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Differential Propositional Calculus

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A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe.

Contents

Work In Progress

Casual introduction

Consider the situation represented by the venn diagram in Figure 1. 

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Figure 1. Local Habitations, And Names

The area of the rectangle represents a universe of discourse, X.\! This might be a population of individuals having various additional properties or it might be a collection of locations that various individuals occupy. The area of the "circle" represents the individuals that have the property q\! or the locations that fall within the corresponding region Q.\! Four individuals, h, i, j, k,\! are singled out by name. It happens that i\! and j\! currently reside in region Q\! while h\! and k\! do not.

Now consider the situation represented by the venn diagram in Figure 2. 

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Figure 2. Same Names, Different Habitations

Figure 2 differs from Figure 1 solely in the circumstance that the object j\! is outside the region Q\! while the object k\! is inside the region Q.\! So far, there is nothing that says that our encountering these Figures in this order is other than purely accidental, but if we interpret the present sequence of frames as a "moving picture" representation of their natural order in a temporal process, then it would be natural to say that h\! and i\! have remained as they were with regard to quality q\! while j\! and k\! have changed their standings in that respect. In particular, j\! has moved from the region where q\! is true\! to the region where q\! is false\! while k\! has moved from the region where q\! is false\! to the region where q\! is true.\!

Figure 1′ reprises the situation shown in Figure 1, but this time interpolates a new quality that is specifically tailored to account for the relation between Figure 1 and Figure 2. 

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Figure 1′. Back, To The Future

This new quality, \operatorname{d}q,\! is an example of a differential quality, since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by means of a "circle" that distinguishes two halves of the universe of discourse, in this case, the portions of X\! outside and inside the region \operatorname{d}Q.\!

Figure 1 represents a universe of discourse, X,\! together with a basis of discussion, \{ q \},\! for expressing propositions about the contents of that universe. Once the quality q\! is given a name, say, the symbol "q\!", we have the basis for a formal language that is specifically cut out for discussing X\! in terms of q,\! and this formal language is more formally known as the propositional calculus with alphabet \{\!"q\!"\}.\!

In the context marked by X\! and \{ q \}\! there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: false,\! \lnot q,\! q,\! true.\! Referring to the sample of points in Figure 1, false\! holds of no points, \lnot q\! holds of h\! and k,\! q\! holds of i\! and j,\! and true\! holds of all points in the sample.

Figure 1′ preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, \{ q, \operatorname{d}q \}.\! In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, \{\!"q\!",\! "\operatorname{d}q\!"\}.\! Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows:

  • \overline{q}\ \overline{\operatorname{d}q} describes h\!

  • \overline{q}\ \operatorname{d}q describes k\!

  • q\ \overline{\operatorname{d}q} describes i\!

  • q\ \operatorname{d}q describes j\!

Table 3 exhibits the rules of inference that give the differential quality \operatorname{d}q\! its meaning in practice.

Table 3. Differential Inference Rules
  From \overline{q}\! and \overline{\operatorname{d}q}\! infer \overline{q}\! next.  
  From \overline{q}\! and \operatorname{d}q\! infer q\! next.  
  From q\! and \overline{\operatorname{d}q}\! infer q\! next.  
  From q\! and \operatorname{d}q\! infer \overline{q}\! next.  


Cactus calculus

Table 4 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable k\!-ary scope.

  • A bracketed list of propositional expressions in the form (e_1, e_2, \ldots, e_{k-1}, e_k) indicates that exactly one of the propositions e_1, e_2, \ldots, e_{k-1}, e_k is false.
  • A concatenation of propositional expressions in the form e_1~e_2~\ldots~e_{k-1}~e_k indicates that all of the propositions e_1, e_2, \ldots, e_{k-1}, e_k are true, in other words, that their logical conjunction is true.
Table 4. Syntax and Semantics of a Propositional Calculus
Expression Interpretation Other Notations
~ \operatorname{True} 1\!
(~) \operatorname{False} 0\!
x\! x\! x\!
(x)\! \operatorname{Not}\ x

\begin{matrix} x' \\ \tilde{x} \\ \lnot x \\ \end{matrix}

x\ y\ z x\ \operatorname{and}\ y\ \operatorname{and}\ z x \land y \land z
((x)(y)(z))\! x\ \operatorname{or}\ y\ \operatorname{or}\ z x \lor y \lor z
(x\ (y))\!

\begin{matrix} x\ \operatorname{implies}\ y \\ \operatorname{If}\ x\ \operatorname{then}\ y \\ \end{matrix}

x \Rightarrow y\!
(x, y)\!

\begin{matrix} x\ \operatorname{not~equal~to}\ y \\ x\ \operatorname{exclusive~or}\ y \\ \end{matrix}

\begin{matrix} x \neq y \\ x + y \\ \end{matrix}

((x, y))\!

\begin{matrix} x\ \operatorname{is~equal~to}\ y \\ x\ \operatorname{if~and~only~if}\ y \\ \end{matrix}

\begin{matrix} x = y \\ x \Leftrightarrow y \\ \end{matrix}

(x, y, z)\!

\begin{matrix} \operatorname{Just~one~of} \\ x, y, z \\ \operatorname{is~false}. \\ \end{matrix}

\begin{matrix} x'y~z~ & \lor \\ x~y'z~ & \lor \\ x~y~z' & \\ \end{matrix}

((x),(y),(z))\!

\begin{matrix} \operatorname{Just~one~of} \\ x, y, z \\ \operatorname{is~true}. \\ & \\ \operatorname{Partition~all} \\ \operatorname{into}\ x, y, z. \\ \end{matrix}

\begin{matrix} x~y'z' & \lor \\ x'y~z' & \lor \\ x'y'z~ & \\ \end{matrix}

\begin{matrix} ((x, y), z) \\ & \\ (x, (y, z)) \\ \end{matrix}

\begin{matrix} \operatorname{Oddly~many~of} \\ x, y, z \\ \operatorname{are~true}. \\ \end{matrix}

x + y + z\!


\begin{matrix} x~y~z~ & \lor \\ x~y'z' & \lor \\ x'y~z' & \lor \\ x'y'z~ & \\ \end{matrix}

(w, (x),(y),(z))\!

\begin{matrix} \operatorname{Partition}\ w \\ \operatorname{into}\ x, y, z. \\ & \\ \operatorname{Genus}\ w\ \operatorname{comprises} \\ \operatorname{species}\ x, y, z. \\ \end{matrix}

\begin{matrix} w'x'y'z' & \lor \\ w~x~y'z' & \lor \\ w~x'y~z' & \lor \\ w~x'y'z~ & \\ \end{matrix}


All other propositional connectives can be obtained through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation for more complicated bracket expressions. The briefest expression for logical truth is the empty word, abstractly denoted \varepsilon\! or \lambda\! in formal languages, where it forms the identity element for concatenation. It can be given visible expression in this context by means of the logically equivalent expression "((~))", or, especially if operating in an algebraic context, by a simple "1\!". Also when working in an algebraic mode, the plus sign "+\! may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

\begin{matrix} x + y & = & (x, y) \end{matrix}

\begin{matrix} x + y + z & = & ((x, y), z) & = & (x, (y, z)) \end{matrix}

It is important to note that the last expressions are not equivalent to the triple bracket (x, y, z).\!

For more information about this syntax for propositional calculus, see the entries on minimal negation operators, zeroth order logic, and Table A1 in Appendix 1.

Formal development

The preceding discussion outlined the ideas leading to the differential extension of propositional logic. The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.

Elementary notions

Logical description of a universe of discourse begins with a set of logical signs. For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, \mathfrak{A} = \lbrace\! “a_1\!” , \ldots,\! “a_n\!” \rbrace.\! Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse. Corresponding to the alphabet \mathfrak{A} there is then a set of logical features, \mathcal{A} = \{ a_1, \ldots, a_n \}.

A set of logical features, \mathcal{A} = \{ a_1, \ldots, a_n \}, affords a basis for generating an n\!-dimensional universe of discourse, written A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ]. It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points A = \langle a_1, \ldots, a_n \rangle and the set of propositions A^\uparrow = \{ f : A \to \mathbb{B} \} that are implicit with the ordinary picture of a venn diagram on n\! features. Accordingly, the universe of discourse A^\circ may be regarded as an ordered pair (A, A^\uparrow) having the type (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})), and this last type designation may be abbreviated as \mathbb{B}^n\ +\!\to \mathbb{B}, or even more succinctly as [ \mathbb{B}^n ]. For convenience, the data type of a finite set on n\! elements may be indicated by either one of the equivalent notations, [n]\! or \mathbf{n}.

Table 5 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

Table 5. Propositional Calculus : Basic Notation
Symbol Notation Description Type
\mathfrak{A} \lbrace\! “a_1\!” , \ldots,\! “a_n\!” \rbrace\! Alphabet [n] = \mathbf{n}
\mathcal{A} \{ a_1, \ldots, a_n \} Basis [n] = \mathbf{n}
A_i\! \{ \overline{a_i}, a_i \}\! Dimension i\! \mathbb{B}
A\! \langle \mathcal{A} \rangle

\langle a_1, \ldots, a_n \rangle
\{ (a_1, \ldots, a_n) \}\! A_1 \times \ldots \times A_n
\textstyle \prod_i A_i\!

Set of cells,

coordinate tuples,
points, or vectors
in the universe
of discourse

\mathbb{B}^n
A^*\! (\operatorname{hom} : A \to \mathbb{B}) Linear functions (\mathbb{B}^n)^* \cong \mathbb{B}^n
A^\uparrow (A \to \mathbb{B}) Boolean functions \mathbb{B}^n \to \mathbb{B}
A^\circ [ \mathcal{A} ]

(A, A^\uparrow)
(A\ +\!\to \mathbb{B})
(A, (A \to \mathbb{B}))
[ a_1, \ldots, a_n ]

Universe of discourse

based on the features
\{ a_1, \ldots, a_n \}

(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))

(\mathbb{B}^n\ +\!\to \mathbb{B})
[\mathbb{B}^n]


Special classes of propositions

A basic proposition, coordinate proposition, or simple proposition in the universe of discourse [a_1, \ldots, a_n] is one of the propositions in the set \{ a_1, \ldots, a_n \}.

Among the 2^{2^n} propositions in [a_1, \ldots, a_n] are several families of 2^n\! propositions each that take on special forms with respect to the basis \{ a_1, \ldots, a_n \}. Three of these families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate n\!-tuples in \mathbb{B}^n and falls into n + 1\! ranks, with a binomial coefficient \tbinom{n}{k} giving the number of propositions that have rank or weight k.\!

  • The linear propositions, \{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}), may be written as sums:

    \sum_{i=1}^n e_i = e_1 + \ldots + e_n  where  e_i = a_i\!  or  e_i = 0\!  for  i = 1\!  to  n.\!

  • The positive propositions, \{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}), may be written as products:

    \prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n  where  e_i = a_i\!  or  e_i = 1\!  for  i = 1\!  to  n.\!

  • The singular propositions, \{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}), may be written as products:

    \prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n  where  e_i = a_i\!  or  e_i = (a_i)\!  for  i = 1\!  to  n.\!

In each case the rank k\! ranges from 0\! to n\! and counts the number of positive appearances of the coordinate propositions a_1, \ldots, a_n\! 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 basic propositions a_i : \mathbb{B}^n \to \mathbb{B} are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis \mathcal{A} = \{ a_1, \ldots, a_n \}. For example, a singular proposition with respect to the basis \mathcal{A} will not remain singular if \mathcal{A} is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options \{ a_i \} \cup \{ (a_i) \} to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.

Differential extensions

An initial universe of discourse, A^\circ, supplies the groundwork for any number of further extensions, beginning with the first order differential extension, \operatorname{E}A^\circ. The construction of \operatorname{E}A^\circ can be described in the following stages:

  • The initial alphabet, \mathfrak{A} = \lbrace\! “a_1\!” , \ldots,\! “a_n\!” \rbrace,\! is extended by a first order differential alphabet, \operatorname{d}\mathfrak{A} = \lbrace\! “\operatorname{d}a_1\!” , \ldots,\! “\operatorname{d}a_n\!” \rbrace,\! resulting in a first order extended alphabet, \operatorname{E}\mathfrak{A}, defined as follows:

    \operatorname{E}\mathfrak{A} = \mathfrak{A}\ \cup\ \operatorname{d}\mathfrak{A} = \lbrace\! “a_1\!” , \ldots,\! “a_n\!”,\!  “\operatorname{d}a_1\!” , \ldots,\! “\operatorname{d}a_n\!” \rbrace.\!

  • The initial basis, \mathcal{A} = \{ a_1, \ldots, a_n \}, is extended by a first order differential basis, \operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}, resulting in a first order extended basis, \operatorname{E}\mathcal{A}, defined as follows:

    \operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.

  • The initial space, A = \langle a_1, \ldots, a_n \rangle, is extended by a first order differential space or tangent space, \operatorname{d}A = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle, at each point of A,\! resulting in a first order extended space or tangent bundle space, \operatorname{E}A, defined as follows:

    \operatorname{E}A = A \times \operatorname{d}A = \langle \operatorname{E}\mathcal{A} \rangle = \langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle = \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.

  • Finally, the initial universe, A^\circ = [ a_1, \ldots, a_n ], is extended by a first order differential universe or tangent universe, \operatorname{d}A^\circ = [ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ], at each point of A^\circ, resulting in a first order extended universe or tangent bundle universe, \operatorname{E}A^\circ, defined as follows:

    \operatorname{E}A^\circ = [ \operatorname{E}\mathcal{A} ] = [ \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} ] = [ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n ].

    This gives \operatorname{E}A^\circ the type:

    [ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).

A proposition in a differential extension of a universe of discourse is called a differential proposition and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe \operatorname{E}A^\circ and the first order differential proposition f : \operatorname{E}A \to \mathbb{B}, we have arrived, in concept at least, at the foothills of differential logic.

Table 6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.

Table 6. Differential Extension : Basic Notation
Symbol Notation Description Type
\operatorname{d}\mathfrak{A} \lbrace\! “\operatorname{d}a_1” , \ldots,\! “\operatorname{d}a_n” \rbrace\! Alphabet of

differential
symbols

[n] = \mathbf{n}
\operatorname{d}\mathcal{A} \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \} Basis of

differential
features

[n] = \mathbf{n}
\operatorname{d}A_i \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} Differential

dimension i\!

\mathbb{D}
\operatorname{d}A \langle \operatorname{d}\mathcal{A} \rangle

\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle
\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}
\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n
\textstyle \prod_i \operatorname{d}A_i

Tangent space

at a point:
Set of changes,
motions, steps,
tangent vectors
at a point

\mathbb{D}^n
\operatorname{d}A^* (\operatorname{hom} : \operatorname{d}A \to \mathbb{B}) Linear functions

on \operatorname{d}A

(\mathbb{D}^n)^* \cong \mathbb{D}^n
\operatorname{d}A^\uparrow (\operatorname{d}A \to \mathbb{B}) Boolean functions

on \operatorname{d}A

\mathbb{D}^n \to \mathbb{B}
\operatorname{d}A^\circ [\operatorname{d}\mathcal{A}]

(\operatorname{d}A, \operatorname{d}A^\uparrow)
(\operatorname{d}A\ +\!\to \mathbb{B})
(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))
[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]

Tangent universe

at a point of A^\circ,
based on the
tangent features
\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}

(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))

(\mathbb{D}^n\ +\!\to \mathbb{B})
[\mathbb{D}^n]


…

Expository examples

…

Consider the logical proposition represented by the following venn diagram: 

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. Proposition q : X \to \mathbb{B}

The following language is useful in describing the facts represented by the venn diagram.

  • The universe of discourse is a set, X,\! represented by the area inside the large rectangle.
  • The boolean domain is a set of two elements, \mathbb{B} = \{ 0, 1 \}, represented by the two distinct shadings of the regions inside the rectangle.
  • According to the conventions observed in this context, the algebraic value 0 is interpreted as the logical value \operatorname{false} and represented by the lighter shading, while the algebraic value 1 is interpreted as the logical value \operatorname{true} and represented by the darker shading.
  • The universe of discourse X\! is the domain of three functions u, v, w : X \to \mathbb{B} called basic, coordinate, or simple propositions.
  • As with any proposition, p : X \to \mathbb{B}, a simple proposition partitions X\! into two fibers, the fiber of 0 under p,\! defined as p^{-1}(0) \subseteq X, and the fiber of 1 under p,\! defined as p^{-1}(1) \subseteq X.
  • Each coordinate proposition is represented by a "circle", or a simple closed curve, that divides the rectangular region into the region exterior to the circle, representing the fiber of 0 under p,\! and the region interior to the circle, representing the fiber of 1 under p.\!
  • The fibers of 1 under the propositions u, v, w\! are the respective subsets U, V, W \subseteq X.

…

Material To Be Collated

Differential Logic : First Approach

Linear Topics : The Differential Theory of Qualitative Equations

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, Cybernetics

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:

\mathbf{a} = (a, b, c),\ \mathbf{p} = (p, q, r),\ \mathbf{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 \operatorname{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 \operatorname{F} to f\!, we write g = \operatorname{F}f.

The first operator, \operatorname{E}, associates with a function f : X \to Y another function \operatorname{E}f, where \operatorname{E}f : X \times X \to Y is defined by the following equation:

\operatorname{E}f(x, y) = f(x + y).

\operatorname{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, \operatorname{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 "shift operator" \operatorname{E} was originally called the "enlargement operator", hence the initial "E" of the usual notation.

The next operator, \operatorname{D}, associates with a function f : X \to Y another function \operatorname{D}f, where \operatorname{D}f : X \times X \to Y is defined by the following equation:

\operatorname{D}f(x, y) = \operatorname{E}f(x, y) - f(x),

or, equivalently,

\operatorname{D}f(x, y) = f(x + y) - f(x).

\operatorname{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:

\operatorname{D}f : X \times X \to Y,
\operatorname{D}f(c, x) = f(c + x) - f(c).

Here, c\! is held constant and \operatorname{D}f(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\!.

\operatorname{D}f : X \times X \to Y,
\operatorname{D}f(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. \operatorname{D}f(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.

\operatorname{D}f : X \times X \to Y,
\operatorname{D}f(x, \operatorname{d}x) = f(x + \operatorname{d}x) - f(x).

This is yet another variant of the previous form, with \operatorname{d}x 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.

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 \to \mathbb{B}. The target concept \mathcal{Q} is one whose extension is a polymorphous set Q\!, the subset Q\! of the universe X\! where the complex feature q : X \to \mathbb{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 capital letters 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
u v w u ∧ v u ∧ w v ∧ w q
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

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 \mathcal{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 \mathbf{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.

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.

Working Note. Need segue here to explain the use of Cactus Language.

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, \ldots, 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 \operatorname{d}x that determines our decision whether or not we will make a difference in how we stand regarding x\!. If \operatorname{d}x 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, \operatorname{d}x_1)" for "x_1\!"

2.\! Substitute "(x_2, \operatorname{d}x_2)" for "x_2\!"

3.\! Substitute "(x_3, \operatorname{d}x_3)" for "x_3\!"

\ldots

k.\! Substitute "(x_k, \operatorname{d}x_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 \operatorname{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 \{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \} 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 \operatorname{D} to the initial proposition \operatorname{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.\!

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, ¶ 360, p. 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 \operatorname{D}q. 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 \to \mathbb{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 \operatorname{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 \{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \} 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 \operatorname{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 \operatorname{D}q are marked with "@" signs, and the boundary crossings along each path are marked with the corresponding differential features among the collection \{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}. 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.

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.

Recapitulation

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 \to \mathbb{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
u v w u ∧ v u ∧ w v ∧ w q
0 0 0 0 0 0 0
0 0 1 0