Differential Logic • Part 3

Author: Jon Awbrey

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• Overview • Part 1 • Part 2 • Part 3 • Document History •

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Development • Field Picture

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, $X^{\circ },$ to considering a larger universe of discourse, $\mathrm {E} X^{\circ }.$ An operator $\mathrm {W}$ of this general type, namely, $\mathrm {W} :X^{\circ }\to \mathrm {E} X^{\circ },$ acts on each proposition $f:X\to \mathbb {B}$ of the source universe ${X^{\circ }}$ to produce a proposition $\mathrm {W} f:\mathrm {E} X\to \mathbb {B}$ of the target universe $\mathrm {E} X^{\circ }.$

The two main operators that we've examined so far are the enlargement or shift operator $\mathrm {E} :X^{\circ }\to \mathrm {E} X^{\circ }$ and the difference operator $\mathrm {D} :X^{\circ }\to \mathrm {E} X^{\circ }.$ The operators $\mathrm {E}$ and $\mathrm {D}$ act on propositions in $X^{\circ },$ that is, propositions of the form $f:X\to \mathbb {B}$ that are said to be about the subject matter of $X,$ and they produce extended propositions of the forms $\mathrm {E} f,\mathrm {D} f:\mathrm {E} X\to \mathbb {B} ,$ propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in $X.$

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 $X,$ a field of a specified type $Y$ over $X$ is formed by associating with each point of $X$ an object of type $Y.$ If that sounds like the same thing as a function from $X$ to the space of things of type $Y$ — 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 $f:X\to \mathbb {B}$ becomes a scalar field, that is, a field of values in $\mathbb {B} .$

For example, consider the logical conjunction $pq:X\to \mathbb {B}$ that is shown in the following venn diagram:

${\text{Conjunction}}~pq:X\to \mathbb {B}$

Each of the operators $\mathrm {E} ,\mathrm {D} :X^{\circ }\to \mathrm {E} X^{\circ }$ takes us from considering propositions $f:X\to \mathbb {B} ,$ 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. With each point of $X$ there is associated an object of the following type: a proposition about changes in $X,$ that is, a proposition $g:\mathrm {d} X\to \mathbb {B} .$ In this frame of reference, if ${X^{\circ }}$ is the universe that is generated by the set of coordinate propositions $\{p,q\},$ then $\mathrm {d} X^{\circ }$ is the differential universe that is generated by the set of differential propositions $\{\mathrm {d} p,\mathrm {d} q\}.$ These differential propositions may be interpreted as indicating ${}^{\backprime \backprime }{\text{change in}}\,p\,{}^{\prime \prime }$ and ${}^{\backprime \backprime }{\text{change in}}\,q\,{}^{\prime \prime },$ respectively.

A differential operator $\mathrm {W} ,$ of the first order class that we have been considering, takes a proposition $f:X\to \mathbb {B}$ and gives back a differential proposition $\mathrm {W} f:\mathrm {E} X\to \mathbb {B} .$ In the field view, we see the proposition $f:X\to \mathbb {B}$ as a scalar field and we see the differential proposition $\mathrm {W} f:\mathrm {E} X\to \mathbb {B}$ as a vector field, specifically, a field of propositions about contemplated changes in $X.$

The field of changes produced by $\mathrm {E}$ on $pq$ is shown in the next venn diagram:

${\text{Enlargement}}~\mathrm {E} (pq):\mathrm {E} X\to \mathbb {B}$

${\begin{array}{rcccccc}\mathrm {E} (pq)&=&p&\cdot &q&\cdot &{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\[4pt]&+&p&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &q&\cdot &{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\end{array}}$

The differential field $\mathrm {E} (pq)$ specifies the changes that need to be made from each point of $X$ in order to reach one of the models of the proposition $pq,$ that is, in order to satisfy the proposition $pq.$

The field of changes produced by $\mathrm {D}$ on $pq$ is shown in the following venn diagram:

${\text{Difference}}~\mathrm {D} (pq):\mathrm {E} X\to \mathbb {B}$

${\begin{array}{rcccccc}\mathrm {D} (pq)&=&p&\cdot &q&\cdot &{\texttt {(}}{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {(}}\mathrm {d} q{\texttt {)}}{\texttt {)}}\\[4pt]&+&p&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {~}}{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {~}}\mathrm {d} q{\texttt {~}}{\texttt {~}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &q&\cdot &{\texttt {~}}{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {(}}\mathrm {d} q{\texttt {)}}{\texttt {~}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {~}}{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {~}}\mathrm {d} q{\texttt {~}}{\texttt {~}}\end{array}}$

The differential field $\mathrm {D} (pq)$ specifies the changes that need to be made from each point of $X$ in order to feel a change in the felt value of the field $pq.$

Proposition and Tacit Extension

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 $f:X\to \mathbb {B} ,$ its enlargement or shift map $\mathrm {E} f:\mathrm {E} X\to \mathbb {B} ,$ and its difference map $\mathrm {D} f:\mathrm {E} X\to \mathbb {B}$ can now be drawn.

To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition $f(p,q)=pq,$ giving the development a slightly different twist at the appropriate point.

The next venn diagram shows once again the proposition $pq,$ 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.

${\text{Proposition}}~pq:X\to \mathbb {B}$

Given a proposition $f:X\to \mathbb {B} ,$ the tacit extension of $f$ to $\mathrm {E} X$ is denoted ${\boldsymbol {\varepsilon }}f:\mathrm {E} X\to \mathbb {B}$ and defined by the equation ${\boldsymbol {\varepsilon }}f=f,$ 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.

The tacit extension of the scalar field $pq:X\to \mathbb {B}$ to the differential field ${\boldsymbol {\varepsilon }}(pq):\mathrm {E} X\to \mathbb {B}$ is shown in the following venn diagram:

${\text{Tacit Extension}}~{\boldsymbol {\varepsilon }}(pq):\mathrm {E} X\to \mathbb {B}$

${\begin{array}{rcccccc}{\boldsymbol {\varepsilon }}(pq)&=&p&\cdot &q&\cdot &{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\[4pt]&+&p&\cdot &q&\cdot &{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\\[4pt]&+&p&\cdot &q&\cdot &{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\[4pt]&+&p&\cdot &q&\cdot &{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\end{array}}$

Enlargement and Difference Maps

Continuing with the example $pq:X\to \mathbb {B} ,$ the next venn diagram shows the enlargement or shift map $\mathrm {E} (pq):\mathrm {E} X\to \mathbb {B}$ in the same style of differential field picture that we drew for the tacit extension ${\boldsymbol {\varepsilon }}(pq):\mathrm {E} X\to \mathbb {B} .$

${\text{Enlargement Map}}~\mathrm {E} (pq):\mathrm {E} X\to \mathbb {B}$

${\begin{array}{rcccccc}\mathrm {E} (pq)&=&p&\cdot &q&\cdot &{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\[4pt]&+&p&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &q&\cdot &{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\end{array}}$

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 ${\boldsymbol {\varepsilon }}f$ and $\mathrm {E} f,$ both of the type $\mathrm {E} X\to \mathbb {B} ,$ 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 ${\boldsymbol {\varepsilon }}f$ and the enlargement $\mathrm {E} f$ are in a certain sense dual to each other. The tacit extension ${\boldsymbol {\varepsilon }}f$ indicates all the arrows out of the region where $f$ is true and the enlargement $\mathrm {E} f$ indicates all the arrows into the region where $f$ is true. The only arc they have in common is the no-change loop ${\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}$ at $pq.$ 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 $\mathrm {D} (pq)={\boldsymbol {\varepsilon }}(pq)+\mathrm {E} (pq)$ that are illustrated below:

${\text{Difference Map}}~\mathrm {D} (pq):\mathrm {E} X\to \mathbb {B}$

${\begin{array}{rcccccc}\mathrm {D} (pq)&=&p&\cdot &q&\cdot &{\texttt {(}}{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {(}}\mathrm {d} q{\texttt {)}}{\texttt {)}}\\[4pt]&+&p&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {~}}{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {~}}\mathrm {d} q{\texttt {~}}{\texttt {~}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &q&\cdot &{\texttt {~}}{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {(}}\mathrm {d} q{\texttt {)}}{\texttt {~}}\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &{\texttt {~}}{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {~}}\mathrm {d} q{\texttt {~}}{\texttt {~}}\end{array}}$

Tangent and Remainder Maps

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 $f=pq:X\to \mathbb {B}$ in the following way.

The next venn diagram shows the differential proposition $\mathrm {d} f=\mathrm {d} (pq):\mathrm {E} X\to \mathbb {B}$ that we get by extracting the cell-wise linear approximation to the difference map $\mathrm {D} f=\mathrm {D} (pq):\mathrm {E} X\to \mathbb {B} .$ This is the logical analogue of what would ordinarily be called the differential of $pq,$ 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 $\mathrm {d} f.$

${\text{Tangent Map}}~\mathrm {d} (pq):\mathrm {E} X\to \mathbb {B}$

Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:

${\begin{array}{rcccccc}\mathrm {d} (pq)&=&p&\cdot &q&\cdot &{\texttt {(}}\mathrm {d} p{\texttt {,}}\mathrm {d} q{\texttt {)}}\\[4pt]&+&p&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &\mathrm {d} q\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &q&\cdot &\mathrm {d} p\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &0\end{array}}$

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:

${\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {,}}\mathrm {d} q{\texttt {)}}&=&{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {(}}\mathrm {d} q{\texttt {)}}&+&{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\\[4pt]dp&=&{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {~}}\mathrm {d} q{\texttt {~}}&+&{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\[4pt]\mathrm {d} q&=&{\texttt {~}}\mathrm {d} p{\texttt {~}}{\texttt {~}}\mathrm {d} q{\texttt {~}}&+&{\texttt {(}}\mathrm {d} p{\texttt {)}}{\texttt {~}}\mathrm {d} q{\texttt {~}}\end{matrix}}$

Capping the series that analyzes the proposition $pq$ in terms of succeeding orders of linear propositions, the final venn diagram in this series shows the remainder map $\mathrm {r} (pq):\mathrm {E} X\to \mathbb {B} ,$ that happens to be linear in pairs of variables.

${\text{Remainder Map}}~\mathrm {r} (pq):\mathrm {E} X\to \mathbb {B}$

Reading the arrows off the map produces the following data:

${\begin{array}{rcccccc}\mathrm {r} (pq)&=&p&\cdot &q&\cdot &\mathrm {d} p~\mathrm {d} q\\[4pt]&+&p&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &\mathrm {d} p~\mathrm {d} q\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &q&\cdot &\mathrm {d} p~\mathrm {d} q\\[4pt]&+&{\texttt {(}}p{\texttt {)}}&\cdot &{\texttt {(}}q{\texttt {)}}&\cdot &\mathrm {d} p~\mathrm {d} q\end{array}}$

In short, $\mathrm {r} (pq)$ is a constant field, having the value $\mathrm {d} p~\mathrm {d} q$ at each cell.

Least Action Operators

We have been contemplating functions of the type $f:X\to \mathbb {B}$ and studying the action of the operators $\mathrm {E}$ and $\mathrm {D}$ 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 $X_{1}\times \ldots \times X_{k}\to Y_{1}\times \ldots \times Y_{n}$ and abstract types $\mathbb {B} ^{k}\to \mathbb {B} ^{n}.$ 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 zeroth order logic.

For example, consider the proposition $f$ of concrete type $f:P\times Q\times R\to \mathbb {B}$ and abstract type $f:\mathbb {B} ^{3}\to \mathbb {B}$ that is written ${\texttt {(}}p,q,r{\texttt {)}}$ in cactus syntax. Taken as an assertion in what Peirce called the existential interpretation, the proposition ${\texttt {(}}p,q,r{\texttt {)}}$ says that just one of $p,q,r$ is false. It is instructive to consider this assertion in relation to the logical conjunction $pqr$ of the same propositions. A venn diagram of ${\texttt {(}}p,q,r{\texttt {)}}$ looks like this:

In relation to the center cell indicated by the conjunction $pqr,$ the region indicated by ${\texttt {(}}p,q,r{\texttt {)}}$ 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, $pqr.$

More generally speaking, in a $k$-dimensional universe of discourse that is based on the alphabet of features ${\mathcal {X}}=\{x_{1},\ldots ,x_{k}\},$ 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 $e_{1}\cdot \ldots \cdot e_{k},$ where $e_{j}=x_{j}~{\text{or}}~e_{j}={\texttt {(}}x_{j}{\texttt {)}},$ for $j=1~{\text{to}}~k.$ The proposition ${\texttt {(}}e_{1},\ldots ,e_{k}{\texttt {)}}$ indicates the disjunctive region consisting of the cells that are just next door to $e_{1}\cdot \ldots \cdot e_{k}.$

Goal-Oriented Systems

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 $X,$ giving us all of the states that the system can be in, plus it has a goal space $G\subseteq X,$ 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 ${\mathcal {T}}$ of all possible actions, each of which is a transformation of the form $T:X\to X,$ 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 $X,$ 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.

o---------------------------------------------------------------------o | | | X | | o-------------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | o G o | | | | | | | | | | | | | | | o<---------T---------o | | | | | | | | | | | | | | o o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | o-------------------o | | | | | o---------------------------------------------------------------------o Figure 23. Elements of a Cybernetic System

Further Reading

• Differential Propositional Calculus
  • Part 1 • Part 2 • Appendices

• Differential Logic and Dynamic Systems
  • Part 1 • Part 2 • Part 3 • Part 4 • Part 5

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• Overview • Part 1 • Part 2 • Part 3 • Document History •

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