Talk:Differential Logic : Introduction
Contents
 1 Frankl 5
 2 Frankl 6
 3 Frankl 7
 4 Frankl 8
 5 Frankl 9
 6 Frankl 10
 7 Frankl 11
 8 Frankl 12
 9 Frankl 13
 10 Differential Operator Tables • Conjunction PQR
 11 Venn Diagrams • Three Variables P Q R
 12 Work Area • 2 Variables
 13 Work Area • 3 Variables
 14 Work Area • Table Images
 15 Work Area • Dispositions
 16 Work Area • Exposition
 17 Work Area • Logical Cacti
Frankl 5
Putting all thought of the Frankl Conjecture out of our minds for the moment, let's return to the proposition in Example 1 and work through its differential analysis from scratch.
Example 1
(1) 
Consider the proposition in boolean terms the function such that as illustrated by the venn diagram in Figure 1.
The enlargement of is the boolean function defined by the following equation:
Given that is the boolean product of its three arguments, may be written as follows:
Difficulties of notation in differential logic are greatly eased by introducing the family of minimal negation operators on finite numbers of boolean variables. For our immediate purposes we need only the minimal negation operators on one and two variables.
 The minimal negation operator on one variable is notated with monospace parentheses as and is simply another notation for the logical negation
 The minimal negation operator on two variables is notated with monospace delimiters as and is simply another notation for the exclusive disjunction
In this notation the previous expression for takes the following form:
A canonical form for may be derived by means of a boolean expansion, in effect, a case analysis that evaluates at each triple of values for the base variables and forms the disjunction of the partial evaluations. Each term of the boolean expansion corresponds to a cell of the venn diagram and is formed by multiplying the value of that cell by a coefficient that amounts to the value of on that cell.
For example, in the case where all three base variables are true, the corresponding coefficient is computed as follows:

Collecting the cases yields the boolean expansion of via the following computation:
Step 1

Step 2

Frankl 6
(3) 
Figure 3 shows the eight terms of the enlarged proposition as arcs, arrows, or directed edges in the venn diagram of the original proposition Each term of the enlargement corresponds to an arc into the cell where is true from one of the eight cells of the venn diagram.
For ease of reference, here is the expansion of from the previous post:

Two examples suffice to convey the general idea of the enlarged venn diagram:
 The term is shown as a looped arc starting in the cell where is true and returning back to it. The differential factor corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
 The term is shown as an arc from the cell where is true to the cell where is true. The differential factor corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.
Frankl 7
We continue with the differential analysis of the proposition in Example 1.
Example 1
(1) 
A proposition defined on one universe of discourse has natural extensions to larger universes of discourse. As a matter of course in a given context of discussion, some of these extensions come to be taken for granted as the most natural extensions to make in passing from one universe to the next and they tend to be assumed automatically, by default, in the absence of explicit notice to the contrary. These are the tacit extensions that apply in that context.
Differential logic, at the first order of analysis, treats extensions from boolean spaces of type to enlarged boolean spaces of type In this setting but we use different letters merely to distinguish base and differential features.
In our present example, the tacit extension of is the boolean function defined by the following equation:
The boolean expansion of takes the following form:

In other words, is simply on the base variables extended by a tautology — commonly known as a “Don't Care” condition — on the differential variables
Frankl 8
(4) 
Figure 4 shows the eight terms of the tacit extension as arcs, arrows, or directed edges in the venn diagram of the original proposition Each term of the tacit extension corresponds to an arc that starts from the cell where is true and ends in one of the eight cells of the venn diagram.
For ease of reference, here is the expansion of from the previous post:

Two examples suffice to convey the general idea of the extended venn diagram:
 The term is shown as a looped arc starting in the cell where is true and returning back to it. The differential factor corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
 The term is shown as an arc going from the cell where is true to the cell where is true. The differential factor corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.
Frankl 9
“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 dressingroom, he passed a punchingball 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
We continue with the differential analysis of the proposition in Example 1.
Example 1
(1) 
The difference operator is defined as the difference between the enlargement operator and the tacit extension operator
The difference map is the result of applying the difference operator to the function When the sense is clear, we may refer to simply as the difference of
In boolean spaces there is no difference between the sum and the difference so the difference operator is equally well expressed as the exclusive disjunction or symmetric difference In this case the difference map can be computed according to the formula
The action of on our present example, can be computed from the data on hand according to the following prescription.
The enlargement map computed in Post 5 and graphed in Post 6, is shown again here:

The tacit extension computed in Post 7 and graphed in Post 8, is shown again here:

The difference map is the sum of the enlargement map and the tacit extension
Here we adopt a paradigm of computation for that aids not only in organizing the stages of the work but also in highlighting the diverse facets of logical meaning that may be read off the result.
The terms of the enlargement map are obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.
The terms of the tacit extension are obtained from the next table below by multiplying the base factor at the head of the first column by each of the differential factors that appear beneath it in the body of the table.
Finally, the terms of the difference map are obtained by overlaying the displays for and and taking their boolean sum entry by entry.
Notice that the “loop” or “no change” term cancels out, leaving 14 terms in the end.
Frankl 10
(5) 
Figure 5 shows the 14 terms of the difference map as arcs, arrows, or directed edges in the venn diagram of the original proposition The arcs of are directed into the cell where is true from each of the other cells. The arcs of are directed from the cell where is true into each of the other cells.
The expansion of computed in the previous post is shown again below with the terms arranged by number of positive differential features, from lowest to highest.


Frankl 11
Let's take a moment from the differential analysis of the proposition in Example 1 to form a handy compendium of the results obtained so far.
Example 1
(1) 
Enlargement Map E(pqr) of the Conjunction pqr
(3) 
Tacit Extension ε(pqr) of the Conjunction pqr
(4) 
Difference Map D(pqr) of the Conjunction pqr
(5) 
Frankl 12
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.
We continue with the differential analysis of the proposition in Example 1.
Example 1
(1) 
Like any moderately complex proposition, the difference map of a proposition has many equivalent logical expressions and can be read in many different ways.
(5) 
The expansion of computed in Post 9 and further discussed in Post 10 is shown again below with the terms arranged by number of positive differential features, from lowest to highest.


The terms of the difference map may be obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.
The full boolean expansion of may be condensed to a degree by collecting terms that share the same base factors, as shown in the following display:

This amounts to summing terms along columns of the previous table, as shown at the bottom margin of the next table:
Collecting terms with the same differential factors produces the following expression:

This is roughly what one would get by summing along rows of the previous tables.
Frankl 13
We have come to a fork in the road. I wish I could take the wit's advice and Take It! but the fork has more tines than I can count and the horse knows the way Not! to ride off in all directions at once. But I can see two classes of choices clearly enough to consider the possibilities of at least those two.
The work we've been doing so far amounts to applying the logical analogue of the finite difference calculus to the proposition in Example 1.
Algebraically,

Consequently,

Consequently,


Figures 46a through 46d illustrate the proposition rounded out in our usual array of prospects. This proposition of is what we refer to as the (first order) differential of and normally regard as the differential proposition corresponding to

Option 1. Tangent Map as Local Linear Approximation

Table A9. Tangent Proposition as Pointwise Linear Approximation



















Differential Operator Tables • Conjunction PQR
Version 1
See Frankl 9





Version 2
See Frankl 10





Version 3





Version 4





Venn Diagrams • Three Variables P Q R
Venn Diagram • Empty Frame P Q R
Venn Diagram • Conjunction PQR
Venn Diagram • Frankl Figure 2
Work Area • 2 Variables
Difference Map of Conjunction
D(uv) • Version 1

With the tacit extension map and the enlargement map well in place, the difference map can be computed along the lines displayed in Table 41, ending up with an expansion of over the cells of



Alternatively, the difference map can be expanded over the cells of 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 in the middle portion of the Table.

Even more simply, the same result is reached by matching up the propositional coefficients of and along the cells of and adding the pairs under boolean addition, that is, “mod 2”, where 1 + 1 = 0, as shown in Table 43.



The difference map can also be given a dispositional interpretation. First, recall that exhibits the dispositions to change from anywhere in to anywhere at all in the universe of discourse and exhibits the dispositions to change from anywhere in the universe to anywhere in Next, observe that each of these classes of dispositions may be divided in accordance with the case of versus that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to and have in common the dispositions to preserve their symmetric difference is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of in one direction or the other. In other words, we may conclude that expresses the collective disposition to make a definite change with respect to no matter what value it holds in the current state of affairs.

Figures 44a through 44d illustrate the difference proposition
D(uv) • Version 2

With the enlargement map and the tacit extension map well in place, the difference map can be computed along the lines displayed in Table 41, ending up with an expansion of over the cells of



Alternatively, the difference map can be expanded over the cells of 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 in the middle portion of the Table.

Differential of Conjunction

Figures 46a through 46d illustrate the proposition rounded out in our usual array of prospects. This proposition of is what we refer to as the (first order) differential of and normally regard as the differential proposition corresponding to
Work Area • 3 Variables
Difference Map of Conjunction
Version 0


Version 1 (a)


Version 1 (b)


Version 2


Version 3


Version 4


Summary Tables

Various Arrays




Column Sum

Row Sum


Work Area • Table Images
Version 1.0
Version 2.0
Version 3.0
Version 4.0
Work Area • Dispositions
The difference map may be given a dispositional interpretation. The enlargement map exhibits the dispositions to change from anywhere at all in the universe of discourse to anywhere in the region where is true and the tacit extension exhibits the dispositions to change from anywhere in the region where is true to anywhere at all in the universe of discourse.
In the interest of a compact notation for concepts like “the region where is true” and “the region where is false” let's turn the game symbol to the purpose in a way that makes the following formulas equivalent:
Definition. For a booleanvalued function the following are equivalent:
Thus we have that is the region where is true and is the region where is false.
The sets of dispositions to and from the region where is true may each be divided in accordance with the case of versus that applies to their points of destination and departure. Since the dispositions corresponding to and have in common the dispositions to preserve their symmetric difference is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of in one direction or the other. In other words, we may conclude that expresses the collective disposition to make a definite change with respect to no matter what value it holds in the current state of affairs.

Work Area • Exposition
What is the function of a boolean function?
To answer this question let's begin by recalling a couple of basic definitions:
 A booleanvalued function is a function of the form where is an arbitrary set and where is a boolean domain, typically, .
 A boolean function is a function of the form where is a nonnegative integer. In the case where a boolean function is simply a constant element of
For most intents and purposes, its function is to indicate a portion of the universe of discourse, namely, the portion where its value is 1, and by contrast to counterindicate the portion where its value is 0. That is why boolean functions are often called indicator functions. There are two types of situation where indicators functions come into play.
 In the more general type of situation we have a booleanvalued function that indicates the subset of where In formal terms, the indicated subset is described as
 In the more special type of situation we have a boolean function ...
The function of is to indicate a portion of the universe of discourse that portion where its value is and by contrast to counterindicate that portion where its value is
In many situations the universe of discourse is described in terms of a finite boolean coordinate basis. This amounts to a finite set of boolean coordinate functions Relative to this coordinates basis the indicative function of may be relegated to the boolean function
Work Area • Logical Cacti
 Theme One Program • Exposition • Logical Cacti
 http://web.archive.org/web/20150109155110/http://stderr.org/pipermail/inquiry/2005February/thread.html#2348
 http://web.archive.org/web/20150109155110/http://stderr.org/pipermail/inquiry/2005February/thread.html#2360
 http://web.archive.org/web/20150109152359/http://stderr.org/pipermail/inquiry/2005February/002360.html
 http://web.archive.org/web/20150109152401/http://stderr.org/pipermail/inquiry/2005February/002361.html
Original Version
Up till now we've been working to hammer out a twoedged sword of syntax, honing the syntax of painted and rooted cacti and expressions (PARCAE), and turning it to use in taming the syntax of twolevel formal languages.
But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs that can denote objective propositions about some universe of objects.
One of the difficulties that we face in this discussion is that the words interpretation, meaning, semantics, and so on will have so many different meanings from one moment to the next of their use. A dedicated neologician might be able to think up distinctive names for all of the aspects of meaning and all of the approaches to them that will concern us here, but I will just have to do the best that I can with the common lot of ambiguous terms, leaving it to context and the intelligent interpreter to sort it out as much as possible.
As it happens, the language of cacti is so abstract that it can bear at least two different interpretations as logical sentences denoting logical propositions. The two interpretations that I know about are descended from the ones that Charles Sanders Peirce called the entitative and the existential interpretations of his systems of graphical logics. For our present aims, I shall briefly introduce the alternatives and then quickly move to the existential interpretation of logical cacti.
Table A illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
 
 
 
 
 
 
 
 

Table B illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
 
 
 
 
 
 
 
 

For the time being, the main things to take away from Tables A and B are the ideas that the compositional structure of cactus graphs and expressions can be articulated in terms of two different kinds of connective operations, and that there are two distinct ways of mapping this compositional structure into the compositional structure of propositional sentences, say, in English:
1.  The node connective joins a number of component cacti at a node: 
C_1 ... C_k @  
2.  The lobe connective joins a number of component cacti to a lobe: 
C_1 C_2 C_k oo...o \ / \ / \ / \ / @ 
Table 15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.
Table 15. Existential & Entitative Interpretations of Cactus Structures ooooo  Cactus Graph  Cactus String  Existential  Entitative     Interpretation  Interpretation  ooooo       @  " "  true  false       ooooo       o            @  ( )  false  true       ooooo       C_1 ... C_k      @  C_1 ... C_k  C_1 & ... & C_k  C_1 v ... v C_k       ooooo       C_1 C_2 C_k   Just one  Not just one   oo...o      \ /   of the C_j,  of the C_j,   \ /      \ /   j = 1 to k,  j = 1 to k,   \ /      @  (C_1, ..., C_k)  is not true.  is true.       ooooo 
It is possible to specify abstract rules of equivalence (AROEs) between cacti, rules for transforming one cactus into another that are formal in the sense of being indifferent to the above choices for logical or semantic interpretations, and that partition the set of cacti into formal equivalence classes.
A reduction is an equivalence transformation that is applied in the direction of decreasing graphical complexity.
A basic reduction is a reduction that applies to one of the two families of basic connectives.
Table 16 schematizes the two types of basic reductions in a purely formal, interpretationindependent fashion.
Table 16. Basic Reductions oo    C_1 ... C_k   @ = @     if and only if     C_j = @ for all j = 1 to k    oo    C_1 C_2 C_k   oo...o   \ /   \ /   \ /   \ /   @ = @     if and only if     o      C_j = @ for exactly one j in [1, k]    oo 
The careful reader will have noticed that we have begun to use graphical paints like "a", "b", "c" and schematic proxies like "C_1", "C_j", "C_k" in a variety of novel and unjustified ways.
The careful writer would have already introduced a whole bevy of technical concepts and proved a whole crew of formal theorems to justify their use before contemplating this stage of development, but I have been hurrying to proceed with the informal exposition, and this expedition must leave steps to the reader's imagination.
Of course I mean the active imagination. So let me assist the prospective exercise with a few hints of what it would take to guarantee that these practices make sense.
Partial Rewrites
Table 13 illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
Even though I do most of my thinking in the existential interpretation, I will continue to speak of these forms as logical graphs, because I think it is an important fact about them that the formal validity of the axioms and theorems is not dependent on the choice between the entitative and the existential interpretations.
The first extension is the reflective extension of logical graphs (RefLog). It is obtained by generalizing the negation operator "" in a certain way, calling "" the controlled, moderated, or reflective negation operator of order 1, then adding another such operator for each finite
In sum, these operators are symbolized by bracketed argument lists as follows: "", "", "", …, where the number of slots is the order of the reflective negation operator in question.
The cactus graph and the cactus expression shown here are both described as a spike.
oo    o      @    oo  ( )  oo 
The rule of reduction for a lobe is:
oo    x_1 x_2 ... x_k   oo ... o   \ /   \ /   \ /   \ /   \ /   \ /   \ /   \ /   @ = @    oo 
if and only if exactly one of the is a spike.
In Ref Log, an expression of the form expresses the fact that exactly one of the is true. Expressions of this form are called universal partition expressions, and they parse into a type of graph called a painted and rooted cactus (PARC):
oo    e_1 e_2 ... e_k   o o o        oo ... o   \ /   \ /   \ /   \ /   \ /   \ /   \ /   \ /   @    oo 
oo    ( x1, x2, ..., xk ) = [blank]     iff     Just one of the arguments   x1, x2, ..., xk = ()    oo 
The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
Existential Interpretation:  Just one of the k argument is false. 
Entitative Interpretation:  Not just one of the k arguments is true. 
Tables



 

 

 

 

 



 

 

 

 


