Differential Logic : Dynamics And Logic
Author: Jon Awbrey
I am going to excerpt some of my previous explorations on differential logic and dynamic systems and bring them to bear on the sorts of discrete dynamical themes that we find of interest in the NKS Forum. This adaptation draws on the “Cactus Rules”, “Propositional Equation Reasoning Systems”, and “Reductions Among Relations” threads, and will in time be applied to the “Differential Analytic Turing Automata” thread:
Contents
 1 Note 1
 2 Note 2
 3 Note 3
 4 Note 4
 5 Note 5
 6 Note 6
 7 Note 7
 8 Note 8
 9 Note 9
 10 Note 10
 11 Note 11
 12 Note 12
 13 Note 13
 14 Note 14
 15 Note 15
 16 Note 16
 17 Note 17
 18 Note 18
 19 Note 19
 20 Note 20
 21 Note 21
 22 Note 22
 23 Note 23
 24 Note 24
 25 Note 25
 26 Note 26
 27 Further Reading
 28 Document History
Note 1
One of the first things that you can do, once you have a moderately efficient calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.
Let us start with a proposition of the form that is graphed as two labels attached to a root node:
Written as a string, this is just the concatenation .
The proposition may be taken as a boolean function having the abstract type where is read in such a way that means and means
In this style of graphical representation, the value looks like a blank label and the value looks like an edge.
Back to the proposition Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition is true, as shown in the following Figure:
Now ask yourself: What is the value of the proposition at a distance of and from the cell where you are standing?
Don't think about it — just compute:
The cactus formula and its corresponding graph arise by substituting for and for in the boolean product or logical conjunction and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
Next question: What is the difference between the value of the proposition over there, at a distance of and and the value of the proposition where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
There is one thing that I ought to mention at this point: Computed over plus and minus are identical operations. This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where is true? Well, substituting for and for in the graph amounts to erasing the labels and as shown here:
And this is equivalent to the following graph:
Note 2
We have just met with the fact that the differential of the and is the or of the differentials.

It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.
If the form of the above statement reminds you of De Morgan'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–De Morgan 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 has been the lack of a syntax that was adequate to handle the complexity of expressions that evolve.
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
A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like or The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
Let be the set of values 
Let be the set of values 
Then interpret the usual propositions about as functions of the concrete type
We are going to consider various operators on these functions. Here, an operator is a function that takes one function into another function
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
The difference operator written here as 
The enlargement" operator written here as 
These days, is more often called the shift operator.
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space its (first order) differential extension is constructed according to the following specifications:

where:

The interpretations of these new symbols can be diverse, but the easiest option for now is just to say that means "change " and means "change ".
Drawing a venn diagram for the differential extension requires four logical dimensions, but it is possible to project a suggestion of what the differential features and are about on the 2dimensional base space by drawing arrows that cross the boundaries of the basic circles in the venn diagram for reading an arrow as if it crosses the boundary between and in either direction and reading an arrow as if it crosses the boundary between and in either direction.
Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways that propositions are formed on ordinary logical variables alone. For example, the proposition says the same thing as in other words, that there is no change in without a change in
Note 3
Given the proposition over the space the (first order) enlargement of is the proposition over the differential extension that is defined by the following formula:

In the example the enlargement is computed as follows:

Given the proposition over the (first order) difference of is the proposition over that is defined by the formula or, written out in full:

In the example the difference is computed as follows:

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 that is, at the place where and This evaluation is written in the form or and we arrived at the locally applicable law that is stated and illustrated as follows:

The picture shows the analysis of the inclusive disjunction into the following exclusive disjunction:

The differential proposition that results may be interpreted to say “change or change 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 require a complete and detailed description of ways to escape it.
Note 4
Last time we computed what is variously called the difference map, the difference proposition, or the local proposition of the proposition at the point where and
In the universe the four propositions 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 respectively.
Thus we can write so long as we know the frame of reference in force.
In the example the value of the difference proposition at each of the four points in may be computed in graphical fashion as shown below:
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.
The Figure shows the points of the extended universe that are indicated by the difference map namely, the following six points or singular propositions::

The information borne by should be clear enough from a survey of these six points — they tell you what you have to do from each point of in order to change the value borne by that is, the move you have to make in order to reach a point where the value of the proposition is different from what it is where you started.
Note 5
We have been studying the action of the difference operator on propositions of the form as illustrated by the example that is known in logic as the conjunction of and The resulting difference map is a (first order) differential proposition, that is, a proposition of the form
Abstracting from the augmented venn diagram that shows how the models or satisfying interpretations of distribute over the extended universe of discourse the difference map can be represented in the form of a digraph or directed graph, one whose points are labeled with the elements of and whose arrows are labeled with the elements of as shown in the following Figure.

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning. We will encounter more and more of these alternative readings as we go.
Note 6
The enlargement or shift operator exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example,
A suitably generic definition of the extended universe of discourse is afforded by the following setup:

For a proposition of the form the (first order) enlargement of is the proposition that is defined by the following equation:

The differential variables are boolean variables of the same basic type as the ordinary variables Although it is conventional to distinguish the (first order) differential variables with the operative prefix "" this way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.
In the example of logical conjunction, the enlargement is formulated as follows:

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 following result:

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the above expression for in the same way that we did for Toward that end, the value of at each may be computed in graphical fashion as shown below:
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition

Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element is drawn as a loop at the point

We may understand the enlarged proposition as telling us all the different ways to reach a model of the proposition from each point of the universe
Note 7
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type and abstract type A few Tables are set here that detail the actions of and on each of these functions, allowing us to view the results in several different ways.
Tables A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.










































Note 8
The next four Tables expand the expressions of and in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, partitioned into seven natural classes by a group action.

































 






























Note 9




























































Note 10
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 longengrained 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 and 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 need 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 A3.






























The shift operator can be understood as enacting a substitution operation on the propositional form that is given as its argument. In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:

Evaluating at particular values of and for example, and where and are values in produces the following result:

The notation is a little awkward, but the data of Table A3 should make the sense clear. The important thing to observe is that has the effect of transforming each proposition into a proposition As it happens, the action of each is onetoone and onto, so the gang of four operators 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 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:



 
It happens that there are just two possible groups of 4 elements. One is the cyclic group (from German Zyklus), which this is not. The other is the Klein fourgroup (from German Vier), which this 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, operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get: Amazing!
Note 11
We have been contemplating functions of the type and studying the action of the operators and 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 snowcovered 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 and abstract types 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 of concrete type and abstract type that is written in cactus syntax. Taken as an assertion in what Peirce called the existential interpretation, the proposition says that just one of is false. It is instructive to consider this assertion in relation to the logical conjunction of the same propositions. A venn diagram of looks like this:
In relation to the center cell indicated by the conjunction the region indicated by is comprised of the adjacent or bordering cells. Thus they are the cells that are just across the boundary of the center cell, reached as if by way of Leibniz's minimal changes from the point of origin, in this case,
More generally speaking, in a dimensional universe of discourse that is based on the alphabet of features the same form of boundary relationship is manifested for any cell of origin that one chooses to indicate. One way to indicate a cell is by forming a logical conjunction of positive and negative basis features, that is, by constructing an expression of the form where for The proposition indicates the disjunctive region consisting of the cells that are just next door to
Note 12
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, "Issues of Pragmaticism", (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 fourgroup We encountered this group in one of its concrete representations, namely, as a transformation group that acts on a set of objects, in this 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, for example, in the form of the group operation table copied here:



 
This table is abstractly the same as, or isomorphic to, the versions with the operators and the transformations that we took up earlier. That is to say, the story is the same, only the names have been changed. An abstract group can have a variety of significantly and superficially different representations. But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called regular representations, that are always readily available, as they can be generated from the mere data of the abstract operation table itself.
To see how a regular representation is constructed from the abstract operation table, 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, as a logical disjunction or boolean sum whose terms represent the ordered pairs of transactions that are produced by each group element in turn. This forms one of the two possible regular representations of the group, in this case the one that is called the postregular representation or the right regular representation. It has long been conventional to organize the terms of this logical aggregate in the form of a matrix:
Reading "" as a logical disjunction:

And so, by expanding effects, we get:

More on the pragmatic maxim as a representation principle later.
Note 13
The abovementioned fact about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:
Every group is isomorphic to a subgroup of the group of automorphisms of a suitably chosen set . 
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 where you can imagine that symbol playing a role. 
This idea of contextual definition by way of conduct transforming operators 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, From Frege to Gödel, p. 216). Today we'd call these constructions term models. This, again, is the big idea behind Schönfinkel's combinators and hence of lambda calculus, and I reckon you know where that leads.
Note 14
The next few excursions in this series will provide a scenic tour of various ideas in group theory that will turn out to be of constant guidance in several of the settings that are associated with our topic.
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 describes the action of an "elementary dual relative" in this way:
Elementary simple relatives are connected together in systems of four. For if be taken to denote the elementary relative which multiplied into gives then this relation existing as elementary, we have the four elementary relatives 
C.S. Peirce, Collected Papers, CP 3.123. 
Peirce is well aware that it is not at all necessary to arrange the elementary relatives of a relation into arrays, matrices, or tables, but when he does so he tends to prefer organizing 2adic relations in the following manner:

For example, given the set suppose that we have the 2adic relative term and the associated 2adic relation the general pattern of whose common structure is represented by the following matrix:

For at least a little while longer, I will keep explicit the distinction between a relative term like and a relation like but it is best to view both these entities as involving different applications of the same information, and so we could just as easily write the following form:

By way of making up a concrete example, let us say that or is given as follows:

In sum, then, the relative term and the relation are both represented by the following matrix:

I think this much will serve to fix notation and set up the remainder of the discussion.
Note 15
In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives as the indices 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:

The various representations of are nothing more than alternative ways of specifying its basic ingredients, namely, the following aggregate of elementary relatives:

Recognizing that is the identity transformation otherwise known as the 2adic relative term can be parsed as an element of the socalled group ring, all of which makes this element just a special sort of linear transformation.
Up to this point, we are still reading the elementary relatives of the form in the way that Peirce read them in logical contexts: is the relate, is the correlate, and in our current example or more exactly, is taken to say that is a marker for 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 the relate and the correlate, the elementary relative now means that gets changed into In this scheme of reading, the transformation is a permutation of the aggregate or what we would now call the set in particular, it is the permutation that is otherwise notated as follows:

This is consistent with the convention that Peirce uses in the paper “On a Class of Multiple Algebras” (CP 3.324–327).
Note 16
We've been exploring the applications of a certain technique for clarifying abstruse concepts, a roughcut version of the pragmatic maxim that I've been accustomed to refer to as the operationalization of ideas. The basic idea is to replace the question of What it is, which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of What it does, which most people know at least a modicum about.
In the case of regular representations of groups we found a nonplussing 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 once again:



 
A group operation table is really just a device for recording a certain 3adic relation, to be specific, the set of triples of the form satisfying the equation
In the case of where is the underlying set we have the 3adic relation whose triples are listed below:

It is part of the definition of a group that the 3adic relation is actually a function 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 we can define a couple of substitution operators:
1.  puts any specified into the empty slot of the rheme with the effect of producing the saturated rheme that evaluates to 
2.  puts any specified into the empty slot of the rheme with the effect of producing the saturated rheme that evaluates to 
In (1) we consider the effects of each in its practical bearing on contexts of the form as ranges over and the effects are such that takes into for in all of which is notated as The pairs can be found by picking an from the left margin of the group operation table and considering its effects on each in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular anterepresentation:

In (2) we consider the effects of each in its practical bearing on contexts of the form as ranges over and the effects are such that takes into for in all of which is notated as The pairs can be found by picking an from the top margin of the group operation table and considering its effects on each in turn as these run down the left margin. This aspect of pragmatic definition we recognize as the regular postrepresentation:

If the anterep looks the same as the postrep, now that I'm writing them in the same dialect, that is because is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.
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 nonabelian (noncommutative) group. This is a group of six elements, say, 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, usually notated as or more abstractly and briefly, as or The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in






Here is the operation table for given in abstract fashion:
By the way, we will meet with the symmetric group 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).
Note 18
By way of collecting a shortterm payoff for all the work that we did on the regular representations of the Klein 4group let us write out as quickly as possible in relative form a minimal budget of representations for the symmetric group on three letters, 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 obscured details of Peirce's early “Algebra + Logic” papers.
Writing the permutations or substitutions of in relative form generates what is generally thought of as a natural representation of

I have without stopping to think about it written out this natural representation of in the style that comes most naturally to me, to wit, the “right” way, whereby an ordered pair configured as constitutes the turning of into It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.
Note 19
To construct the regular representations of we begin with the data of its operation table:
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 3adic relation is actually a function 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 we can define a couple of substitution operators:
1.  puts any specified into the empty slot of the rheme with the effect of producing the saturated rheme that evaluates to 
2.  puts any specified into the empty slot of the rheme with the effect of producing the saturated rheme that evaluates to 
In (1) we consider the effects of each in its practical bearing on contexts of the form as ranges over and the effects are such that takes into for in all of which is notated as The pairs can be found by picking an from the left margin of the group operation table and considering its effects on each in turn as these run along the right margin. This produces the regular anterepresentation of like so:

In (2) we consider the effects of each in its practical bearing on contexts of the form as ranges over and the effects are such that takes into for in all of which is notated as The pairs can be found by picking an on the right margin of the group operation table and considering its effects on each in turn as these run along the left margin. This produces the regular postrepresentation of like so:

If the anterep looks different from the postrep, it is just as it should be, as is nonabelian (noncommutative), 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.
Note 20
the way of heaven and earth  
— i ching, hexagram 32 
The Reader may be wondering what happened to the announced subject of Dynamics And Logic. What happened was a bit like this:
We made the observation that the shift operators form a transformation group that acts on the set of propositions of the form Group theory is a very attractive subject, but it did not draw us so far from our intended course as one might initially think. For one thing, groups, especially the groups that are named after the Norwegian mathematician Marius Sophus Lie (1842–1899), have turned out to be of critical utility in the solution of differential equations. For another thing, group operations provide us with an ample supply of triadic relations that have been extremely wellstudied over the years, and thus they give us no small measure of useful guidance in the study of sign relations, another brand of 3adic relations that have significance for logical studies, and in our acquaintance with which we have barely begun to break the ice. Finally, I couldn't resist taking up the links between group representations, amounting to the very archetypes of logical models, and the pragmatic maxim.
Note 21
We've seen a couple of groups, and 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 socalled matrix representation of a group.
Recalling the manner of our acquaintance with the symmetric group we began with the bigraph (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set






These permutations were then converted to relative form as logical sums of elementary relatives:

From the relational representation of one easily derives a linear representation of the group by viewing each permutation as a linear transformation that maps the elements of a suitable vector space onto each other. Each of these linear transformations is in turn represented by a 2dimensional array of coefficients in resulting in the following set of matrices for the group:






The key to the mysteries of these matrices is revealed by observing that their coefficient entries are arrayed and overlaid on a placemat marked like so:

Note 22
Let us summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far. We've been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, to considering a larger universe of discourse, An operator of this general type, namely, acts on each proposition of the source universe to produce a proposition of the target universe
The two main operators that we've examined so far are the enlargement or shift operator and the difference operator The operators and act on propositions in that is, propositions of the form that are said to be about the subject matter of and they produce extended propositions of the forms propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in
At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us as we venture higher into the ever more rarefied air of abstractions.
One good picture comes to us by way of the field concept. Given a space a field of a specified type over is formed by associating with each point of an object of type If that sounds like the same thing as a function from to the space of things of type — it is nothing but — and yet it does seem helpful to vary the mental images and to take advantage of the figures of speech that spring to mind under the emblem of this field idea.
In the field picture, a proposition becomes a scalar field, that is, a field of values in
Let us take a moment to view an old proposition in this new light, for example, the logical conjunction pictured in Figure 22a.
Each of the operators takes us from considering propositions here viewed as scalar fields over to considering the corresponding differential fields over analogous to what are usually called vector fields over
The structure of these differential fields can be described this way. With each point of there is associated an object of the following type: a proposition about changes in that is, a proposition In this frame of reference, if is the universe that is generated by the set of coordinate propositions then is the differential universe that is generated by the set of differential propositions These differential propositions may be interpreted as indicating and respectively.
A differential operator of the first order class that we have been considering, takes a proposition and gives back a differential proposition In the field view, we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in
The field of changes produced by on is shown in Figure 22b.

The differential field specifies the changes that need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition
The field of changes produced by on is shown in Figure 22c.

The differential field specifies the changes that need to be made from each point of in order to feel a change in the felt value of the field
Note 23
I want to continue developing the basic tools of differential logic, which arose from exploring the connections between dynamics and logic, but I also wanted to give some hint of the applications that have motivated this work all along. One of these applications is to cybernetic systems, whether we see these systems as agents or cultures, individuals or species, organisms or organizations.
A cybernetic system has goals and actions for reaching them. It has a state space giving us all of the states that the system can be in, plus it has a goal space the set of states that the system “likes” to be in, in other words, the distinguished subset of possible states where the system is regarded as living, surviving, or thriving, depending on the type of goal that one has in mind for the system in question. As for actions, there is to begin with the full set of all possible actions, each of which is a transformation of the form but a given cybernetic system will most likely have but a subset of these actions available to it at any given time. And even if we begin by thinking of actions in very general and very global terms, as arbitrarily complex transformations acting on the whole state space we quickly find a need to analyze and approximate them in terms of simple transformations acting locally. The preferred measure of “simplicity” will of course vary from one paradigm of research to another.
A generic enough picture at this stage of the game, and one that will remind us of these fundamental features of the cybernetic system even as things get far more complex, is afforded by Figure 23.
oo    X   oo   / \   / \   / \   / \   / \   / \   / \   o G o                o<To               o o   \ /   \ /   \ /   \ /   \ /   \ /   \ /   oo      oo Figure 23. Elements of a Cybernetic System 
Note 24
Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition its enlargement or shift map and its difference map can now be drawn.
To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition giving the development a slightly different twist at the appropriate point.
Figure 241 shows the proposition once again, which we now view as a scalar field — analogous to a potential hill in physics, but in logic tantamount to a potential plateau — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.
Given a proposition the tacit extension of to is denoted and defined by the equation so it's really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.
Figure 242 shows the tacit extension of the scalar field to the differential field

Note 25
Continuing with the example Figure 251 shows the enlargement or shift map in the same style of differential field picture that we drew for the tacit extension

A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields and both of the type is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.
In this case one notices that the tacit extension and the enlargement are in a certain sense dual to each other. The tacit extension indicates all the arrows out of the region where is true and the enlargement indicates all the arrows into the region where is true. The only arc they have in common is the nochange loop at If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of that are illustrated in Figure 252.

Note 26
If we follow the classical line that singles out linear functions as ideals of simplicity, then we may complete the analytic series of the proposition in the following way.
Figure 261 shows the differential proposition that we get by extracting the cellwise linear approximation to the difference map This is the logical analogue of what would ordinarily be called the differential of but since I've been attaching the adjective differential to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name tangent map for
Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:

To understand the extended interpretations, that is, the conjunctions of basic and differential features that are being indicated here, it may help to note the following equivalences:

Capping the series that analyzes the proposition in terms of succeeding orders of linear propositions, Figure 262 shows the remainder map that happens to be linear in pairs of variables.
Reading the arrows off the map produces the following data:

In short, is a constant field, having the value at each cell.
Further Reading
A more detailed presentation of Differential Logic can be found here:
Document History
Differential Logic • Ontology List 2002
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Dynamics And Logic • Inquiry List 2004
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