possible reading.

Diff Log A 2004 • Discussion Note 7


| Consider what effects that might conceivably
| have practical bearings you conceive the
| objects of your conception to have.  Then,
| your conception of those effects is the
| whole of your conception of the object.
|
| Peirce, "Maxim of Pragmatism",
| 'Collected Papers', CP 5.438.

Re: DLOG A9.  http://suo.ieee.org/ontology/msg05372.html  
In: DLOG A.   http://suo.ieee.org/ontology/thrd1.html#05359

Back to the picture that we had last time:

                  o y = C_J (x)
                 /
                /
    x o--------@
      .         \
      .          \
      .           o z
      .
      .
      .
      .           o C_J (E_J (x))
      .          /
      .         /
      o--------@
   E_J (x)      \
                 \
                  o

If my "recursive interpretation of the maxim of pragmatic thinking" (RIOTMOPT)
is suited to work out, then it would have to be the case that the "effects" of
an object are somehow simpler than the object itself, and simpler in the sense
that their concept is easier to conceive than the concept of the object itself.
After all, that is just the sort of recourse that one has in recursion, namely,
that one's attempt to answer a harder question has "recourse" to the resources
of one's answers to easier but related questions.

It seems that we must inquire further into the precise nature of these
effects, of the sort that "might conceivably have practical bearings".

Introducing a variant formulation of the pragmatic maxim,
Peirce puts a gloss on the theme of "practical bearings":

| Such reasonings and all reasonings turn upon the idea that if one exerts
| certain kinds of volition, one will undergo in return certain compulsory
| perceptions.  Now this sort of consideration, namely, that certain lines
| of conduct will entail certain kinds of inevitable experiences is what
| is called a "practical consideration".  Hence is justified the maxim,
| belief in which constitutes pragmatism;  namely,
|
| In order to ascertain the meaning of an intellectual conception one should
| consider what practical consequences might conceivably result by necessity
| from the truth of that conception;  and the sum of these consequences will
| constitute the entire meaning of the conception.
|
| C.S. Peirce, "Pragmatism" (c. 1905), 'Collected Papers', CP 5.9

Very roughly, then, let's try to formalize "effects" as referring to
ordered pairs of the form <Volition, Perception>, one might even say,
to invert a 2-gone paradigm, pairs of the shape <Response, Stimulus>,

Taking up Peirce's more developed description of the subject matter,
I will attempt to represent a practical consideration of effects,
or an effect that an agent conceives to have practical bearings,
as a comprehensive connection between a domain of conduct and
a domain of experience, or even just an extensive collection
of ordered pairs between certain "lines of conduct" (LOC's)
and certain "kinds of inevitable experiences" (KOIE's).

Diff Log A 2004 • Discussion Note 8


Perforce a necessity that custom deems practical, empirical researchers
and not just applied mathematicians are accustomed to approach the more
refractory objects of inquiry by strategic orders of approximation, and
thus to lay down the successive strata of a representation that renders
every object, that may be simple enough in its own right, complex to us.

We come to the question of whether the analysis is terminable, and thus
convergent to a definite result, or interminable, and thus inconclusive
in its indications of the object.  It is in the setting out of analytic
representational series that we see the importance of closure operators,
because they clue us in to how an otherwise infinite representation may
wrap up in a finite term.

Here is one way to see this.  A typical form of analytic expansion will
generally be conducted with respect to an operator Q in such a way that
the successive levels of analysis correlate with increasing powers of Q,
as Q^0, Q^1, Q^2, Q^3, ..., and so on.  If the operator Q is subject to
a law that makes all higher powers redundant after some point, then one
has the power to sum up what is logically an infinite series in what is
computably a finite term.

Some of the simpler operator laws that might turn up,
at least, among those that are not entirely trivial,
are those of the forms:  Q^2 = 0, Q^2 = 1, Q^2 = Q.

A "closure operator" C is one that obeys a rule of the last mentioned shape,
since requiring C(C(x)) = C(x) for all x is the same as saying that C^2 = C.
In algebraic language, one refers to such an operator as being "idempotent".

Cf.  Kelley, 'General Topology'.  http://suo.ieee.org/ontology/msg03874.html

I'll discuss this further, later on,
in the context of concrete examples.

Diff Log A 2004 • Discussion Note 9


HT = Hugh Trenchard

Re: DLOG A Discussion 7.  http://suo.ieee.org/ontology/msg05402.html

HT:  Hi Jon.  As usual I can supply only a superficial comment rather than
     a really rigorous analysis of your discussion.  However, I see some
     parallels here to the concept of emergence, which as we all know
     is described simplistically as the whole is more than the sum
     of its parts.  The concept actually runs counter to what
     Peirce said in the quote below, that "the sum of these
     consequences will constitute the entire meaning of
     the conception".

HT: Even so, I see from your commentary that if your formalism is to work
    "... the "effects" of an object are somehow simpler than the object
    itself, and simpler in the sense that their concept is easier to
    conceive than the concept of the object itself".  This appears
    to me to describe the concept of emergence.

HT: I know that you are adhering to a particular line of inquiry, and
    I'm sure my comments are, more often than not, little more than a
    distraction to you, but I think it may be a not entirely un-useful
    observation that what you are doing appears to be consistent (albeit
    on a very superficial analysis) with other current lines of inquiry,
    namely regarding descriptions of the emergence of novel forms of order
    which arise from the interaction of component parts of a complex system.

When I find myself using the word "emergence" it is usually to point out
some surprising phenomenon that has escaped an adequate explanation by the
prevailing frameworks of representation and theory.  There are times when it
seems like so many people have become so sure of their dominant paradigms that
they regard it as some kind of "emergency" whenever one of them is revealed
as falling short of reality, but I guess that I've gotten so used to their
fallibility that I tend to consider this the normal course of inquiry.

In that light, the word "emergence" is sometimes used in a way that bears
a number of misleading connotations.  Most serious I think is this one:
Are we really so sure that the phenomenon has emerged in recent ages,
or is it simply that we are just beginning to notice something that
has been there all the time?  The answer may depend on the case,
but it needs to be asked, in any case.

So I tend to view the issue this way:  There is the reality, and then there
are the many different representations that we have of the given reality.
It's a convenient figure of speech, but one that we deploy at the risk
of self-deception, to say that we analyze an objective reality into
its "atoms", "constituents", "elements", "effects", "parts", etc.
But any form of analysis that we pick is conducted on the level
of our chosen representation, to which we may be very "partial",
and not without good reason, but which is always very "partial",
and often just wrong, in regard to the objective reality itself.

That is why it is critically -- you might even say "Critique-ally" --
important to recall that the concept is a sign, a symbol, a tool
whose good is to "grasp", to "seize as one", to "throw together",
to "unify a manifold of sense impressions", as the various Greek,
Latin, and Anglo-Saxon-Teutonic etymologies will inform us if we
stop to examine them closely enough.  Remembering this should be
enough to remind us that neither the analysis nor the synthesis
of the signs that we bring to bear on the case will necessarily
touch the real constitution of the object, the phenomenon, or
the reality itself.  That does not render the representation
useless, not by a long shot, it is simply the nature of our
representation of reality to function in this partial way.

Diff Log A 2004 • Discussion Note 10


| Such reasonings and all reasonings turn upon the idea that if one exerts
| certain kinds of volition, one will undergo in return certain compulsory
| perceptions.  Now this sort of consideration, namely, that certain lines
| of conduct will entail certain kinds of inevitable experiences is what
| is called a "practical consideration".  Hence is justified the maxim,
| belief in which constitutes pragmatism;  namely,
|
| In order to ascertain the meaning of an intellectual conception one should
| consider what practical consequences might conceivably result by necessity
| from the truth of that conception;  and the sum of these consequences will
| constitute the entire meaning of the conception.
|
| C.S. Peirce, "Pragmatism" (c. 1905), 'Collected Papers', CP 5.9

This particular formulation of the pragmatic maxim is personally important
to me, as it's the version that linked up my studies in algebra, automata,
computer simulation, cybernetics, dynamical systems, and formal languages
all through the 1980's to the streams of psychology literature that I was
exploring in parallel at the same time.

Here is how I was accustomed to picture a "machine with input" in those days:

         |  a b c d           |  a b c d           |  a b c d
   T_1 : |              T_2 : |              T_3 : |
         V  c d d b           V  b a d c           V  d c d b

This parameterized set of transformations is given more compactly in Table 1.

Table 1.  Finite Transducer, or Machine with Input
o---------o---------o---------o---------o---------o
|    |    |         |         |         |         |
|    |    |    a    |    b    |    c    |    d    |
|    V    |         |         |         |         |
o=========o=========o=========o=========o=========o
|         |         |         |         |         |
|   T_1   |    c    |    d    |    d    |    b    |
|         |         |         |         |         |
|   T_2   |    b    |    a    |    d    |    c    |
|         |         |         |         |         |
|   T_3   |    d    |    c    |    d    |    b    |
|         |         |         |         |         |
o---------o---------o---------o---------o---------o

I copied this out of Ashby's 'Cybernetics', 4/1,
except for replacing his letter "R" with my "T".

Diff Log A 2004 • Discussion Note 11


Cf: Pragmatic Maxim.  http://suo.ieee.org/ontology/msg05407.html

Taking its analogy to various formal principles of closure, recursion, and
representation as rough guides, we have been exploring the implications of
the pragmatic maxim, so far giving special attention to these two versions:

| Consider what effects that might conceivably
| have practical bearings you conceive the
| objects of your conception to have.  Then,
| your conception of those effects is the
| whole of your conception of the object.
|
| C.S. Peirce, "Issues of Pragmaticism", CP 5.438, (1878/1905).

| Such reasonings and all reasonings turn upon the idea that if one exerts
| certain kinds of volition, one will undergo in return certain compulsory
| perceptions.  Now this sort of consideration, namely, that certain lines
| of conduct will entail certain kinds of inevitable experiences is what
| is called a "practical consideration".  Hence is justified the maxim,
| belief in which constitutes pragmatism;  namely,
|
| In order to ascertain the meaning of an intellectual conception one should
| consider what practical consequences might conceivably result by necessity
| from the truth of that conception;  and the sum of these consequences will
| constitute the entire meaning of the conception.
|
| C.S. Peirce, "Pragmatism", CP 5.9, (c. 1905).

Among the more glaring differences between these two versions we find
the words "sum" and "truth" explicitly figuring in the latter variant.
I will argue, in good time, that both of these notions are implicitly
contained in the sense of the maxim no matter how it may be expressed,
but for the time being I'll merely make a note of the point and stick
with my plan to deal with first things first.

After several days of relaxed reflection, I think that I can now bring some of
the points on Hugh Trenchard's "Emergent Phenomena Tangent" back home to roost.

The way I see it, we have a reality, say x, and then we have a representation
of that reality, say y.  If the representation y is "analytic" or "articulate"
in any sense of those words, then it will analyze or articulate the reality x
in terms of y's components, say, y_1, y_2, y_3, just for a start.  So we have
a picture like this:

                         x              y
                         o------------->o
                                       /|\
                                      / | \
                                     /  |  \
                                    /   |   \
                                   o    o    o
                                  y_1  y_2  y_3

One thing that we ought to appreciate at this point is that the
components of the representation are things that may or may not
correspond in any simple way to components of the reality, even
if the reality in question can be said to decompose in some way.
Such a direct correspondence is optional for representations in
general, and the utility of the representation, broadly treated,
is independent of our being able to "project" or to "reify" the
parts of y as the pieces of x.  Just by way of a grammar school
example, think of the knuckle-rapping cautions against reifying
the subject-predicate structure of natural language syntax into
ontological categories.  At least, that was the rule when I was
in school, and I remember it well.

Diff Log A 2004 • Discussion Note 12


I return to my picture of the relation between a reality x
and a representation y whose components are y_1, y_2, y_3.

JA: The way I see it, we have a reality, say x, and then we have a representation
    of that reality, say y.  If the representation y is "analytic" or "articulate"
    in any sense of those words, then it will analyze or articulate the reality x
    in terms of y's components, say, for example, y_1, y_2, y_3, just for a start.
    So we have a picture like this:

                         x              y
                         o------------->o
                                       /|\
                                      / | \
                                     /  |  \
                                    /   |   \
                                   o    o    o
                                  y_1  y_2  y_3

Let me now draw what I consider to be a critical distinction with respect
to the question of emergence, that is, to put it in the roughest possible
terms, the issue of whether "the whole is more than the sum of its parts".

If by "whole" we mean the object reality x, and if by "parts" we mean the
parts of speech, so to speak, of the representation y, then the emergence
of x beyond the y_j is hardly surprising, indeed, it's a corollary of the
fact that the representation y is approximate, and thus it proves nothing
about the potential emergence of y beyond its own components on the plane
of the given representation.  Of course, this issue will only be confused
still more by the unreflective reification of representational components.

On the representational plane, however, the utility of the representation
generally depends on each representation being determined by its components.
What results from this requirement of useful representations is an obligation
to render more explicit what we mean by "parts" and by "sum" in a given setting.
For example, if y is simply the set {y_1, y_2, y_3}, then y is something that can
be said to exist "over and above" its elements, at least, in some sense, even in
the case of a singleton set, say, z = {z_1}.  But any claim of "emergence" for
the relationship of a set to its elements would most likely be discounted as
trivial, since the set is defined as something that is fully determined by
its elements.

Diff Log A 2004 • Discussion Note 13


HT = Hugh Trenchard
JA = Jon Awbrey

Re: DLOG A Discussion 12.  http://suo.ieee.org/ontology/msg05421.html

Copied with corrections here:

JA: The way I see it, we have a reality, say x, and then we have a representation
    of that reality, say y.  If the representation y is "analytic" or "articulate"
    in any sense of those words, then it will analyze or articulate the reality x
    in terms of y's components, say, for example, y_1, y_2, y_3, just for a start.
    So we have a picture like this:

                         x              y
                         o------------->o
                                       /|\
                                      / | \
                                     /  |  \
                                    /   |   \
                                   o    o    o
                                  y_1  y_2  y_3

JA: Let me now draw what I consider to be a critical distinction with respect
    to the question of emergence, that is, to put it in the roughest possible
    terms, the issue of whether "the whole is more than the sum of its parts".

JA: If by "whole" we mean the object reality x, and if by "parts" we mean the
    parts of speech, so to speak, of the representation y, then the emergence
    of x beyond the y_j is hardly surprising, indeed, it's a corollary of the
    fact that the representation y is approximate, and thus it proves nothing
    about the potential emergence of y beyond its own components on the plane
    of the given representation.  Of course, this issue will only be confused
    still more by the unreflective reification of representational components.

JA: On the representational plane, however, the utility of the representation
    generally depends on each representation being determined by its components.
    What results from this requirement of useful representations is an obligation
    to render more explicit what we mean by "parts" and by "sum" in a given setting.
    For example, if y is simply the set {y_1, y_2, y_3}, then y is something that can
    be said to exist "over and above" its elements, at least, in some sense, even in
    the case of a singleton set, say, z = {z_1}.  But any claim of "emergence" for
    the relationship of a set to its elements would most likely be discounted as
    trivial, since the set is defined as something that is fully determined by
    its elements.

HT responds:

HT: I haven't given as much thought to this topic
    as I might have liked this weekend.  However,
    let me just throw another scenario out for
    discussion.

HT: It looks to me you've described a static scenario in which the components of
    reality x do not interact with one another -- they are simply properties of
    reality x.  This would be akin to describing an apple as consisting of the
    following properties:  it is roughly spherical, it is red, and it is juicy.

One of the things that I'm trying to convey here is just
how self-conscious we need to be about the gap between
the reality and the many alternative representations.

Notice how empty the space under x in my picture is.

I have not said anything positive about the components of x,
and only emphasized how cautious we have to be about imputing
the structure of the representation to the reality.  Now it's
true that a representation is always kind of static in a way,
verbs as words are not more fluid than nouns, and a program
is a fixed code even if we run it to simulate some process.
But that is a property of representations, and I have not
projected any of it onto the unknown x.

HT: I'm not clear what you mean by "parts of speech" as being the
    components which comprise the whole of reality x, but if they
    are linguistic/semiotic elements which represent properties
    of reality x, then the apple analogy seems to apply.

That was a figure of speech.  My intent was to stress
the fact that the y_j are parts of y, not parts of x.

HT: However, what happens when you complicate
    the scenario so as to describe components
    which interact?

As things stand, it would be complicating the
scenario a bit just to posit components for x,
for example, as depicted in the following way:

                         x              y
                         o------------->o
                        ...            /|\
                       . . .          / | \
                      .  .  .        /  |  \
                     .   .   .      /   |   \
                    .    .    .    o    o    o
                   x_1  x_2  x_3  y_1  y_2  y_3

Now, even if I were so bold to as to risk such a step
at this point in the development of my representation,
what would that really mean?  Fiat Lux?  Not my apple.
In fact, I never deplaned the plane of representation,
much less in a way that would harm any realities with
my filmy images.  Yes, it is probably a very safe bet
that everyone in this theatre has already bought into
the "physical symbol system thesis" (PSST), and so we
can admit that even our imaginings are bound by surly
bounds of earthly constraints, but talking about that
is still another matter, logically speaking, from the
matter of reality x that I chase in the present frame.

So I continue to recommend a great deal of caution with respect to this step,
to advise that we consider very carefully just what it means to take it, and
to suggest that there may be more ways to take it than are dreamt of til now.

Diff Log A 2004 • Discussion Note 14


HT = Hugh Trenchard
JA = Jon Awbrey

Re: DLOG A Discussion 12.  http://suo.ieee.org/ontology/msg05421.html

HT: From some of the discussions you've presented,
    it seems that Peircean logic does contemplate
    formalizing interactions between components.

That would be a misapprehension on several scores:

First, as to what I've been busy about under this
subject line, and along the lines of inquiry that
I've been tracing in its name for the last decade
and a half or so, it is very much about extending
our most rudimentary logical formalisms in a way
that would be more adequate to dealing with the
problems of change and diversity as we run into
them in qualitative settings.

Second, as to the character of Peirce's "logic as semiotics" (LAS),
that I am merely drawing on as the best resource for my own effort.

Third, as to the purpose of logic in general, a normative science
dedicated to the objective of optimizing the practice of thinking,
in other words, that investigates the question of how we may best
conduct our thoughts, on the condition that we desire to achieve
a number of more or less well-appreciated purposes of reasoning.

HT: I am reminded, as I mentioned previously, of your
    discussion relating to measures of uncertainties
    and options at succeeding junctures ("what to do,
    what to hope ...").  Even in the basic scenario
    you presented, which seems to imply some degree
    of causal interactivity, I recall that compound
    uncertainty was multiplicative (though additive
    by logarithm), which even in that sense begins
    to look more like "emergence" than simply the
    whole being the sum of its parts.  But beyond
    that, I wonder if there is potential to combine
    "trees", if you will, of compound uncertainty,
    that lead to patterns of compound uncertainty
    which wereperhaps not predictable by the initial
    set of "decisions" at the base of the "trees".

I continue to detect in this interpretation of what I wrote on that thread
some assumptions that I never intended to convey, but I do not know how to
revise my story in a way that would fix the evident deficits in my account.

Part of the problem seems to come in with my use of trees to represent what
is really the stepwise analysis of a present situation, and not necessarily
a sequential process of decision-making that happens on the plane of action.
It may be that my try at a triple play, stretching to cover all three bases
of Kant's 3-fold inquiry "What's true?  What's to do?  What's to hope?" has
strained the limits of my agility for capturing all three in a general idea.
Anyway, I can see no way home, nor any way to retire the field at this time.

HT: I realize that emergence in the sense I have applied it, occurs
    primarily in the context of complex dynamical systems, and may
    simply be inapplicable to Peircean logic.  But I do wonder if
    one might stumble upon the existence of fractals or power
    laws in the patterns of relatively complex uncertainty
    trees.  I would suggest that if such patterns exist,
    then there is a fundamental relationship between
    Peircean logic and complexity (in the sense of
    the science of complexity).

Here I get that talking past each other feeling again.  Just about all the
disciplines that we lump together under the "formal sciences" are designed
to address complexity of one ilk or another, and if there is anything that
all of them have in common, it is a high level of care about what it takes
to attribute any property at all to an objective reality, as distinguished
from the careless attribution of a relative property to one of the parties
to the relation in question.  Now Peirce is remarkable for his early sense
of the subtleties that are involved in such questions, but the insights he
foresaw are hardly unique today, indeed, they are prevalent in most of the
formal sciences that I have ever encountered.

HT: I think I know what the general response is going to be --
    but here again, I just throw this out as food for thought.

Well, anticipated or not I will go ahead and try to give a succinct summary
of how I see the question of emergence.  I am not questioning the value of
the word in pointing to phenomena that escape explanation in terms of our
prevailing "frames of inquiry" (FOI's), indeed, it seems to follow almost
automatically from the approximate nature of all human FOI's that there
will always be these kinds of emergent complications arising from time
to Timbuktu.  Given all of the things that we ignore in order to form
a FOI in the first place, it's a global no-brainer that some of the
details we neglected will rise up and bite us back sooner or later.
But it does not address that very acute danger if we use the terms
Chaos, Complexity, Emergence, Manifoldness, Multiplicity, Variety,
Uncertainty, ..., in ways that consign them to the very long list
of mystifying vital principles that function not to urge inquiry
but solely to pacify its urgings.  It's all about how the words
are used in practice, not essentially a question of spelling.

The question is the next question, after you've succeeded
in calling attention to an emerging phenomenon, then what?

Diff Log A 2004 • Discussion Note 15


RM = Richard Martin

RM: On a different plane I am flying.  Flight, whether of a bird or airplane,
    is an often mentioned emergent property.  So if I assign flight's reality
    to x, which assignments to the components of y might yield the emergence
    of flight?

For my piece I'm still building boats in the basement root-cellars of logic,
so I'll not be uplifting you with any new theories of aerodynamics.  Let's
just say that x is the flight of the bumblebee, which not to shine you on
appears to demand the emergence of some new height of virtuosity in any
medium that can carry it, and is often mentioned as a case where all
the king's measures and all the king's scores fail to account for
the living actuality of the live performance.  I think I catch
the drift of that.  Now, our personal best theory of how it's
done will be transcrypted in partial differential equations,
with differential coefficients that blur before my eyes in
all their jiffily rushing complexity, and so by the grace
of Lethe I'll just hint at the shape and flow of their
parody in the conventions of that Grimm Marchen style
as y_1, y_2, y_3, and by these three to say we play
at summing up the eternal infinity, if more in the
breach than in the observables.

Diff Log A 2004 • Discussion Note 16


HT = Hugh Trenchard

Re: DLOG A Discussion 14.  http://suo.ieee.org/ontology/msg05425.html
In: Differential Logic A.  http://suo.ieee.org/ontology/thrd1.html#05359

It seems that many of these consternations
are accountable to defects of my exposition
that I do not know how to remedy right now,
but I will try to address some of the less
entangled ones.

HT: Thanks for the succinct exposition.  I think I have understood the whole
    time that Peircean logic (albeit formalized logic at any level) involves
    a set of axioms and procedures which are fundamentally abstract and do
    not necessarily allow for the overlap of other or competing formalisms
    (although I take your point that Peircean logic is not entirely unique).

I do not think that "lack of overlap" follows from abstraction.
Indeed, one of the reasons for scrambling up the concrete rocks
of experience to some higher level of abstraction is just so we
can better see the commonalities among domains that appear to be
disparate when our nose is to their grindstones.  And one of the
reasons that I started this work on differential logic was to get
a better focus on the principles from mathematical systems theory
that might be analogized and generalized to deal with outstanding
issues of change and diversity in logic-based intelligent systems.

Now, I really do understand the impatience to get on to the interesting bits.
The literature that you mention, or the precursors of it, evokes memories of
the very sort of excitements that filled my days in the 70's and 80's of the
late great 1900's.  But -- famous first words -- that was only the beginning.

But there is a prerequisite subject that we should have taken in school
in order to sew up this very overlap, to forge the links of necessity
between the quantitative-probabilistic and the qualitative-logical,
but we can't have taken it because it hadn't been invented yet,
and so I've been busy working on that.

HT: And I do think that current studies of emergent phenomena involve
    very particular formalisms -- that it isn't just about how and
    where we might wish to insert our own interpretation of what
    we want to call an emergent phenomenon.  One only needs to
    review such articles as "Collective Induced Computation"
    by Delgado and Sole, or Per Bak or countless others to
    see how emergent phenomena may be analyzed mathematically.

The fact that we are bound to use particular formalisms and
systems of concepts to talk and to think about phenomena of
any kind does not entail that "all is vanity and subjective".
But there are general strategies that all formal sciences use
to identify the objective properties that may be lurking under
the bushes of appearance.

As a very typical example, consider the way that the Chomsky-Schutzenberger
complexity of a formal language is defined.  A given formal language will
have many formal grammars, or, alternatively, many automata, that accept
or generate it.  Now, formal grammars and automata can be classified by
ostensible complexity-like properties that they wear on their sleeves,
that is, you can easily identify a context-free grammar or a pushdown
automaton simply by inspecting the form of its given presentation.
A formal language gets to be called "context-free" if there is
'some' context-free grammar or pushdown automaton that accepts
or generates it, but that only gives an upper bound on the
complexity of the language.  In order for it to be called
"properly context-free" one has to prove that there are
'no' finite-state automata or grammars that do the job.

This definition of formal language complexity exemplifies
a very common pattern for defining invariant properties.

If one thinks of a grammar as a "theory" of a language, that is,
as a particular way of describing a language in effective terms,
then the attribution of a property as an invariant of a language,
as with the case of graduated complexity properties, entails the
idea that the property in question can be demonstrated to belong
to the language in question from the standpoint of every relevant
description, grammar, or theory of the language.

Extracting the general pattern, the attribution of a property that
merits being named an "invariant" or an "objective" property of the
object under survey will typically involve a logical quantification,
like "all" or "some", ranging over a set of theoretical perspectives.

Diff Log A 2004 • Discussion Note 17


Let me see if I can still remember how it came about that
the pragmatic maxim emerged in the context of pouring the
foundations for differential logic.

I was looking at sets of propositions, conceived as sets of functions
of the form f : B^k -> B.  Then I contemplated the action of a couple
of operators on these sets of propositions, operators that were meant
as logical analogues of the usual finite difference operators E and D.
Of course, E and D refer to whole parameterized families of operators,
where "E" denotes the enlargement or shift operators, and "D" denotes
the difference or delta operators.  I wrote all this out for the case
of k = 2 starting at this point:

Re: Differential Logic A6.  http://suo.ieee.org/ontology/msg05368.html

Table 4.  E(f) Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
|      |            |            |            |            |            |
|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
|      |            |            |            |            |            |
| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
|      |            |            |            |            |            |
| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
|      |            |            |            |            |            |
| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
|      |            |            |            |            |            |
| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|                   |            |            |            |            |
| Fixed Point Total |      4     |      4     |      4     |     16     |
|                   |            |            |            |            |
o-------------------o------------o------------o------------o------------o

In this context, the set of shift operators acts as a mathematical group
on the set of propositions, splitting it up into disjoint and exhaustive
subsets that are technically known as "group-reduced equivalence classes",
but that are rather more commonly known as "orbits".  Generally speaking,
all of the elements in the same orbit have some common property that is
said to be "preserved" by the action of the group.

For example, consider the case for k = 2 that is presented in Table 4.
The set of shift operators, redescribed as a set of 4 transformations,
T_ij in {T_00, T_01, T_10, T_11}, forms a mathematical group that acts
on the set of 16 propositions, in such a way that the 16 propositions
are partitioned into exactly 7 orbits.  Looking over the propositions
in each orbit we can see that the members of the same orbit all have
similar geometric "shapes" when viewed as figures in a venn diagram.

Why is this important?

Here are some reasons that come to mind:

1.  Group invariants afford us with one of the ways that invariant properties
    of objects, actual and formal, commonly come to be recognized in practice.

2.  Group actions, by gathering together "birds of a feather", that is,
    objects that have isomorphisms between them or that share some order
    of similar structure, into common orbits, act to reduce the complexity
    of the underlying domain in ways that may also reduce the computational
    complexity of working with objects in that domain.

3.  Group representations, that is, the representations of groups that we get
    by considering them as sets of operators on sets of relatively concrete
    objects, are prime examples of how to form operational definitions of
    abstruse concepts.  In this way they provide us with useful guidance
    as to how we might apply the pragmatic maxim in more general cases.
    For example, Peirce gave operational representations of concepts
    like "truth" and "falsity" by means of the same sort of tactic.
    Now, you must not expect the last word on defining a difficult
    concept to come from such an isolated form of representation,
    but even the smallest exemplar of the concept can make its
    contribution to dispelling its more problematic mysteries.

Diff Log A 2004 • Discussion Note 18


One of the times -- or is it most of the times? -- that the gap between
a reality and a representation becomes rather glaringly obvious is when
we have several representations of what is regarded as the same reality,
for instance, as illustrated in the following picture for the reality x
and the alternative representations u and v.

          u              x              v
          o<-------------o------------->o
         /|\                           /|\
        / | \                         / | \
       /  |  \                       /  |  \
      /   |   \                     /   |   \
     o    o    o                   o    o    o
    u_1  u_2  u_3                 v_1  v_2  v_3

There are times when the variant representations are easily brought
into felicitous correspondence with one another, and then there are
times when the different views are so incommensurable that we begin
to wonder whether they really lookout on the same worldscape at all.

In the examples of regular representations of groups that I discussed
a while back, we have already seen cases of variant representations of
the same object that are easily related to each other at various levels
of abstraction.  Just for a stock example of alternative representations
that are not quite so trivially related to each other, one might think of
the complementary interpretations of the same quantum phenomenon in terms
of particles and in terms of waves.

Diff Log A 2004 • Discussion Note 19


HT = Hugh Trenchard

Re: DLOG A Discussion 14.  http://suo.ieee.org/ontology/msg05425.html
In: Differential Logic A.  http://suo.ieee.org/ontology/thrd1.html#05359

HT: I do also realize that unless I can deliver the formal goods
    myself on any possible overlap, there isn't much point in my
    going on about it.

HT: My only purpose in raising the questions is to see if perhaps you yourself
    might see some underlying common mathematical principle which remains to be
    identified.  I see that you do not believe there is one, nor are you about
    to make finding one an objective of yours, and I certainly don't blame you.

I do not see how you see that.

HT: In terms of your last question about where does one go once he has
    identified an emergent phenomenon, one might conceivably do what,
    for example, Delgado and Sole have done, which is to address the
    question of whether there is "a tradeoff between the individual
    complexity and collective behavior in such a way that complex
    emergent properties cannot appear if individuals are too
    much complex", after which they proceeded by rigourous
    statistical analysis to answer the question.

HT: Or, do as Per Bak did: identify the properties of emergent phenomena
    (e.g. self-organized criticality), identify the patterns of order
    which characterize emergent phenomena like power laws (such as
    Zipf's law or Pareto's law), or fractals (as I mentioned in
    a previous email).

HT: Or do as Langland, Bak, Wolfram, and others have done in simulating the
    phenomena by computer analysis to show how complex phenomena does indeed
    derive from very simple rules.  To boot, Wolfram believes practically all
    of science can be examined in terms of cellular automata (see "A New Kind
    of Science").  (Maybe I should be asking Steven Wolfram this question (?)).

HT: These are simply a few examples.

Yes, those are very good examples of what one customarily does next.
But I hope you understand that nobody goes about developing rigorous
mathematical or vigorous statistical models of any phenomenal domain
if they feel compelled to sacrifice their rigor and their vigor to
the spectre of emergence every time they try to use a plus sign.

HT: In terms of power laws, as we all know, they are ubiquitous in
    complex dynamic systems (this is what I am doing in my analysis
    of bicycle racing -- I am trying to confirm the existence of
    a power law relationship between the number and intensity of
    "attacks"), and given their relative omni-presence, I don't
    think it entirely unreasonble to suggest they might appear
    in surprising ways in other formalisms (if such formalisms
    describe in some way physical phenomena), as unlikely as
    it might seem.  Although again, unless I can make some
    sort of intelligent connection myself, there isn't
    much point in pursuing it.

All of the things that you mention here are somewhere
on my list of "How I Got Into This" (HIGIT), but the
first thing that I discovered when I first got into
this, so many blue moons ago, is that a whole lot
of the spadework for building those castles in
the air just hadn't been done yet, so the next
thing for me to do was just to get down tuit.
And "There You Were" as Wayne & Schuster say.

HT: Having said all that, from this point forward I'll stay inside
    the Peircean box, unless we jump to a completely separate box
    should we wish, but never the twain shall meet.

Ok, if you view it as a box you should be very careful about getting intuit.
I sure know I would.  But from our station outside the box, we may reach in
and find a few tools or toys that have the goods that tools and toys may do.
And if it's a vehicle that can take you somewhere you want to go and cannot
get there as quickly on your own power, then sometimes you have to overcome
your fear of flying and make what mileage you can.

HT: p.s. famous last words?

Zoom, Zoom, Zoom

Diff Log A 2004 • Discussion Note 20


| Yea, from the table of my memory
| I'll wipe away all trivial fond records,
| All saws of books, all forms, all pressures past,
| That youth and observation copied there,
| And thy commandment all alone shall live
| Within the book and volume of my brain
| Unmixed with baser matter.
|
| Hamlet, 1.5.98-104

Let me go back to one of the places where I came in,
where I woke up in one of my math psych courses and
began to pay attention to something inside that box
of our classroom one buzzy b(l)oomy midsommer's day.

As adapted from Ashby's 'Cybernetics' 4/1, here 'tis:

Table 1.  Finite Transducer, or Machine with Input
o---------o---------o---------o---------o---------o
|    |    |         |         |         |         |
|    |    |    a    |    b    |    c    |    d    |
|    V    |         |         |         |         |
o=========o=========o=========o=========o=========o
|         |         |         |         |         |
|   T_1   |    c    |    d    |    d    |    b    |
|         |         |         |         |         |
|   T_2   |    b    |    a    |    d    |    c    |
|         |         |         |         |         |
|   T_3   |    d    |    c    |    d    |    b    |
|         |         |         |         |         |
o---------o---------o---------o---------o---------o

As spied from the perspective of my 3rd eye,
it struck me as a table of 3-tuples, to wit:

   <1, a, c>
   <1, b, d>
   <1, c, d>
   <1, d, b>

   <2, a, b>
   <2, b, a>
   <2, c, d>
   <2, d, c>

   <3, a, d>
   <3, b, c>
   <3, c, d>
   <3, d, b>

At first sight, one usually sees the "input parameters", in this example
the identifiers in the set {1, 2, 3}, as signals or signlike data, while
the states are seen as objective conditions in which the machine resides
for moments at a time and transits between on cue from the input signals.

That view of the 3-adic relation M under survey casts it with a type
M c S x O x O, where S is a sign domain and O is an object domain.
Initially, then, the machine M appears to have some sort of dual
type in relation to the typical sign relation L c O x S x S.

It is obviously worth considering 3-adic relations of that type,
along with many others, but the apparent differences here tend
to disappear from more general points of view.  For example,
one may be thinking of the input parameters as relatively
direct data about the state of an external object system,
with the purpose of the machine being nothing more than
to store information about this object system in the
record of its states.

From a still more general point of view, all three domains may be
the state spaces of systems, and here it is possible to think of
cases where there would be 1, 2, or 3 different systems related.

Consider the possibilities ...

Diff Log A 2004 • Discussion Note 21


HT = Hugh Trenchard

Re: DLOG A Discussion 20.  http://suo.ieee.org/ontology/msg05432.html

HT: I am wondering if by your statement "from a still more general point of view,
    all three domains may be the state spaces of systems", you are suggesting that
    there are a number of inherent data configurations or relations within a system
    that may be said to represent or define its information content (i.e. ones not
    obvious or necessarily intended).

Once you adopt the system-theoretic point of view,
then the only realities are states of systems, and
you could even say that there's really only one big
system of which everything else is either a shadow
or a subsystem.

In order to talk about the information content of a system,
you have to go back to the basic sign-theoretic situation
and ask what it means for the state of a sign system S
to convey information about an object system O
to an interpretant system I.

HT: Perhaps that's a trivial summation, but I am wondering further:
    when running the grid relations you've shown through a machine
    which results in a specific output, does the output inherently
    carry the 3-adic relations you're showing?

In a sense the output is the entire state of the system 
at any given time, or whenever somebody says "time's up".
But you can always define a projection from that state
to another range of values that you consider to be
the designated outputs.

HT: Again, trivial perhaps, but maybe this underlying
    relationship could be useful as a ciphering method,
    or potentially as a method for reducing the complexity
    of algorithmic input (complexity here in the sense of
    quantity of rules and descriptions used to represent
    a specific object, set of objects, or system).

Yes, many such journeys are possible.

Did you know that you can download a copy of Ashby,
well, his book anyway, from the Cybernetica portal:

http://pespmc1.vub.ac.be/ASHBBOOK.html

HT: Just more grist for the mill.

| I could be bounded in a nutshell
| and count myself a king of infinite space,
| were it not that I have bad dreams.
|
| Hamlet, 2.2.256-258

Well, I'm going to risk it anyway.

Z Z Z . . .

Diff Log A 2004 • Discussion Note 22


| My tables,
| My tables -- meet it is I set it down
| That one may smile and smile and be a villain.
|
| Hamlet, 1.5.107-109

| Meet it is -- or is it join? --
| That error and information
| Bear our cognate strife
| With us in the middle,
| As ambits torn from
| A singular womb.
|
| But leave the space
| That promises peace,
| With wile enough and
| The wareness to boot:
| 'Twill amend thy selve.
|
| Jon Awbrey, 18 Feb 2004

Diff Log A 2004 • Discussion Note 23


To facilitate the comparison with sign relations, let's reset
our machine table in the form of a relational database for
the 3-adic relation M c X x Y x Z, as shown in Table 2.

Table 2.  Transduction Triples
o---------o---------o---------o
|    X    |    Y    |    Z    |
o---------o---------o---------o
|    1    |    a    |    c    |
|    1    |    b    |    d    |
|    1    |    c    |    d    |
|    1    |    d    |    b    |
o---------o---------o---------o
|    2    |    a    |    b    |
|    2    |    b    |    a    |
|    2    |    c    |    d    |
|    2    |    d    |    c    |
o---------o---------o---------o
|    3    |    a    |    d    |
|    3    |    b    |    c    |
|    3    |    c    |    d    |
|    3    |    d    |    b    |
o---------o---------o---------o

Using the same tactic that we used for the regular representations of groups,
we can express the "meaning" of each input parameter as a logical sum of its
operational effects, where an effect is defined as an ordered pair of states,
let's say in the order <ante-op, post-op>, sometimes written as "ante:post".

   T_1  =  a:c + b:d + c:d + d:b

   T_2  =  a:b + b:a + c:d + d:c

   T_3  =  a:d + b:c + c:d + d:b

In a typical control system scenario, an agent has the job,
given the ability to choose a sequence of input parameters,
to take the system from whatever state it happens to be in
to a specified state, that we may regard as the goal state.

This type of set-up lends itself to a game-theoretic description,
where one player puts the system in any state and another player
has to pick a sequence of input parameters that will bring it to
the designated goal state.  Of course, there are many such games.

Diff Log A 2004 • Discussion Note 24


DL = David Letterman
HT = Hugh Trenchard
JA = Jon Awbrey
WS = Wm Shockshafte

Sub*ject Line.  Automatopoiesis

WS: | My tables,
    | My tables -- meet it is I set it down
    | That one may smile and smile and be a villain.
    |
    | Hamlet, 1.5.107-109

JA: | Meet it is -- or is it join? --
    | That error and information
    | Bear our cognate strife
    | With us in the middle,
    | As ambits torn from
    | A singular womb.
    |
    | But leave the space
    | That promises peace,
    | With wile enough and
    | The wareness to boot:
    | 'Twill amend thy selve.
    |
    | Jon Awbrey, 18 Feb 2004

HT: | Cognate Strife
    |
    | Consider once a duple,
    | And thou hast a tuple.
    | Consider all that,
    | and a recursion create;
    | Such sweet paradox
    | To abandon safe space
    | To consider all that
    | For infinite ends
    | No more fathomable
    | Than the one safe place
    | From where it began
    |
    | Hugh Trenchard
    | February 19, 2004

DL: It's only an exhibition;
    It's not a competition:
    Please!  No wagering!

Diff Log A 2004 • Discussion Note 25


Refreshed from that idyllic interlude,
it's back to the grinstone once again,
to the "finite state automaton" (FSA)
that Table 2 shows as M c X x Y x Z.

Table 2.  Transduction Triples
o---------o---------o---------o
|    X    |    Y    |    Z    |
o---------o---------o---------o
|    1    |    a    |    c    |
|    1    |    b    |    d    |
|    1    |    c    |    d    |
|    1    |    d    |    b    |
o---------o---------o---------o
|    2    |    a    |    b    |
|    2    |    b    |    a    |
|    2    |    c    |    d    |
|    2    |    d    |    c    |
o---------o---------o---------o
|    3    |    a    |    d    |
|    3    |    b    |    c    |
|    3    |    c    |    d    |
|    3    |    d    |    b    |
o---------o---------o---------o

Graphical modes of visualizing FSA's can be extremely
useful when the number of states is relatively small.
Figure 3 shows one way of depicting the machine M,
as an "edge-painted node-labeled digraph".  Here,
the term "painted", say, with "palette" X, means
that each edge is associated with a subset of X,
the term "labeled", say, with "label set" Y = Z,
means that there is a bijective map between the
nodes and the set Y, which is here equal to Z,
and the term "digraph" is the same thing as
saying "directed graph".  When the context
is understood, it will be most convenient
to refer to such a construction as the
"graph" of the machine M, or Graph(M).

o-----------------------------------------------------------o
|                                                           |
|                           o---o                           |
|                          /     \                          |
|                         o       o                         |
|                         |   a   |                         |
|                         o       o                         |
|                        / \     / \                        |
|                       /   o---o   \                       |
|                      /   /    |    \                      |
|                   2 /   /     |     \ 3                   |
|                    ^   /      |      v                    |
|                   /   /       |       \                   |
|                  /   v        |        \                  |
|                 /   / 2       |         \                 |
|                /   /          |          \                |
|           o---o   /           |           o---o           |
|          /     \ /         1  |          /     \          |
|         o       o----------->-|---------o       o         |
|         |   b   |             v 1       |   d   |         |
|         o       o-----------<-|---------o       o         |
|          \     /           13 |        / \     /          |
|           o---o               |       /   o---o           |
|                \              |      /   /                |
|                 \             | 123 /   /                 |
|                  \            |    ^   /                  |
|                   \           |   /   /                   |
|                    ^          |  /   v                    |
|                   3 \         | /   / 2                   |
|                      \        |/   /                      |
|                       \   o---o   /                       |
|                        \ /     \ /                        |
|                         o       o                         |
|                         |   c   |                         |
|                         o       o                         |
|                          \     /                          |
|                           o---o                           |
|                                                           |
o-----------------------------------------------------------o
Figure 3.  FSA as Edge-Painted Node-Labeled Digraph

Looking over the graph, it should tell us the same thing
as any of our previous representations of M, for example:

   T_1  =  a:c + b:d + c:d + d:b

   T_2  =  a:b + b:a + c:d + d:c

   T_3  =  a:d + b:c + c:d + d:b

In so many words:

   1 takes a to c, b and c to d, and d to b.

   2 switches a and b, and switches c and d.

   3 maps a to d and cycles through d, b, c.

Differential Logic 2004 • Series B • Discussion

Diff Log B 2004 • Discussion Note 1


HT = Hugh Trenchard

HT: Just throwing out a thought (as usual):
    if one of the units, or logical variables
    is stable, while the other is active, do we
    see a "radiating tree" something like this:

        active   active   active
             \      |      /
    active--- \  stable   /---active
              /     |     \
             /   active    \
        active            active

HT: I know you were probably going to answer that
    question in the next episode, but I wanted to
    see if I roughly understand the concept.

Consider the logical expression that I write
as "(x , y)" and graph in the following way:

    x   y
    o---o
     \ /
      @

According to the story that we are imagining about the relationship
of the logical syntax to the state of a quasi-physical system, this
expresssion describes the set-up of a neural pool with two mutually
inhibitory units x and y when it finally reaches equilibrium, which
in the brand of model that we have here can only be one of two ways:
(1) x is active and y is stable, or (2) x is stable and y is active.

I can't remember if I had this in mind at the time, but I know that
shortly after I wrote this up for my master's work I recognized that
there were models of competition processes that Grossberg and others
had considered that have this very sort of "winner take all" behavior
at equilibrium.  However, there is a different twist here that I was
just about to get into.

Diff Log B 2004 • Discussion Note 2


HT = Hugh Trenchard

HT: I should have added to my "radial tree" a description
    that "stable" represented either x or y, and "active"
    also either represented x or y, but "active" in either
    case was a variable in the sense that all the "actives"
    radiating around the "stable" represent dynamic change.
    The number of actives in my illustration was purely
    arbitrary and was set up to show that while one of
    either x or y remained in stable state, around it
    revolve either x or y in a state of change.

HT: So I'm not sure if this was already apparent -- but does my explanation
    above change how you may have viewed my previous email?  Of course I mean
    it only as conceptual tool for me to understand how one unit remains stable
    while another is changing.

HT: In any event, I think I take your point that your representation
    already illustrates the point, and is simpler than mine (Occam's
    razor being instructive).

Once upon a time in the kingdom of AI it wasn't considered sporting unless
you tried the methods of your own device out on many problems aside from
those of your own device.  So I took up the dare of the Jets and Sharks
example in that spirit, and had to go along with the assumptions that
came with the turf.  In this (West Side) story, complex propositions
are represented by pools of formal neurons that represent the basic
logical variables.  It doesn't matter all that much what you call
the states, so long as you have two distinguishable states.  And
because one is trying to imagine how a continuous dynamic system
can reliably embody or represent a discrete logical proposition,
there is plenty of room to play around with alternative models.

I was using the word "stable" for a state of dynamic equilibrium,
typically a lower state of activation than the other states that
are available to agents, and I was using the word "active" for a
higher state of activation, typically transient, that there must
be exactly one formal neuron occupying, at equilibrium, in these
specialized pools of mutually inhibitory units.  And even though
the system passes through all sorts of intervening states before
it finally settles down, it is only the equilibrium state of the
neural pool as a whole that counts.

Whatever choice of words eventually works best, the connotation
that we do want to preserve is the similarity between the agent
at rest and the collective of agents in a state of equilibrium,
as this is what gives us a notion of logical value that makes
sense for both simple and complex propositions, and thus what
allows us to combine propositions in the proper logical way.

Diff Log B 2004 • Discussion Note 3


HT = Hugh Trenchard

Thanks for waking me from my documental stupors --
I'm beginning to remember why I put off doing this
for 15 years -- that estimate of 10^3 words/picture
really is a gross under-statement.

HT: I'm not clear on at least a couple of points here.
    Is the "region bordering the origin cell" that part
    of the Venn diagram where all three circles overlap?
    If so, is the "point omitted neighbourhood" those
    regions of the three circles which do not overlap
    with any of their neighbours?

I called that outer region where all the predicates x, y, z
are false the "origin" because its coordinates are <0, 0, 0>.
So the cells adjacent to the origin are the gravely accented
ones in the following venn diagram:

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o`````````````````````````o                |
|                |``````````` X ```````````|                |
|                |`````````````````````````|                |
|                |```````````100```````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o```o----------o--o             |
|            /````\          \`/          /````\            |
|           /``````\   110    o    101   /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    / 111 \    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |``````` Y ```````|       |`````` Z ````````|       |
|       |`````````````````|       |`````````````````|       |
|       o```````010```````o  011  o``````001````````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                            000                            |
|                                                           |
o-----------------------------------------------------------o
Figure 11.  Venn Diagram for ((x),(y),(z))

HT: You note that the phrase "just one of three is true" describes
    the Venn diagram.  This is obviously different from the intuitive
    sense that "there is a part of x and a part of y and a part of z
    that are common, or shared, among x, y, and z".  So if the centre,
    common region, represents "one of three is true" (if I have that
    much right, which I might not), does that mean as between y and z,
    "one of two is true", and as between x and z "one of two is true",
    and as between x and y "one of two is true"?

No, the center cell is the one where all 3 are true,
that is, its coordinates are <x, y, z> = <1, 1, 1>.
The shaded region that is indicated by ((x),(y),(z))
has the 3 cells whose coordinates are 100, 010, 001.
If I had overlapping gels, these would be the pure
primary colors, thus, the cells where just 1 of 3
properties applies, a different 1 in each case.

Diff Log B 2004 • Discussion Note 4


HT = Hugh Trenchard

Re: http://stderr.org/pipermail/inquiry/2004-February/001223.html

HT: Ah -- looks like my initial interpretation was almost
    precisely backwards!  Things are definitely clearer now.
    So the origin, or 000 is, so to speak, the remainder of
    the universe of discourse.  (For example, in expropriation
    law, an area of law I was working in for a while -- I am
    employed as a paralegal -- one speaks of a portion of land
    expropriated from an owner as the land "taken" and the area
    still owned by the land owner as the "remainder").

"Adverse possession" !!!  Slowly I turned ...

Yes, all of those "cells", the undivided regions, are really equal citizens
in the universe of discourse, and it's only the particular form of planar
projection that gives them such different shapes in the venn diagram.
Another popular way to represent three logical dimensions would be
in the form of a 3-cube, like so:

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                        o                        |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|       x  y (z) o    x (y) z    o (x) y  z       |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|       x (y)(z) o   (x) y (z)   o (x)(y) z       |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                    (x)(y)(z)                    |
|                                                 |
o-------------------------------------------------o

In this variety of picture, the cells are the nodes of the cube
and the propositions are all the different ways of coloring the
nodes of the cube in just 2 colors, corresponding to "indicated"
and "undicated", or true and false, under the given proposition.
In general, for an n-cube, there are 2^n nodes for the singular
elements that the logician calls "interpretations", and 2^(2^n)
possible 2-colorings that represent the set of propositions.

HT: And the common elements of the three overlapping circles
    is the point where all three are true, and the "petals"
    not overlapping with anything are the points where one
    of three is true.  Have I got it right now?

Yes, that's the ticket.

HT: That still leaves my other question though -- if there is
    a common region where all three are true, then aren't there
    also three regions where two cells overlap -- which I guess
    now means "both are true" (rather than one of two is true,
    which was my previous interpretation)?

Here we come to the question that is sometimes described
as the distinction between contemplation and conviction,
or the difference between considering a proposition and
asserting it.  This is also bound up with the difference
between mention and use.  Historically speaking, a whole
lot more noise than signal has been emitted on this score.

Consider what we are doing when we draw a venn diagram like this:

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o````````````X````````````o                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````x (y)(z)````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o```o----------o--o             |
|            /````\          \`/          /````\            |
|           /``````\ x  y (z) o  x (y) z /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    /x y z\    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |````````Y````````|       |```````Z`````````|       |
|       |`````````````````|(x)y z |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       o````(x) y (z)````o       o```(x)(y) z``````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                         (x)(y)(z)                         |
|                                                           |
o-----------------------------------------------------------o
Figure 11.  Venn Diagram for ((x),(y),(z))

One thing that we are doing is according additional attention to
the gravely accented areas that we have shaded in like so [```].
That is done in order to represent one particular proposition
out of the 2^(2^3) = 2^8 = 256 possible propositions that are
available to us in this 3-dimensional universe of discourse.
Our practical reasons for representing such a proposition
may be various.  We may just be thinking about it, we may
believe that it's true, we may wish that it were true,
we may wish to convince somebody else that it's true
whether we believe it ourselves or not, and so on.
I'll bet you can think of a slew of other cases.

The mathematical way of handling the issue is in terms
of concepts that are known as "fibers" and "quotients".

Diff Log B 2004 • Discussion Note 5


I know that it's only Thursday, but it feels like
it ought to be Friday, so let me take a break and
touch base, if lightly in passing, on a couple of
issues that occasionally emerge in this framework,
even if my better judgment and worsted experience
tells me that I will most likely have to continue
putting them off until I'm older if not wiser.

Put Off 1.  Infinite Expressions

I wasn't quite sure what Hugh was getting at
with that "radiating tree" business, and the
mismatch between our character formats makes
my reconstruction of his illustrative Figure
just a bit hypothetical, but something about
the general drift and shape of what he wrote
reminds me of the problem area that concerns
the meaning of "infinite expressions", or as
it's sometimes met, "re-entrant expressions".

HT: Just throwing out a thought (as usual):
    if one of the units, or logical variables
    is stable, while the other is active, do we
    see a "radiating tree" something like this:

       active     active     active
             \       |      /
              \      |     /
       active--o  stable  o--active
              /      |     \
             /       |      \
       active     active     active

I will just put a flag in this for now,
and maybe revisit it later in the day.

I didn't say which day ...

Diff Log B 2004 • Discussion Note 6


HT = Hugh Trenchard

HT: I have this feeling that you were a teacher or instructor of
    some kind once.  You have the teacher's knack for planting
    seeds of thought in the minds of his students.  I think
    my "radiating tree" was an illustration of what you'd
    said about mutually inhibitory units in a neural pool
    at equilibrium.  In my tree "stable" and "active" could
    be either x or y, but there is only one "stable" because
    it does not change, while revolving around it is a series
    of "actives" -- these are "radiating" or revolving only
    because they are active.

HT: Now that may not be an accurate illustration of your point, but you
    inspired me to look up "re-entrant/infinite expressions" and one
    article online I found particularly interesting (granted I
    didn't look at very many!).

    www.eeng.dcu.ie/~alife/bmcm9401/vern2.pdf

HT: The article touches largely on the notion of self-reference in the
    context of autopoeitic systems.  While the article will require
    substantially more in-depth study (and much of it is simply far
    beyond my league), some of the logical discussion did have at
    least some superficial resemblances to my radiating tree.
    As I say, however, I am hopelessly unable at this time
    at least to say anything more on it than that.

HT: Have you any references that might be a sort of textbook introduction
    to re-entrant/infinite expressions?  Perhaps if I start with something
    basic, I can progress to a point where I can see if my radiating trees
    have any connection to the re-entrant/infinite expressions.

HT: "The barber shaves only those men who do not shave themselves".

This is in that distributed chapter known as "how I got into this ...
and may one day get around to again" (HIGIT ... AMODGATA).  There's
Spencer Brown's 'Laws of Form', Peter Aczel's 'Non-Well-Founded Sets',
Barwise & Etchemendy's 'The Liar', Barwise & Moss's 'Vicious Circles',
Manes & Arbib's 'Algebraic Approaches to Program Semantics', and then
there's Mandelbrot, just to name a few that come to mind right off.

These days, I tend to view all of this stuff as being a question --
not so much of loopiness -- but more of the relationship between
the finite and the infinite.  And what I found is mainly that we
are not nearly as good at the finite as we like to imagine, and
and so there's been all this work to do on that side of things.

Oh, did you see that there's now a discussion forum for NKS? --

http://forum.wolframscience.com/

Differential Logic 2004 • Document History

Differential Logic 2004 • Series A • History

Diff Log A 2004 • Ontology List • History

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05359.html
  2. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05360.html
  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05365.html
  4. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05366.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05367.html
  6. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05368.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05370.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05371.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05372.html
  10. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05373.html
  11. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05374.html
  12. http://web.archive.org/web/20070305081414/http://suo.ieee.org/ontology/msg05375.html
  13. http://web.archive.org/web/20070305081424/http://suo.ieee.org/ontology/msg05376.html
  14. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05377.html
  15. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05378.html
  16. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05379.html
  17. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05381.html
  18. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05383.html
  19. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05384.html
  20. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05385.html
  21. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05386.html

Diff Log A 2004 • Inquiry List • History

  1. http://stderr.org/pipermail/inquiry/2004-February/001132.html
  2. http://stderr.org/pipermail/inquiry/2004-February/001133.html
  3. http://stderr.org/pipermail/inquiry/2004-February/001138.html
  4. http://stderr.org/pipermail/inquiry/2004-February/001139.html
  5. http://stderr.org/pipermail/inquiry/2004-February/001140.html
  6. http://stderr.org/pipermail/inquiry/2004-February/001141.html
  7. http://stderr.org/pipermail/inquiry/2004-February/001143.html
  8. http://stderr.org/pipermail/inquiry/2004-February/001144.html
  9. http://stderr.org/pipermail/inquiry/2004-February/001145.html
  10. http://stderr.org/pipermail/inquiry/2004-February/001146.html
  11. http://stderr.org/pipermail/inquiry/2004-February/001147.html
  12. http://stderr.org/pipermail/inquiry/2004-February/001148.html
  13. http://stderr.org/pipermail/inquiry/2004-February/001149.html
  14. http://stderr.org/pipermail/inquiry/2004-February/001150.html
  15. http://stderr.org/pipermail/inquiry/2004-February/001151.html
  16. http://stderr.org/pipermail/inquiry/2004-February/001152.html
  17. http://stderr.org/pipermail/inquiry/2004-February/001154.html
  18. http://stderr.org/pipermail/inquiry/2004-February/001156.html
  19. http://stderr.org/pipermail/inquiry/2004-February/001157.html
  20. http://stderr.org/pipermail/inquiry/2004-February/001158.html
  21. http://stderr.org/pipermail/inquiry/2004-February/001159.html

Differential Logic 2004 • Series B • History

Diff Log B 2004 • Ontology List • History

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05387.html
  2. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05389.html
  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05393.html
  4. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05394.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05395.html
  6. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05396.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05397.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05398.html
  9. http://web.archive.org/web/20070228111146/http://suo.ieee.org/ontology/msg05399.html
  10. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05440.html
  11. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05444.html
  12. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05445.html
  13. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05448.html
  14. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05449.html
  15. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05450.html
  16. http://web.archive.org/web/20070225025757/http://suo.ieee.org/ontology/msg05451.html
  17. http://web.archive.org/web/20070225025807/http://suo.ieee.org/ontology/msg05452.html
  18. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05453.html
  19. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05455.html
  20. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05456.html

Diff Log B 2004 • Inquiry List • History

  1. http://stderr.org/pipermail/inquiry/2004-February/001160.html
  2. http://stderr.org/pipermail/inquiry/2004-February/001161.html
  3. http://stderr.org/pipermail/inquiry/2004-February/001165.html
  4. http://stderr.org/pipermail/inquiry/2004-February/001166.html
  5. http://stderr.org/pipermail/inquiry/2004-February/001167.html
  6. http://stderr.org/pipermail/inquiry/2004-February/001168.html
  7. http://stderr.org/pipermail/inquiry/2004-February/001169.html
  8. http://stderr.org/pipermail/inquiry/2004-February/001170.html
  9. http://stderr.org/pipermail/inquiry/2004-February/001171.html
  10. http://stderr.org/pipermail/inquiry/2004-February/001209.html
  11. http://stderr.org/pipermail/inquiry/2004-February/001213.html
  12. http://stderr.org/pipermail/inquiry/2004-February/001214.html
  13. http://stderr.org/pipermail/inquiry/2004-February/001217.html
  14. http://stderr.org/pipermail/inquiry/2004-February/001218.html
  15. http://stderr.org/pipermail/inquiry/2004-February/001219.html
  16. http://stderr.org/pipermail/inquiry/2004-February/001220.html
  17. http://stderr.org/pipermail/inquiry/2004-February/001221.html
  18. http://stderr.org/pipermail/inquiry/2004-February/001222.html
  19. http://stderr.org/pipermail/inquiry/2004-February/001225.html
  20. http://stderr.org/pipermail/inquiry/2004-February/001226.html

Differential Logic 2004 • Series C • History

Diff Log C 2004 • Ontology List • History

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05406.html

Diff Log C 2004 • Inquiry List • History

  1. http://stderr.org/pipermail/inquiry/2004-February/001178.html

Differential Logic 2004 • Discussion History

Differential Logic 2004 • Series A • Discussion History

Diff Log A 2004 • Ontology List • Discussion History

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05380.html
  2. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05390.html
  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05391.html
  4. http://web.archive.org/web/20070302155257/http://suo.ieee.org/ontology/msg05392.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05400.html
  6. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05401.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05402.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05403.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05404.html
  10. http://web.archive.org/web/20070228111246/http://suo.ieee.org/ontology/msg05405.html
  11. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05420.html
  12. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05421.html
  13. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05423.html
  14. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05425.html
  15. http://web.archive.org/web/20070228111623/http://suo.ieee.org/ontology/msg05426.html
  16. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05427.html
  17. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05429.html
  18. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05430.html
  19. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05431.html
  20. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05432.html
  21. http://web.archive.org/web/20070228111748/http://suo.ieee.org/ontology/msg05433.html
  22. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05434.html
  23. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05437.html
  24. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05438.html
  25. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05441.html

Diff Log A 2004 • Inquiry List • Discussion History

  1. http://stderr.org/pipermail/inquiry/2004-February/001153.html
  2. http://stderr.org/pipermail/inquiry/2004-February/001162.html
  3. http://stderr.org/pipermail/inquiry/2004-February/001163.html
  4. http://stderr.org/pipermail/inquiry/2004-February/001164.html
  5. http://stderr.org/pipermail/inquiry/2004-February/001172.html
  6. http://stderr.org/pipermail/inquiry/2004-February/001173.html
  7. http://stderr.org/pipermail/inquiry/2004-February/001174.html
  8. http://stderr.org/pipermail/inquiry/2004-February/001175.html
  9. http://stderr.org/pipermail/inquiry/2004-February/001176.html
  10. http://stderr.org/pipermail/inquiry/2004-February/001177.html
  11. http://stderr.org/pipermail/inquiry/2004-February/001192.html
  12. http://stderr.org/pipermail/inquiry/2004-February/001193.html
  13. http://stderr.org/pipermail/inquiry/2004-February/001194.html
  14. http://stderr.org/pipermail/inquiry/2004-February/001195.html
  15. http://stderr.org/pipermail/inquiry/2004-February/001196.html
  16. http://stderr.org/pipermail/inquiry/2004-February/001197.html
  17. http://stderr.org/pipermail/inquiry/2004-February/001198.html
  18. http://stderr.org/pipermail/inquiry/2004-February/001199.html
  19. http://stderr.org/pipermail/inquiry/2004-February/001200.html
  20. http://stderr.org/pipermail/inquiry/2004-February/001201.html
  21. http://stderr.org/pipermail/inquiry/2004-February/001202.html
  22. http://stderr.org/pipermail/inquiry/2004-February/001203.html
  23. http://stderr.org/pipermail/inquiry/2004-February/001206.html
  24. http://stderr.org/pipermail/inquiry/2004-February/001207.html
  25. http://stderr.org/pipermail/inquiry/2004-February/001210.html

Differential Logic 2004 • Series B • Discussion History

Diff Log B 2004 • Ontology List • Discussion History

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05446.html
  2. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05447.html
  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg05454.html

Diff Log B 2004 • Inquiry List • Discussion History

  1. http://stderr.org/pipermail/inquiry/2004-February/001215.html
  2. http://stderr.org/pipermail/inquiry/2004-February/001216.html
  3. http://stderr.org/pipermail/inquiry/2004-February/001223.html
  4. http://stderr.org/pipermail/inquiry/2004-February/001224.html
  5. http://stderr.org/pipermail/inquiry/2004-February/001227.html
  6. http://stderr.org/pipermail/inquiry/2004-February/001229.html