# Differential Logic • Various Sketches

Author: Jon Awbrey

## Work In Progress

### Expository Examples

Consider the logical proposition represented by the following venn diagram:

 $\text{Figure 1. Proposition}~ q : X \to \mathbb{B}~\!$

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 $\mathrm{false}$ and represented by the lighter shading, while the algebraic value 1 is interpreted as the logical value $\mathrm{true}$ 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 from “Differential Logic and Dynamic Systems”

Excerpts from Differential Logic and Dynamic Systems

### A Functional Conception of Propositional Calculus

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

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

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

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

 $\text{Symbol}\!$ $\text{Notation}\!$ $\text{Description}\!$ $\text{Type}\!$ $\mathfrak{A}\!$ $\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!$ $\text{Alphabet}\!$ $[n] = \mathbf{n}\!$ $\mathcal{A}\!$ $\{ a_1, \ldots, a_n \}\!$ $\text{Basis}\!$ $[n] = \mathbf{n}\!$ $A_i\!$ $\{ \texttt{(} a_i \texttt{)}, a_i \}\!$ $\text{Dimension}~ i\!$ $\mathbb{B}\!$ $A\!$ $\begin{matrix} \langle \mathcal{A} \rangle \\[2pt] \langle a_1, \ldots, a_n \rangle \\[2pt] \{ (a_1, \ldots, a_n) \} \\[2pt] A_1 \times \ldots \times A_n \\[2pt] \textstyle \prod_{i=1}^n A_i \end{matrix}$ $\begin{matrix} \text{Set of cells}, \\[2pt] \text{coordinate tuples}, \\[2pt] \text{points, or vectors} \\[2pt] \text{in the universe} \\[2pt] \text{of discourse} \end{matrix}$ $\mathbb{B}^n\!$ $A^*\!$ $(\mathrm{hom} : A \to \mathbb{B})\!$ $\text{Linear functions}\!$ $(\mathbb{B}^n)^* \cong \mathbb{B}^n\!$ $A^\uparrow\!$ $(A \to \mathbb{B})\!$ $\text{Boolean functions}\!$ $\mathbb{B}^n \to \mathbb{B}\!$ $A^\bullet\!$ $\begin{matrix} [\mathcal{A}] \\[2pt] (A, A^\uparrow) \\[2pt] (A ~+\!\to \mathbb{B}) \\[2pt] (A, (A \to \mathbb{B})) \\[2pt] [a_1, \ldots, a_n] \end{matrix}$ $\begin{matrix} \text{Universe of discourse} \\[2pt] \text{based on the features} \\[2pt] \{ a_1, \ldots, a_n \} \end{matrix}$ $\begin{matrix} (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\[2pt] (\mathbb{B}^n ~+\!\to \mathbb{B}) \\[2pt] [\mathbb{B}^n] \end{matrix}$

#### Reality at the Threshold of Logic

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

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

 $\text{Linear Space}\!$ $\text{Liminal Space}\!$ $\text{Logical Space}\!$ $\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}$ $\begin{matrix}\underline\mathcal{X} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}$ $\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}$ $\begin{matrix} X_i & = & \langle x_i \rangle \\ & \cong & \mathbb{K} \end{matrix}$ $\begin{matrix} \underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \} \\ & \cong & \mathbb{B} \end{matrix}$ $\begin{matrix} A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \} \\ & \cong & \mathbb{B} \end{matrix}$ $\begin{matrix} X \\ = & \langle \mathcal{X} \rangle \\ = & \langle x_1, \ldots, x_n \rangle \\ = & X_1 \times \ldots \times X_n \\ = & \prod_{i=1}^n X_i \\ \cong & \mathbb{K}^n \end{matrix}$ $\begin{matrix} \underline{X} \\ = & \langle \underline\mathcal{X} \rangle \\ = & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\ = & \underline{X}_1 \times \ldots \times \underline{X}_n \\ = & \prod_{i=1}^n \underline{X}_i \\ \cong & \mathbb{B}^n \end{matrix}$ $\begin{matrix} A \\ = & \langle \mathcal{A} \rangle \\ = & \langle a_1, \ldots, a_n \rangle \\ = & A_1 \times \ldots \times A_n \\ = & \prod_{i=1}^n A_i \\ \cong & \mathbb{B}^n \end{matrix}$ $\begin{matrix} X^* & = & (\ell : X \to \mathbb{K}) \\ & \cong & \mathbb{K}^n \end{matrix}$ $\begin{matrix} \underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B}) \\ & \cong & \mathbb{B}^n \end{matrix}$ $\begin{matrix} A^* & = & (\ell : A \to \mathbb{B}) \\ & \cong & \mathbb{B}^n \end{matrix}$ $\begin{matrix} X^\uparrow & = & (X \to \mathbb{K}) \\ & \cong & (\mathbb{K}^n \to \mathbb{K}) \end{matrix}$ $\begin{matrix} \underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B}) \\ & \cong & (\mathbb{B}^n \to \mathbb{B}) \end{matrix}$ $\begin{matrix} A^\uparrow & = & (A \to \mathbb{B}) \\ & \cong & (\mathbb{B}^n \to \mathbb{B}) \end{matrix}$ $\begin{matrix} X^\bullet \\ = & [\mathcal{X}] \\ = & [x_1, \ldots, x_n] \\ = & (X, X^\uparrow) \\ = & (X ~+\!\to \mathbb{K}) \\ = & (X, (X \to \mathbb{K})) \\ \cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K})) \\ = & (\mathbb{K}^n ~+\!\to \mathbb{K}) \\ = & [\mathbb{K}^n] \end{matrix}$ $\begin{matrix} \underline{X}^\bullet \\ = & [\underline\mathcal{X}] \\ = & [\underline{x}_1, \ldots, \underline{x}_n] \\ = & (\underline{X}, \underline{X}^\uparrow) \\ = & (\underline{X} ~+\!\to \mathbb{B}) \\ = & (\underline{X}, (\underline{X} \to \mathbb{B})) \\ \cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\ = & (\mathbb{B}^n ~+\!\to \mathbb{B}) \\ = & [\mathbb{B}^n] \end{matrix}$ $\begin{matrix} A^\bullet \\ = & [\mathcal{A}] \\ = & [a_1, \ldots, a_n] \\ = & (A, A^\uparrow) \\ = & (A ~+\!\to \mathbb{B}) \\ = & (A, (A \to \mathbb{B})) \\ \cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\ = & (\mathbb{B}^n ~+\!\to \mathbb{B}) \\ = & [\mathbb{B}^n] \end{matrix}$

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

I now proceed to explain these concepts in more detail. The most important ideas developed in Table 5 are these:

• The idea of a universe of discourse, which includes both a space of points and a space of maps on those points.
• The idea of passing from a more complex universe to a simpler universe by a process of thresholding each dimension of variation down to a single bit of information.

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

 $\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!$ $\begin{matrix} \underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i, \\[4pt] \underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i. \end{matrix}$

In other notations that are sometimes used, the operator $\chi (\ldots)$ or the corner brackets $\lceil\ldots\rceil$ can be used to denote a characteristic function, that is, a mapping from statements to their truth values in $\mathbb{B}.$ Finally, it is not uncommon to use the name of the relation itself as a predicate that maps $n\!$-tuples into truth values. Thus we have the following notational variants of the above definition:

 $\begin{matrix} \underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}). \end{matrix}$

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

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

Primitive assertions of the form $\underline{x}_i (\mathbf{x})$ may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state $\mathbf{x}$ of a contemplated system or a statistical ensemble of systems. Parentheses $\texttt{(} \ldots \texttt{)}\!$ may be used to indicate logical negation. Eventually one discovers the usefulness of the $k\!$-ary just one false operators of the form $\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!$ as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), $\underline{X} \cong \mathbb{B}^n,$ and a space of functions (regions, propositions), $\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).$ Together these form a new universe of discourse $\underline{X}^\bullet$ of the type $(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),$ which we may abbreviate as $\mathbb{B}^n\ +\!\to \mathbb{B}$ or most succinctly as $[\mathbb{B}^n].$

The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells $\underline\mathbf{x},$ the defining features $\underline{x}_i,$ and the potential shadings $f : \underline{X} \to \mathbb{B}$ all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.

Finally, let $X^*\!$ denote the space of linear functions, $(\ell : X \to \mathbb{K}),$ which has in the finite case the same dimensionality as $X,\!$ and let the same notation be extended across the Table.

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

### A Differential Extension of Propositional Calculus

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

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

#### Differential Propositions : Qualitative Analogues of Differential Equations

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

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

The tangent space to $A\!$ at one of its points $x,\!$ sometimes written $\mathrm{T}_x(A),$ takes the form $\mathrm{d}A$ $=\!$ $\langle \mathrm{d}\mathcal{A} \rangle$ $=\!$ $\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!$ Strictly speaking, the name cotangent space is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.

Proceeding as we did with the base space $A,\!$ the tangent space $\mathrm{d}A$ at a point of $A\!$ can be analyzed as a product of distinct and independent factors:

 $\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!$

Here, $\mathrm{d}A_i\!$ is a set of two differential propositions, $\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!$ where $\texttt{(} \mathrm{d}a_i \texttt{)}\!$ is a proposition with the logical value of $\text{not} ~ \mathrm{d}a_i.\!$ Each component $\mathrm{d}A_i\!$ has the type $\mathbb{B},\!$ operating under the ordered correspondence $\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!$ However, clarity is often served by acknowledging this differential usage with a superficially distinct type $\mathbb{D},\!$ whose intension may be indicated as follows:

 $\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!$

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

#### An Interlude on the Path

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

A sense of the relation between $\mathbb{B}$ and $\mathbb{D}$ may be obtained by considering the path classifier (or the equivalence class of curves) approach to tangent vectors. Consider a universe $[\mathcal{X}].\!$ Given the boolean value system, a path in the space $X = \langle \mathcal{X} \rangle$ is a map $q : \mathbb{B} \to X.$ In this context the set of paths $(\mathbb{B} \to X)$ is isomorphic to the cartesian square $X^2 = X \times X,$ or the set of ordered pairs chosen from $X.\!$

We may analyze $X^2 = \{ (u, v) : u, v \in X \}$ into two parts, specifically, the ordered pairs $(u, v)\!$ that lie on and off the diagonal:

 $\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}$

This partition may also be expressed in the following symbolic form:

 $\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}$

The separate terms of this formula are defined as follows:

 $\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!$
 $\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!$

Thus we have:

 $\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}$

We may now use the features in $\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}$ to classify the paths of $(\mathbb{B} \to X)$ by way of the pairs in $X^2.\!$ If $X \cong \mathbb{B}^n,$ then a path $q\!$ in $X\!$ has the following form:

 $\begin{matrix} q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n. \end{matrix}$

Intuitively, we want to map this $(\mathbb{B}^2)^n$ onto $\mathbb{D}^n$ by mapping each component $\mathbb{B}^2$ onto a copy of $\mathbb{D}.$ But in the presenting context ${}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}$ is just a name associated with, or an incidental quality attributed to, coefficient values in $\mathbb{B}$ when they are attached to features in $\mathrm{d}\mathcal{X}.$

Taking these intentions into account, define $\mathrm{d}x_i : X^2 \to \mathbb{B}$ in the following manner:

 $\begin{array}{lcrcl} \mathrm{d}x_i(u, v) & = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)} \\ & = & x_i(u) & + & x_i(v) \\ & = & x_i(v) & - & x_i(u). \end{array}$

In the above transcription, the operator bracket of the form $\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!$ is a cactus lobe, in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in $\mathrm{GF}(2),$ and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.

The above definition of $\mathrm{d}x_i : X^2 \to \mathbb{B}$ is equivalent to defining $\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!$ in the following way:

 $\begin{array}{lcrcl} \mathrm{d}x_i (q) & = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)} \\ & = & x_i(q_0) & + & x_i(q_1) \\ & = & x_i(q_1) & - & x_i(q_0). \end{array}$

In this definition $q_b = q(b),\!$ for each $b\!$ in $\mathbb{B}.$ Thus, the proposition $\mathrm{d}x_i$ is true of the path $q = (u, v)\!$ exactly if the terms of $q,\!$ the endpoints $u\!$ and $v,\!$ lie on different sides of the question $x_i.\!$

The language of features in $\langle \mathrm{d}\mathcal{X} \rangle,$ indeed the whole calculus of propositions in $[\mathrm{d}\mathcal{X}],$ may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions $g : \mathrm{d}X \to \mathbb{B}.$ For example, the paths corresponding to $\mathrm{diag}(X)$ fall under the description $\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!$ which says that nothing changes against the backdrop of the coordinate frame $\{ x_1, \ldots, x_n \}.\!$

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

#### The Extended Universe of Discourse

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

Next we define the extended alphabet or bundled alphabet $\mathrm{E}\mathcal{A}$ as follows:

 $\begin{array}{lclcl} \mathrm{E}\mathcal{A} & = & \mathcal{A} \cup \mathrm{d}\mathcal{A} & = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}. \end{array}$

This supplies enough material to construct the differential extension $\mathrm{E}A,$ or the tangent bundle over the initial space $A,\!$ in the following fashion:

 $\begin{array}{lcl} \mathrm{E}A & = & \langle \mathrm{E}\mathcal{A} \rangle \\[4pt] & = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle \\[4pt] & = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle, \end{array}$

and also:

 $\begin{array}{lcl} \mathrm{E}A & = & A \times \mathrm{d}A \\[4pt] & = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n. \end{array}$

This gives $\mathrm{E}A$ the type $\mathbb{B}^n \times \mathbb{D}^n.$

Finally, the tangent universe $\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!$ is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features $\mathrm{E}\mathcal{A},$ and this fact is summed up in the following notation:

 $\begin{array}{lclcl} \mathrm{E}A^\bullet & = & [\mathrm{E}\mathcal{A}] & = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n]. \end{array}$

This gives the tangent universe $\mathrm{E}A^\bullet\!$ the type:

 $\begin{array}{lcl} (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) & = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})). \end{array}$

A proposition in the tangent universe $[\mathrm{E}\mathcal{A}]$ is called a differential proposition and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.

With these constructions, the differential extension $\mathrm{E}A$ and the space of differential propositions $(\mathrm{E}A \to \mathbb{B}),\!$ we have arrived, in main outline, at one of the major subgoals of this study. Table 8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.

 $\text{Symbol}\!$ $\text{Notation}\!$ $\text{Description}\!$ $\text{Type}\!$ $\mathrm{d}\mathfrak{A}\!$ $\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!$ $\begin{matrix} \text{Alphabet of} \\[2pt] \text{differential symbols} \end{matrix}$ $[n] = \mathbf{n}\!$ $\mathrm{d}\mathcal{A}\!$ $\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!$ $\begin{matrix} \text{Basis of} \\[2pt] \text{differential features} \end{matrix}$ $[n] = \mathbf{n}\!$ $\mathrm{d}A_i\!$ $\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!$ $\text{Differential dimension}~ i\!$ $\mathbb{D}\!$ $\mathrm{d}A\!$ $\begin{matrix} \langle \mathrm{d}\mathcal{A} \rangle \\[2pt] \langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle \\[2pt] \{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \} \\[2pt] \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n \\[2pt] \textstyle \prod_i \mathrm{d}A_i \end{matrix}$ $\begin{matrix} \text{Tangent space at a point:} \\[2pt] \text{Set of changes, motions,} \\[2pt] \text{steps, tangent vectors} \\[2pt] \text{at a point} \end{matrix}$ $\mathbb{D}^n\!$ $\mathrm{d}A^*\!$ $(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!$ $\text{Linear functions on}~ \mathrm{d}A\!$ $(\mathbb{D}^n)^* \cong \mathbb{D}^n\!$ $\mathrm{d}A^\uparrow\!$ $(\mathrm{d}A \to \mathbb{B})\!$ $\text{Boolean functions on}~ \mathrm{d}A\!$ $\mathbb{D}^n \to \mathbb{B}\!$ $\mathrm{d}A^\bullet\!$ $\begin{matrix} [\mathrm{d}\mathcal{A}] \\[2pt] (\mathrm{d}A, \mathrm{d}A^\uparrow) \\[2pt] (\mathrm{d}A ~+\!\to \mathbb{B}) \\[2pt] (\mathrm{d}A, (\mathrm{d}A \to \mathbb{B})) \\[2pt] [\mathrm{d}a_1, \ldots, \mathrm{d}a_n] \end{matrix}$ $\begin{matrix} \text{Tangent universe at a point of}~ A^\bullet, \\[2pt] \text{based on the tangent features} \\[2pt] \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} \end{matrix}$ $\begin{matrix} (\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B})) \\[2pt] (\mathbb{D}^n ~+\!\to \mathbb{B}) \\[2pt] [\mathbb{D}^n] \end{matrix}$

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

It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table 9 provides a suggestion of how these further extensions can be carried out.

 $\begin{array}{lllll} \mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A} \\ \mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A} \end{array}$ $\begin{array}{lll} \mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \} \\ \mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \} \end{array}$ $\begin{array}{lll} \mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} \\ \mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A} \\ \mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A} \\ \mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A} \end{array}$

#### Intentional Propositions

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

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

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

 $\begin{array}{lllll} \mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A} \\ \mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime \\ \mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime} \\ \cdots & & \cdots & \end{array}$ $\begin{array}{lll} \mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \} \end{array}$ $\begin{array}{lll} \mathrm{Q}^0 \mathcal{A} & = & \mathcal{A} \\ \mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \\ \mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}'' \\ \cdots & & \cdots \\ \mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A} \end{array}$

The resulting augmentations of our logical basis determine a series of discursive universes that may be called the intentional extension of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators $\mathrm{d}^k$ and $\mathrm{E}^k,$ and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain $X\!$ through an indefinite number of higher reaches, a particular collection of domains based on $X\!$ will be referred to as a realm of $X,\!$ and when the succession exhibits a temporal aspect, as a reign of $X.\!$

For the purposes of this discussion, an intentional proposition is defined as a proposition in the universe of discourse $\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],$ in other words, a map $q : \mathrm{Q}X \to \mathbb{B}.$ The sense of this definition may be seen if we consider the following facts. First, the equivalence $\mathrm{Q}X = X \times X'$ motivates the following chain of isomorphisms between spaces:

 $\begin{array}{lllcl} (\mathrm{Q}X \to \mathbb{B}) & \cong & (X & \times & ~X' \to \mathbb{B}) \\[4pt] & \cong & (X & \to & (X' \to \mathbb{B})) \\[4pt] & \cong & (X' & \to & (X~ \to \mathbb{B})). \end{array}$

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

In sum, the intentional proposition $q\!$ indicates a method for the systematic selection of local goals. As a general form of description, a map of the type $q : \mathrm{Q}^i X \to \mathbb{B}\!$ may be referred to as an "$i^\text{th}$ order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.

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

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

#### Life on Easy Street

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

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

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

### Back to the Beginning : Exemplary Universes

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

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

#### A One-Dimensional Universe

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

Let $\mathcal{X} = \{ x_1 \} = \{ A \}$ be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter ${}^{\backprime\backprime} A {}^{\prime\prime}\!$ is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element $A = x_1\!$ may be interpreted as a simple proposition or a coordinate projection $A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.$ The space $X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}$ of points (cells, vectors, interpretations) has cardinality $2^n = 2^1 = 2\!$ and is isomorphic to $\mathbb{B} = \{ 0, 1 \}.$ Moreover, $X\!$ may be identified with the set of singular propositions $\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.$ The space of linear propositions $X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}$ is algebraically dual to $X\!$ and also has cardinality $2.\!$ Here, ${}^{\backprime\backprime} 0 {}^{\prime\prime}\!$ is interpreted as denoting the constant function $0 : \mathbb{B} \to \mathbb{B},$ amounting to the linear proposition of rank $0,\!$ while $A\!$ is the linear proposition of rank $1.\!$ Last but not least we have the positive propositions $\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!$ of rank $1\!$ and $0,\!$ respectively, where ${}^{\backprime\backprime} 1 {}^{\prime\prime}\!$ is understood as denoting the constant function $1 : \mathbb{B} \to \mathbb{B}.$ In sum, there are $2^{2^n} = 2^{2^1} = 4$ propositions altogether in the universe of discourse, comprising the set $X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).$

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

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

 $\begin{matrix} \text{From} & \texttt{(} A \texttt{)} & \text{and} & \texttt{(} \mathrm{d}A \texttt{)} & \text{infer} & \texttt{(} A \texttt{)} & \text{next.} \\[8pt] \text{From} & \texttt{(} A \texttt{)} & \text{and} & \mathrm{d}A & \text{infer} & A & \text{next.} \\[8pt] \text{From} & A & \text{and} & \texttt{(} \mathrm{d}A \texttt{)} & \text{infer} & A & \text{next.} \\[8pt] \text{From} & A & \text{and} & \mathrm{d}A & \text{infer} & \texttt{(} A \texttt{)} & \text{next.} \end{matrix}$

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

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

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

#### Example 1. A Square Rigging

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

By way of example, suppose that we are given the initial condition $A = \mathrm{d}A\!$ and the law $\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!$ Since the equation $A = \mathrm{d}A\!$ is logically equivalent to the disjunction $A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!$ we may infer two possible trajectories, as displayed in Table 11. In either case the state $A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!$ is a stable attractor or a terminal condition for both starting points.

 $\text{Time}\!$ $\text{Trajectory 1}\!$ $\text{Trajectory 2}\!$ $\begin{matrix} 0 \\[4pt] 1 \\[4pt] 2 \\[4pt] 3 \\[4pt] 4 \end{matrix}$ $\begin{matrix} A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)} \\[4pt] \texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A \\[4pt] A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)} \\[4pt] A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)} \\[4pt] {}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel} \end{matrix}$ $\begin{matrix} \texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A \\[4pt] \texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A \\[4pt] A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)} \\[4pt] A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)} \\[4pt] {}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel} \end{matrix}\!$

Because the initial space $X = \langle A \rangle\!$ is one-dimensional, we can easily fit the second order extension $\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!$ within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.

 $\text{Figure 12.} ~~ \text{The Anchor}\!$

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

 $\text{Figure 13.} ~~ \text{The Tiller}\!$

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

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

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

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

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

#### Back to the Feature

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

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

 $\begin{array}{lll} \mathrm{E}X & = & \langle A, \mathrm{d}A \rangle \\[4pt] & = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \} \\[4pt] & = & \{ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~ \texttt{(} A \texttt{)} \mathrm{d}A,~ A \texttt{(} \mathrm{d}A \texttt{)},~ A ~ \mathrm{d}A \}. \end{array}$

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

To complete the construction of the extended universe of discourse $\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!$ one must add the set of differential propositions $\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!$ to the set of dispositions in $\mathrm{E}X.\!$ There are $2^{2^{2n}} = 16\!$ propositions in $\mathrm{E}X^\uparrow,\!$ as detailed in Table 14.

 $A\colon\!$ $1~1~0~0\!$ $\mathrm{d}A\colon\!$ $1~0~1~0\!$ $f_{0}\!$ $g_{0}\!$ $0~0~0~0\!$ $\texttt{(~)}\!$ $\text{false}\!$ $0\!$ $\begin{matrix} g_{1} \\[4pt] g_{2} \\[4pt] g_{4} \\[4pt] g_{8} \end{matrix}\!$ $\begin{matrix} 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~1~0~0 \\[4pt] 1~0~0~0 \end{matrix}\!$ $\begin{matrix} \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)} \\[4pt] \texttt{(} A \texttt{)} ~ \mathrm{d}A ~ \\[4pt] ~ A ~ \texttt{(} \mathrm{d}A \texttt{)} \\[4pt] ~ A ~~ \mathrm{d}A ~ \end{matrix}\!$ $\begin{matrix} \text{neither}~ A ~\text{nor}~ \mathrm{d}A \\[4pt] \mathrm{d}A ~\text{and not}~ A \\[4pt] A ~\text{and not}~ \mathrm{d}A \\[4pt] A ~\text{and}~ \mathrm{d}A \end{matrix}\!$ $\begin{matrix} \lnot A \land \lnot \mathrm{d}A \\[4pt] \lnot A \land \mathrm{d}A \\[4pt] A \land \lnot \mathrm{d}A \\[4pt] A \land \mathrm{d}A \end{matrix}\!$ $\begin{matrix} f_{1} \\[4pt] f_{2} \end{matrix}\!$ $\begin{matrix} g_{3} \\[4pt] g_{12} \end{matrix}\!$ $\begin{matrix} 0~0~1~1 \\[4pt] 1~1~0~0 \end{matrix}\!$ $\begin{matrix} \texttt{(} A \texttt{)} \\[4pt] A \end{matrix}\!$ $\begin{matrix} \text{not}~ A \\[4pt] A \end{matrix}\!$ $\begin{matrix} \lnot A \\[4pt] A \end{matrix}\!$ $\begin{matrix} g_{6} \\[4pt] g_{9} \end{matrix}\!$ $\begin{matrix} 0~1~1~0 \\[4pt] 1~0~0~1 \end{matrix}\!$ $\begin{matrix} \texttt{(} A \texttt{,} \mathrm{d}A \texttt{)} \\[4pt] \texttt{((} A \texttt{,} \mathrm{d}A \texttt{))} \end{matrix}\!$ $\begin{matrix} A ~\text{not equal to}~ \mathrm{d}A \\[4pt] A ~\text{equal to}~ \mathrm{d}A \end{matrix}\!$ $\begin{matrix} A \ne \mathrm{d}A \\[4pt] A = \mathrm{d}A \end{matrix}\!$ $\begin{matrix} g_{5} \\[4pt] g_{10} \end{matrix}\!$ $\begin{matrix} 0~1~0~1 \\[4pt] 1~0~1~0 \end{matrix}\!$ $\begin{matrix} \texttt{(} \mathrm{d}A \texttt{)} \\[4pt] \mathrm{d}A \end{matrix}\!$ $\begin{matrix} \text{not}~ \mathrm{d}A \\[4pt] \mathrm{d}A \end{matrix}\!$ $\begin{matrix} \lnot \mathrm{d}A \\[4pt] \mathrm{d}A \end{matrix}\!$ $\begin{matrix} g_{7} \\[4pt] g_{11} \\[4pt] g_{13} \\[4pt] g_{14} \end{matrix}\!$ $\begin{matrix} 0~1~1~1 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \end{matrix}\!$ $\begin{matrix} \texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)} \\[4pt] \texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))} \\[4pt] \texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)} \\[4pt] \texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))} \end{matrix}\!$ $\begin{matrix} \text{not both}~ A ~\text{and}~ \mathrm{d}A \\[4pt] \text{not}~ A ~\text{without}~ \mathrm{d}A \\[4pt] \text{not}~ \mathrm{d}A ~\text{without}~ A \\[4pt] A ~\text{or}~ \mathrm{d}A \end{matrix}\!$ $\begin{matrix} \lnot A \lor \lnot \mathrm{d}A \\[4pt] A \Rightarrow \mathrm{d}A \\[4pt] A \Leftarrow \mathrm{d}A \\[4pt] A \lor \mathrm{d}A \end{matrix}\!$ $f_{3}\!$ $g_{15}\!$ $1~1~1~1\!$ $\texttt{((~))}\!$ $\text{true}\!$ $1\!$

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

#### Tacit Extensions

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

Strictly speaking, however, there is a subtle distinction in type between the function $f_i : X \to \mathbb{B}$ and the corresponding function $g_j : \mathrm{E}X \to \mathbb{B},$ even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.

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

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

 $\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!$

On formal occasions, such as the present context of definition, the tacit extension from $\mathcal{X}$ to $\mathcal{Y}$ is explicitly symbolized by the operator $\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),$ where the appropriate alphabets $\mathcal{X}$ and $\mathcal{Y}$ are understood from context, but normally one may leave the "$\boldsymbol\varepsilon\!$" silent.

Let's explore what this means for the present Example. Here, $\mathcal{X} = \{ A \}$ and $\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.$ For each of the propositions $f_i\!$ over $X\!,$ specifically, those whose expression $e_i\!$ lies in the collection $\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!$ the tacit extension $\boldsymbol\varepsilon f\!$ of $f\!$ to $\mathrm{E}X$ can be phrased as a logical conjunction of two factors, $f_i = e_i \cdot \tau ~ ,\!$ where $\tau\!$ is a logical tautology that uses all the variables of $\mathcal{Y} - \mathcal{X}.$ Working in these terms, the tacit extensions $\boldsymbol\varepsilon f\!$ of $f\!$ to $\mathrm{E}X$ may be explicated as shown in Table 15.

 $\begin{matrix} 0 & = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0 \\[8pt] \texttt{(} A \texttt{)} & = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)} \\[8pt] A & = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)} \\[8pt] 1 & = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1 \end{matrix}$

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

#### Example 2. Drives and Their Vicissitudes

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

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

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

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

Given this language, the Example we take up here can be described as the family of $4^\text{th}\!$-gear curves through $\mathrm{E}^4 X\!$ $=\!$ $\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.$ These are the trajectories generated subject to the dynamic law $\mathrm{d}^4 A = 1,\!$ where it is understood in such a statement that all higher order differences are equal to $0.\!$ Since $\mathrm{d}^4 A\!$ and all higher $\mathrm{d}^k A\!$ are fixed, the temporal or transitional conditions (initial, mediate, terminal — transient or stable states) vary only with respect to their projections as points of $\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.$ Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of $\mathrm{E}^3 X.\!$ It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.

 $\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!$

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

 $\begin{matrix} r(q) & = & \displaystyle\sum_k d_k \cdot 2^{-k} & = & \displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k} \\[8pt] = \\[8pt] \displaystyle\frac{s(q)}{t} & = & \displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m} & = & \displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m} \end{matrix}$

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

$\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!$
$\text{Time}\!$ $\text{State}\!$ $A\!$ $\mathrm{d}A\!$
$p_i\!$ $q_j\!$ $\mathrm{d}^0\!A$ $\mathrm{d}^1\!A$ $\mathrm{d}^2\!A$ $\mathrm{d}^3\!A$ $\mathrm{d}^4\!A$

$\begin{matrix} p_0 \\[4pt] p_1 \\[4pt] p_2 \\[4pt] p_3 \\[4pt] p_4 \\[4pt] p_5 \\[4pt] p_6 \\[4pt] p_7 \end{matrix}\!$

$\begin{matrix} q_{01} \\[4pt] q_{03} \\[4pt] q_{05} \\[4pt] q_{15} \\[4pt] q_{17} \\[4pt] q_{19} \\[4pt] q_{21} \\[4pt] q_{31} \end{matrix}\!$

 $\begin{matrix} 0. \\[4pt] 0. \\[4pt] 0. \\[4pt] 0. \\[4pt] 1. \\[4pt] 1. \\[4pt] 1. \\[4pt] 1. \end{matrix}\!$ $\begin{matrix} 0 \\[4pt] 0 \\[4pt] 0 \\[4pt] 1 \\[4pt] 0 \\[4pt] 0 \\[4pt] 0 \\[4pt] 1 \end{matrix}\!$ $\begin{matrix} 0 \\[4pt] 0 \\[4pt] 1 \\[4pt] 1 \\[4pt] 0 \\[4pt] 0 \\[4pt] 1 \\[4pt] 1 \end{matrix}\!$ $\begin{matrix} 0 \\[4pt] 1 \\[4pt] 0 \\[4pt] 1 \\[4pt] 0 \\[4pt] 1 \\[4pt] 0 \\[4pt] 1 \end{matrix}\!$ $\begin{matrix} 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \end{matrix}\!$

$\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!$
$\text{Time}\!$ $\text{State}\!$ $A\!$ $\mathrm{d}A\!$
$p_i\!$ $q_j\!$ $\mathrm{d}^0\!A$ $\mathrm{d}^1\!A$ $\mathrm{d}^2\!A$ $\mathrm{d}^3\!A$ $\mathrm{d}^4\!A$

$\begin{matrix} p_0 \\[4pt] p_1 \\[4pt] p_2 \\[4pt] p_3 \\[4pt] p_4 \\[4pt] p_5 \\[4pt] p_6 \\[4pt] p_7 \end{matrix}\!$

$\begin{matrix} q_{25} \\[4pt] q_{11} \\[4pt] q_{29} \\[4pt] q_{07} \\[4pt] q_{09} \\[4pt] q_{27} \\[4pt] q_{13} \\[4pt] q_{23} \end{matrix}\!$

 $\begin{matrix} 1. \\[4pt] 0. \\[4pt] 1. \\[4pt] 0. \\[4pt] 0. \\[4pt] 1. \\[4pt] 0. \\[4pt] 1. \end{matrix}\!$ $\begin{matrix} 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 0 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 0 \end{matrix}\!$ $\begin{matrix} 0 \\[4pt] 0 \\[4pt] 1 \\[4pt] 1 \\[4pt] 0 \\[4pt] 0 \\[4pt] 1 \\[4pt] 1 \end{matrix}\!$ $\begin{matrix} 0 \\[4pt] 1 \\[4pt] 0 \\[4pt] 1 \\[4pt] 0 \\[4pt] 1 \\[4pt] 0 \\[4pt] 1 \end{matrix}\!$ $\begin{matrix} 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \\[4pt] 1 \end{matrix}\!$

## 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:

 $\mathbf{a} = (a, b, c), \quad \mathbf{p} = (p, q, r), \quad \mathbf{x} = (x, y, z),$

or, in more complicated situations:

 $x = (x_1, x_2, x_3), \quad y = (y_1, y_2, y_3), \quad z = (z_1, z_2, z_3).\!$

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 $\mathrm{F}$ 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 $\mathrm{F}$ to $f\!$, we write $g = \mathrm{F}f.$

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

 $\mathrm{E}f(x, y) ~=~ f(x + y).$

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

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

 $\mathrm{D}f(x, y) ~=~ \mathrm{E}f(x, y) - f(x),$

or, equivalently,

 $\mathrm{D}f(x, y) ~=~ f(x + y) - f(x).$

$\mathrm{D}$ 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:

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

Here, $c\!$ is held constant and $\mathrm{D}f(c, x)$ is regarded mainly as a function of the second variable $x\!$, giving the relative change in $f\!$ at various distances $x\!$ from the center $c\!$.

 $\mathrm{D}f : X \times X \to Y,$ $\mathrm{D}f(x, h) ~=~ f(x + h) - f(x).$

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. $\mathrm{D}f(x, h)$ 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.

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

This is yet another variant of the previous form, with $\mathrm{d}x$ 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 $w\!$)".

 $\text{Figure 1. Polymorphous Set}~ Q$

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 uv uw vw 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 $\mathrm{d}x$ that determines our decision whether or not we will make a difference in how we stand regarding $x\!$. If $\mathrm{d}x$ 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 $(x_1, \mathrm{d}x_1)\!$ for $x_1\!$

$2.\!$ Substitute $(x_2, \mathrm{d}x_2)\!$ for $x_2\!$

$3.\!$ Substitute $(x_3, \mathrm{d}x_3)\!$ for $x_3\!$

$\ldots$

$k.\!$ Substitute $(x_k, \mathrm{d}x_k)\!$ 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 "$u\ v\ w$".

 $\text{Figure 1. Polymorphous Set}~ Q$

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 $\mathrm{E}$ 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 $\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}$ 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 $\mathrm{D}$ to the initial proposition $\mathrm{q}$, 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 $\mathrm{D}q$. 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 "$u\ v\ w\!$".

 $\text{Figure 1. Polymorphous Set}~ Q$

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 $\mathrm{E}$ 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 $\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}$ 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 $\mathrm{D}$ 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 $\mathrm{D}q$ are marked with "@" signs, and the boundary crossings along each path are marked with the corresponding differential features among the collection $\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}$. 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 $q\!$.

 $\text{Figure 1. Polymorphous Set}~ Q$

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 uv uw vw 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

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

1. The points of the space $X\!$ that are assigned the coordinates:
$(u, v, w)\!$ = $(0, 1, 1)\!$ or $(1, 0, 1)\!$ or $(1, 1, 0)\!$ or $(1, 1, 1)\!$.
2. The points of the space $X\!$ that have the conjunctive descriptions:
(u) v w or u (v) w or u v (w) or u v w, where "(x)" is "not x".

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

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

I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of the sample proposition, the truth-function $q(u, v, w)\!$ that is given by the following expression:

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

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

For instance, consider the proposition $q(u, v, w)\!$, as before, and then consider its tacit extension $q(u, v, w, \mathrm{d}u, \mathrm{d}v, \mathrm{d}w)\!$, the latter of which may be indicated more explicitly as "$\mathrm{e}q\!$".

1. Proposition $q\!$ is abstractly typed as $q : \mathbb{B}^3 \to \mathbb{B}.$

Proposition $q\!$ is concretely typed as $q : X \to \mathbb{B}.$

2. Proposition $\mathrm{e}q\!$ is abstractly typed as $\mathrm{e}q : \mathbb{B}^3 \times \mathrm{d}\mathbb{B}^3 \to \mathbb{B}.$

Proposition $\mathrm{e}q\!$ is concretely typed as $\mathrm{e}q : X \times \mathrm{d}X \to \mathbb{B}.$

Succinctly, $\mathrm{e}q : \mathrm{E}X \to \mathbb{B}.$

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

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

Applying the operator $\mathrm{E}$ to the operand proposition $q\!$ yields:

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

The enlarged proposition $\mathrm{E}q$ is minimally interpretable as a function on the six variables of $\{ u, v, w, \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}.$ In other words, $\mathrm{E}q : \mathrm{E}X \to \mathbb{B},$ or $\mathrm{E}q : X \times \mathrm{d}X \to \mathbb{B}.$

Conjoining a query on the center cell, $c = u\ v\ w\!$, yields:

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

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

The models of $\mathrm{E}q \cdot c$ can be described in the usual ways as follows:

• The points of the space $\mathrm{E}X$ that have the following coordinate descriptions:
 = <1, 1, 1, 0, 0, 0>, <1, 1, 1, 0, 0, 1>, <1, 1, 1, 0, 1, 0>, <1, 1, 1, 1, 0, 0>.
• The points of the space $\mathrm{E}X$ that have the following conjunctive expressions:
 u v w (du)(dv)(dw), u v w (du)(dv) dw , u v w (du) dv (dw), u v w du (dv)(dw).

In summary, $\mathrm{E}q \cdot c$ informs us that we can get from $c\!$ to a model of $q\!$ by changing our position with respect to $u, v, w\!$ according to the following description:

 Change just one or none among $\{ u, v, w \}.\!$

I think that it would be worth our time to diagram the models of the enlarged or shifted proposition, $\mathrm{E}q,$ at least, the selection of them that we find issuing from the center cell $c.\!$

Figure 4 is an extended venn diagram for the proposition $\mathrm{E}q \cdot c,$ where the shaded area gives the models of $q\!$ and the "@" signs mark the terminal points of the requisite feature alterations.

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

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

Let $\mathcal{X} = \{ x_1, \ldots, x_k \}$ be a finite set of variables, regarded as a formal alphabet of formal symbols but listed here without quotation marks. Starting from this initial alphabet, the following items may then be defined:

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

$\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_k \}.$

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

$\mathrm{E}\mathcal{X} = \mathcal{X} \cup \mathrm{d}\mathcal{X},$

$\mathrm{E}\mathcal{X} = \{ x_1, \dots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k \}.$

Before we continue with the differential analysis of the source proposition $q\!$, we need to pause and take another look at just how it shapes up in the light of the extended universe $\mathrm{E}X,$ in other words, to examine in detail its tacit extension $\mathrm{e}q.\!$

The models of $\mathrm{e}q\!$ in $\mathrm{E}X\!$ can be comprehended as follows:

• Working in the summary coefficient form of representation, if the coordinate list $\mathbf{x}\!$ is a model of $q\!$ in $X,\!$ then one can construct a coordinate list $\mathrm{e}\mathbf{x}\!$ as a model for $\mathrm{e}q\!$ in $\mathrm{E}X\!$ just by appending any combination of values for the differential variables in $\mathrm{d}\mathcal{X}.$

For example, to focus once again on the center cell $c,\!$ which happens to be a model of the proposition $q\!$ in $X,\!$ one can extend $c\!$ in eight different ways into $\mathrm{E}X,\!$ and thus get eight models of the tacit extension $\mathrm{e}q\!$ in $\mathrm{E}X.\!$

It is a trivial exercise to write these out, but it is useful to do so at least once in order to see the patterns of data involved.

The tacit extensions of $c\!$ that are models of $\mathrm{e}q\!$ in $\mathrm{E}X\!$ are as follows:

 = <1, 1, 1, 0, 0, 0>, <1, 1, 1, 0, 0, 1>, <1, 1, 1, 0, 1, 0>, <1, 1, 1, 0, 1, 1>, <1, 1, 1, 1, 0, 0>, <1, 1, 1, 1, 0, 1>, <1, 1, 1, 1, 1, 0>, <1, 1, 1, 1, 1, 1>.
• Working in the conjunctive product form of representation, if the conjunctive proposition $x\!$ is a model of $q\!$ in $X,\!$ then one can construct a conjunctive proposition $\mathrm{e}x\!$ as a model for $\mathrm{e}q\!$ in $\mathrm{E}X\!$ just by appending any combination of values for the differential variables in $\mathrm{d}\mathcal{X}.$

The tacit extensions of $c\!$ that are models of $\mathrm{e}q\!$ in $\mathrm{E}X\!$ are as follows:

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

In short, $\mathrm{e}q \cdot c$ just enumerates all of the possible changes in $\mathrm{E}X\!$ that derive from, issue from, or stem from the cell $c\!$ in $X.\!$

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

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

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

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

$\mathrm{D}q = \mathrm{E}q - q = \mathrm{E}q - \mathrm{e}q.$
Since "+" and "-" signify the same operation over $\mathbb{B},$ we have:
$\mathrm{D}q = \mathrm{E}q + q = \mathrm{E}q + \mathrm{e}q.$
Since "+" = "exclusive-or", cactus syntax expresses this as:
 Eq q Eq eq o---o o---o \ / \ / Dq = @ = @ Dq = ( Eq , q ) = ( Eq , eq ).

Recall that a $k$-place bracket "$(x_1, x_2, \ldots, x_k)\!$" is interpreted (in the existential interpretation) to mean "Exactly one of the $x_j\!$ is false", thus the two-place bracket is equivalent to the exclusive-or.

The result of applying the difference operator $\mathrm{D}$ to the source proposition $q,\!$ conjoined with a query on the center cell $c,\!$ is:

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

The models of the difference proposition $\mathrm{D}q \cdot uvw\!$ are:

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

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

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

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

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

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

Another way of looking at this situation is by letting the (first order) differential features $\mathrm{d}u, \mathrm{d}v, \mathrm{d}w$ be viewed as the features of another universe of discourse, called the tangent universe to $X\!$ with respect to the interpretation $c\!$ and represented as $\mathrm{d}X \cdot c$ . In this setting, $\mathrm{D}q \cdot c$ , the difference proposition of $q\!$ at the interpretation $c\!$ , where $c = u\ v\ w$ , is marked by the shaded region in Figure 4.

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

Taken in the context of the tangent universe to $X\!$ at $c = u\ v\ w$ , written $\mathrm{d}X \cdot c$ or $\mathrm{d}X \cdot u\ v\ w$ , the shaded area of Figure 4 indicates the models of the difference proposition $\mathrm{d}q \cdot u\ v\ w$ , specifically:

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

#### Example 2. Jets and Sharks

Reference. Awbrey, J., and Awbrey, S. (1989), “Theme One : A Program of Inquiry”, unpublished manuscript, 09 Aug 1989. Online.

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

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

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

With a query on the name "ken" we obtain all of the propositional features associated with Ken, as shown in the following output.

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

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

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

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

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

#### Interlude

"The burden of genius is undeliverable"

From a poster, as I once misread it,
Marlboro, Vermont, c. 1976

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

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

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

#### Notes on Cactus Language

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

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

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

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

 ( jets , sharks )

Drawn as the corresponding cactus graph, we have:

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

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

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

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

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

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

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

I'm going to let that settle a while.

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

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

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

• If we make the identifications:

$\mathrm{charged} = \mathrm{true}\ (= \mathrm{indicated}),\!$

$\mathrm{resting} = \mathrm{false}\ (= \mathrm{otherwise}),\!$

then the physical function $f : \mathbf{B} \times \mathbf{B} \to \mathbf{B}$ is tantamount to the logical function that is commonly known as logical equivalence, or just plain equality:

 Table 6. Equality Function o---------o---------o---------o | x | y | (x , y) | o=========o=========o=========o | true | true | true | o---------o---------o---------o | true | false | false | o---------o---------o---------o | false | true | false | o---------o---------o---------o | false | false | true | o---------o---------o---------o
• If we make the identifications:

$\mathrm{resting} = \mathrm{true}\ (= \mathrm{indicated}),\!$

$\mathrm{charged} = \mathrm{false}\ (= \mathrm{otherwise}),\!$

then the physical function $f : \mathbf{B} \times \mathbf{B} \to \mathbf{B}$ is tantamount to the logical function that is commonly known as logical difference, or exclusive disjunction:

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

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

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

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

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

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

Given $\mathbb{B} = \{ \mathrm{charged}, \mathrm{resting} \},$ the Table presents the appearance of a function $f : \mathbb{B} \times \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ where $f(x, y, z) = (x, y, z).\!$

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

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

The cactus lobe operators $(\ ), (x_1), (x_1, x_2), (x_1, x_2, x_3), \ldots, (x_1, \ldots, x_k)\!$ are often referred to as boundary operators and one of the reasons for this can be seen most easily in the venn diagram for the $k\!$-argument boundary operator $(x_1, \ldots, x_k).\!$ Figure 10 shows the venn diagram for the 3-fold boundary form $(x, y, z).\!$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 ( jets , sharks ) (( 20's ),( 30's ),( 40's )) (( junior_high ),( high_school ),( college )) (( single ),( married ),( divorced )) (( bookie ),( burglar ),( pusher ))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Differential Logic : Graphical Exposition

#### Differential Propositions

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

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

Start with a proposition of the form $x ~\mathrm{and}~ y.\!$ This is graphed as two labels attached to a root node:

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

Written as a string, this is just the concatenation $xy.~\!$

The proposition $xy\!$ may be taken as a boolean function $f(x, y)\!$ having the abstract type $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!$ where $\mathbb{B} = \{ 0, 1 \}~\!$ is read in such a way that $0\!$ means $\mathrm{false}\!$ and $1\!$ means $\mathrm{true}.\!$

In this style of graphical representation, the value $\mathrm{true}\!$ looks like a blank label and the value $\mathrm{false}\!$ looks like an edge.

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

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

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

Now ask yourself: What is the value of the proposition $xy\!$ at a distance of $\mathrm{d}x$ and $\mathrm{d}y$ from the cell $xy\!$ where you are standing?

Don't think about it — just compute:

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

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

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

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

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

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

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

Computed over $\mathbb{B},$ plus and minus are the very same operation This will make the relationship between the differential and the integral parts of the resulting calculus slightly stranger than usual, but never mind that now.

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

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

And this is equivalent to the following graph:

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

We have just met with the fact that the differential of the $\mathrm{and}$ is the $\mathrm{or}$ of the differentials. Briefly summarized:

 $x ~\mathrm{and}~ y ~\xrightarrow{\mathrm{Diff}}~ \mathrm{d}x ~\mathrm{or}~ \mathrm{d}y\!$
 o-------------------------------------------------o | | | dx dy | | o o | | \ / | | o | | x y | | | @ --Diff--> @ | | | o-------------------------------------------------o | x y --Diff--> ((dx) (dy)) | o-------------------------------------------------o

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

If the form of the above statement reminds you of 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 C.S. Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which being a syntax adequate to handle the complexity of expressions that evolve.

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

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

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

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

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

Let $X\!$ be the set of values $\{ (\!|x|\!), x \} = \{ \mathrm{not}\ x, x \}.$
Let $Y\!$ be the set of values $\{ (\!|y|\!), y \} = \{ \mathrm{not}\ y, y \}.$

Then interpret the usual propositions about $x, y\!$ as functions of the concrete type $f : X \times Y \to \mathbb{B}.$

We are going to consider various operators on these functions. Here, an operator $\mathrm{F}$ is a function that takes one function $f\!$ into another function $\mathrm{F}f.$

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

The difference operator $\Delta,\!$ written here as $\mathrm{D}.$
The enlargement operator $\Epsilon,\!$ written here as $\mathrm{E}.$

These days, $\mathrm{E}$ is more often called the shift operator.

In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse. We mount up from the space $U = X \times Y$ to its differential extension:

 $\mathrm{E}U = U \times \mathrm{d}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y$ $\mathrm{d}X = \{ (\!|\mathrm{d}x|\!), \mathrm{d}x \}$ $\mathrm{d}Y = \{ (\!|\mathrm{d}y|\!), \mathrm{d}y \}$

The interpretations of these new symbols can be diverse, but the easiest for now is just to say that $\mathrm{d}x$ means "change $x\!$ " and $\mathrm{d}y$ means "change $y\!$ ".

To draw the differential extension $\mathrm{E}U$ of our present universe $U = X \times Y$ as a venn diagram, it would take us four logical dimensions $X, Y, \mathrm{d}X, \mathrm{d}Y,$ but we can project a suggestion of what it's about on the universe $X \times Y$ by drawing arrows that cross designated borders, labeling the arrows as $\mathrm{d}x$ when crossing the border between $x\!$ and $(\!|x|\!)$ and as $\mathrm{d}y$ when crossing the border between $y\!$ and $(\!|y|\!),$ in either direction, in either case.

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

We can form propositions from these differential variables in the same way that we would any other logical variables, for instance, interpreting the proposition $(\!| \mathrm{d}x\ (\!| \mathrm{d}y |\!) |\!)$ to say "$\mathrm{d}x \Rightarrow \mathrm{d}y$ ", in other words, however you wish to take it, whether indicatively or injunctively, as saying something to the effect that there is "no change in $x\!$ without a change in $y\!$ ".

Given the proposition $f(x, y)\!$ in $U = X \times Y\!,$ the (first order) enlargement of $f\!$ is the proposition $\mathrm{E}f$ in $\mathrm{E}U$ that is defined by the formula $\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = f(x + \mathrm{d}x, y + \mathrm{d}y).$

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

 $\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) ~=~ (x + \mathrm{d}x)(y + \mathrm{d}y).$
 o-------------------------------------------------o | | | x dx y dy | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | Ef = (x, dx) (y, dy) | o-------------------------------------------------o

Given the proposition $f(x, y)\!$ in $U = X \times Y,$ the (first order) difference of $f\!$ is the proposition $\mathrm{D}f$ in $\mathrm{E}U$ that is defined by the formula $\mathrm{D}f = \mathrm{E}f - f,$ or, written out in full, $\mathrm{D}f(x, y, \mathrm{d}x, \mathrm{d}y) = f(x + \mathrm{d}x, y + \mathrm{d}y) - f(x, y).$

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

 $\mathrm{D}f(x, y, \mathrm{d}x, \mathrm{d}y) = (x + \mathrm{d}x)(y + \mathrm{d}y) - xy.$
 o-------------------------------------------------o | | | x dx y dy | | o---o o---o | | \ | | / | | \ | | / | | \| |/ x y | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | Df = ((x, dx)(y, dy), xy) | o-------------------------------------------------o

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

 $f = xy = x\ \mathrm{and}\ y \Rightarrow \mathrm{D}f|_{xy} = (\!|(\!| \mathrm{d}x |\!)(\!| \mathrm{d}y |\!)|\!) ~=~ \mathrm{d}x\ \mathrm{or}\ \mathrm{d}y.$
 o-------------------------------------------------o | | | | | o-----------o o-----------o | | / \ / \ | | / x o y \ | | / /%\ \ | | / /%%%\ \ | | o o%%%%%o o | | | |%%%%%| | | | | dy (dx) |%%%%%| dx (dy) | | | | o<----------|--o--|---------->o | | | | |%%|%%| | | | | |%%|%%| | | | o o%%|%%o o | | \ \%|%/ / | | \ \|/ / | | \ | / | | \ /|\ / | | o-----------o | o-----------o | | | | | dx|dy | | | | | v | | o | | | o-------------------------------------------------o | | | dx dy | | o o | | \ / | | o | | | | | @ | | | o-------------------------------------------------o | Df|xy = ((dx) (dy)) | o-------------------------------------------------o

The picture illustrates the analysis of the inclusive disjunction $(\!|(\!| \mathrm{d}x |\!)(\!| \mathrm{d}y |\!)|\!)$ into the following exclusive disjunction:

 $\mathrm{d}x\ (\!| \mathrm{d}y |\!) + \mathrm{d}y\ (\!| \mathrm{d}x |\!) + \mathrm{d}x\ \mathrm{d}y.$

The latter proposition may be interpreted as saying "change $x\!$ or change $y\!$ or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.

We have just computed what will variously be called the difference map, the difference proposition, or the local proposition $\mathrm{D}f_p$ for the proposition $f(x, y) = xy\!$ at the point $p\!$ where $x = 1\!$ and $y = 1.\!$

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

Thus, we can write $\mathrm{D}f_p = \mathrm{D}f|_p = \mathrm{D}f|_{(1, 1)} = \mathrm{D}f|_{xy},$ so long as we know the frame of reference in force.

Sticking with the example $f(x, y) = xy,\!$ let us compute the value of the difference proposition $\mathrm{D}f$ at all of the points.

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

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

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

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

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

This really just constitutes a depiction of the interpretations in $\mathrm{E}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y$ that satisfy the difference proposition $\mathrm{D}f,$ namely, these:

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

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

We have been studying the action of the difference operator $\mathrm{D},$ also known as the localization operator, on the proposition $f : X \times Y \to \mathbb{B}$ that is commonly known as the conjunction $xy.~\!$ We described $\mathrm{D}f$ as a (first order) differential proposition, that is, a proposition of the type $\mathrm{D}f : X \times Y \times \mathrm{d}X \times \mathrm{d}Y \to \mathbb{B}.$ Abstracting from the augmented venn diagram that illustrates how the models, or the satisfying interpretations, of $\mathrm{D}f$ distribute within the extended universe $\mathrm{E}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y,$ we can depict $\mathrm{D}f$ in the form of a digraph or directed graph, one whose points are labeled with the elements of $U = X \times Y$ and whose arrows are labeled with the elements of $\mathrm{d}U = \mathrm{d}X \times \mathrm{d}Y.$

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

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

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

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

For $U = X_1 \times \ldots \times X_k$,
let $\mathrm{E}U = U \times \mathrm{d}U = X_1 \times \ldots \times X_k \times \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k.$

For a proposition $f : X_1 \times \ldots \times X_k \to \mathbb{B},$ the (first order) enlargement of $f\!$ is the proposition $\mathrm{E}f : \mathrm{E}U \to \mathbb{B}$ that is defined by:

 $\mathrm{E}f(x_1, \ldots x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k) ~=~ f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k).$

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

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

 $\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = (x + \mathrm{d}x)(y + \mathrm{d}y).$

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

 $\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = xy + x\ \mathrm{d}y + y\ \mathrm{d}x + \mathrm{d}x\ \mathrm{d}y.$

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

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

Given the sort of data that arises from this form of analysis, we can now fold the disjoined ingredients back into a boolean expansion or a DNF that is equivalent to the proposition $\mathrm{E}f.$

 $\mathrm{E}f = xy ~ \mathrm{E}f_{x y} + x (\!| y |\!) ~ \mathrm{E}f_{x (\!| y |\!)} + (\!| x |\!) y ~ \mathrm{E}f_{(\!| x |\!) y} + (\!| x |\!)(\!| y |\!) ~ \mathrm{E}f_{(\!| x |\!)(\!| y |\!)}.$

Here is a summary of the result, illustrated by means of a digraph picture, where the no change element $(\!| \mathrm{d}x |\!)(\!| \mathrm{d}y |\!)$ is drawn as a loop at the point $xy.~\!$

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

We may understand the enlarged proposition $\mathrm{E}f$ as telling us all the different ways to reach a model of $f\!$ from any point of the universe $U.\!$

#### Propositional Forms on Two Variables

To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type $X \times Y \to \mathbb{B}\!$ and abstract type $\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!$ For future reference, I will set here a few Tables that detail the actions of $\mathrm{E}\!$ and $\mathrm{D}\!$ on each of these functions, allowing us to view the results in several different ways.

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

Table 1. Propositional Forms on Two Variables
$\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$

 $x\!$ : $y\!$ :
 1 1 0 0 1 0 1 0

 $f_{0}\!$ $f_{1}\!$ $f_{2}\!$ $f_{3}\!$ $f_{4}\!$ $f_{5}\!$ $f_{6}\!$ $f_{7}\!$
 $f_{0000}\!$ $f_{0001}~\!$ $f_{0010}\!$ $f_{0011}\!$ $f_{0100}\!$ $f_{0101}\!$ $f_{0110}\!$ $f_{0111}\!$
 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1
 $(~)\!$ $(x)(y)\!$ $(x)\ y\!$ $(x)\!$ $x\ (y)\!$ $(y)\!$ $(x,\ y)\!$ $(x\ y)\!$
 $\mathrm{false}$ $\mathrm{neither}\ x\ \mathrm{nor}\ y$ $y\ \mathrm{without}\ x$ $\mathrm{not}\ x$ $x\ \mathrm{without}\ y$ $\mathrm{not}\ y$ $x\ \mathrm{not~equal~to}\ y$ $\mathrm{not~both}\ x\ \mathrm{and}\ y$
 $0\!$ $\lnot x \land \lnot y$ $\lnot x \land y$ $\lnot x$ $x \land \lnot y$ $\lnot y$ $x \ne y$ $\lnot x \lor \lnot y$
 $f_{8}\!$ $f_{9}\!$ $f_{10}\!$ $f_{11}\!$ $f_{12}\!$ $f_{13}\!$ $f_{14}\!$ $f_{15}\!$
 $f_{1000}\!$ $f_{1001}\!$ $f_{1010}\!$ $f_{1011}\!$ $f_{1100}\!$ $f_{1101}\!$ $f_{1110}\!$ $f_{1111}\!$
 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1
 $x\ y\!$ $((x,\ y))\!$ $y\!$ $(x\ (y))\!$ $x\!$ $((x)\ y)\!$ $((x)(y))\!$ $((~))\!$
 $x\ \mathrm{and}\ y$ $x\ \mathrm{equal~to}\ y$ $y\!$ $\mathrm{not}\ x\ \mathrm{without}\ y$ $x\!$ $\mathrm{not}\ y\ \mathrm{without}\ x$ $x\ \mathrm{or}\ y$ $\mathrm{true}$
 $x \land y$ $x = y\!$ $y\!$ $x \Rightarrow y$ $x\!$ $x \Leftarrow y$ $x \lor y$ $1\!$

Table 2 exhibits the same information in a different order, grouping the sixteen functions into seven natural classes.

Table 2. Propositional Forms on Two Variables
$\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$

 $x\!$ : $y\!$ :
 1 1 0 0 1 0 1 0

$f_{0}\!$

$f_{0000}\!$

0 0 0 0

$(~)\!$

$\mathrm{false}$

$0\!$

 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $f_{0001}~\!$ $f_{0010}\!$ $f_{0100}\!$ $f_{1000}\!$
 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $\mathrm{neither}\ x\ \mathrm{nor}\ y$ $y\ \mathrm{without}\ x$ $x\ \mathrm{without}\ y$ $x\ \mathrm{and}\ y$
 $\lnot x \land \lnot y$ $\lnot x \land y$ $x \land \lnot y$ $x \land y$
 $f_{3}\!$ $f_{12}\!$
 $f_{0011}\!$ $f_{1100}\!$
 0 0 1 1 1 1 0 0
 $(x)\!$ $x\!$
 $\mathrm{not}\ x$ $x\!$
 $\lnot x$ $x\!$
 $f_{6}\!$ $f_{9}\!$
 $f_{0110}\!$ $f_{1001}\!$
 0 1 1 0 1 0 0 1
 $(x,\ y)\!$ $((x,\ y))\!$
 $x\ \mathrm{not~equal~to}\ y$ $x\ \mathrm{equal~to}\ y$
 $x \ne y$ $x = y\!$
 $f_{5}\!$ $f_{10}\!$
 $f_{0101}\!$ $f_{1010}\!$
 0 1 0 1 1 0 1 0
 $(y)\!$ $y\!$
 $\mathrm{not}\ y$ $y\!$
 $\lnot y$ $y\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $f_{0111}\!$ $f_{1011}\!$ $f_{1101}\!$ $f_{1110}\!$
 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $\mathrm{not~both}\ x\ \mathrm{and}\ y$ $\mathrm{not}\ x\ \mathrm{without}\ y$ $\mathrm{not}\ y\ \mathrm{without}\ x$ $x\ \mathrm{or}\ y$
 $\lnot x \lor \lnot y$ $x \Rightarrow y$ $x \Leftarrow y$ $x \lor y$

$f_{15}\!$

$f_{1111}\!$

1 1 1 1

$((~))\!$

$\mathrm{true}$

$1\!$

The next four Tables expand the expressions of $\mathrm{E}f$ and $\mathrm{D}f$ in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.

Table 3. $\mathrm{E}f$ Expanded Over Differential Features $\{ \mathrm{d}x, \mathrm{d}y \}$
$f\!$
 $\mathrm{T}_{11}f$ $\mathrm{E}f|_{\mathrm{d}x\ \mathrm{d}y}$
 $\mathrm{T}_{10}f$ $\mathrm{E}f|_{\mathrm{d}x(\mathrm{d}y)}$
 $\mathrm{T}_{01}f$ $\mathrm{E}f|_{(\mathrm{d}x)\mathrm{d}y}$
 $\mathrm{T}_{00}f$ $\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $x\ y\!$ $x\ (y)\!$ $(x)\ y\!$ $(x)(y)\!$
 $x\ (y)\!$ $x\ y\!$ $(x)(y)\!$ $(x)\ y\!$
 $(x)\ y\!$ $(x)(y)\!$ $x\ y\!$ $x\ (y)\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $x\!$ $(x)\!$
 $x\!$ $(x)\!$
 $(x)\!$ $x\!$
 $(x)\!$ $x\!$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $((x,\ y))\!$ $(x,\ y)\!$
 $((x,\ y))\!$ $(x,\ y)\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $y\!$ $(y)\!$
 $(y)\!$ $y\!$
 $y\!$ $(y)\!$
 $(y)\!$ $y\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((x)(y))\!$ $((x)\ y)\!$ $(x\ (y))\!$ $(x\ y)\!$
 $((x)\ y)\!$ $((x)(y))\!$ $(x\ y)\!$ $(x\ (y))\!$
 $(x\ (y))\!$ $(x\ y)\!$ $((x)(y))\!$ $((x)\ y)\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
$f_{15}\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$
Fixed Point Total : $4\!$ $4\!$ $4\!$ $16\!$

Table 4. $\mathrm{D}f$ Expanded Over Differential Features $\{ \mathrm{d}x, \mathrm{d}y \}$
$f\!$ $\mathrm{D}f|_{\mathrm{d}x\ \mathrm{d}y}$ $\mathrm{D}f|_{\mathrm{d}x(\mathrm{d}y)}$ $\mathrm{D}f|_{(\mathrm{d}x)\mathrm{d}y}$ $\mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $((x,\ y))\!$ $(x,\ y)\!$ $(x,\ y)\!$ $((x,\ y))\!$
 $(y)\!$ $y\!$ $(y)\!$ $y\!$
 $(x)\!$ $(x)\!$ $x\!$ $x\!$
 $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $((~))\!$ $((~))\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $(~)\!$ $(~)\!$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(~)\!$ $(~)\!$
 $((~))\!$ $((~))\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((x,\ y))\!$ $(x,\ y)\!$ $(x,\ y)\!$ $((x,\ y))\!$
 $y\!$ $(y)\!$ $y\!$ $(y)\!$
 $x\!$ $x\!$ $(x)\!$ $(x)\!$
 $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
$f_{15}\!$ $((~))\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$

Table 5. $\mathrm{E}f$ Expanded Over Ordinary Features $\{ x, y \}\!$
$f\!$ $\mathrm{E}f|_{xy}$ $\mathrm{E}f|_{x(y)}$ $\mathrm{E}f|_{(x)y}$ $\mathrm{E}f|_{(x)(y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$ $(\mathrm{d}x)\ \mathrm{d}y$ $(\mathrm{d}x) (\mathrm{d}y)$
 $\mathrm{d}x\ (\mathrm{d}y)$ $\mathrm{d}x\ \mathrm{d}y$ $(\mathrm{d}x) (\mathrm{d}y)$ $(\mathrm{d}x)\ \mathrm{d}y$
 $(\mathrm{d}x)\ \mathrm{d}y$ $(\mathrm{d}x) (\mathrm{d}y)$ $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$
 $(\mathrm{d}x) (\mathrm{d}y)$ $(\mathrm{d}x)\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$ $\mathrm{d}x\ \mathrm{d}y$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $\mathrm{d}x$ $(\mathrm{d}x)$
 $\mathrm{d}x$ $(\mathrm{d}x)$
 $(\mathrm{d}x)$ $\mathrm{d}x$
 $(\mathrm{d}x)$ $\mathrm{d}x$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(\mathrm{d}x,\ \mathrm{d}y)$ $((\mathrm{d}x,\ \mathrm{d}y))$
 $((\mathrm{d}x,\ \mathrm{d}y))$ $(\mathrm{d}x,\ \mathrm{d}y)$
 $((\mathrm{d}x,\ \mathrm{d}y))$ $(\mathrm{d}x,\ \mathrm{d}y)$
 $(\mathrm{d}x,\ \mathrm{d}y)$ $((\mathrm{d}x,\ \mathrm{d}y))$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $\mathrm{d}y$ $(\mathrm{d}y)$
 $(\mathrm{d}y)$ $\mathrm{d}y$
 $\mathrm{d}y$ $(\mathrm{d}y)$
 $(\mathrm{d}y)$ $\mathrm{d}y$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((\mathrm{d}x)(\mathrm{d}y))$ $((\mathrm{d}x)\ \mathrm{d}y)$ $(\mathrm{d}x\ (\mathrm{d}y))$ $(\mathrm{d}x\ \mathrm{d}y)$
 $((\mathrm{d}x)\ \mathrm{d}y)$ $((\mathrm{d}x)(\mathrm{d}y))$ $(\mathrm{d}x\ \mathrm{d}y)$ $(\mathrm{d}x\ (\mathrm{d}y))$
 $(\mathrm{d}x\ (\mathrm{d}y))$ $(\mathrm{d}x\ \mathrm{d}y)$ $((\mathrm{d}x)(\mathrm{d}y))$ $((\mathrm{d}x)\ \mathrm{d}y)$
 $(\mathrm{d}x\ \mathrm{d}y)$ $(\mathrm{d}x\ (\mathrm{d}y))$ $((\mathrm{d}x)\ \mathrm{d}y)$ $((\mathrm{d}x)(\mathrm{d}y))$
$f_{15}\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$

Table 6. $\mathrm{D}f$ Expanded Over Ordinary Features $\{ x, y \}\!$
$f\!$ $\mathrm{D}f|_{xy}$ $\mathrm{D}f|_{x(y)}$ $\mathrm{D}f|_{(x)y}$ $\mathrm{D}f|_{(x)(y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$ $(\mathrm{d}x)\ \mathrm{d}y$ $((\mathrm{d}x)(\mathrm{d}y))$
 $\mathrm{d}x\ (\mathrm{d}y)$ $\mathrm{d}x\ \mathrm{d}y$ $((\mathrm{d}x)(\mathrm{d}y))$ $(\mathrm{d}x)\ \mathrm{d}y$
 $(\mathrm{d}x)\ \mathrm{d}y$ $((\mathrm{d}x)(\mathrm{d}y))$ $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$
 $((\mathrm{d}x)(\mathrm{d}y))$ $(\mathrm{d}x)\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$ $\mathrm{d}x\ \mathrm{d}y$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $\mathrm{d}x$ $\mathrm{d}x$
 $\mathrm{d}x$ $\mathrm{d}x$
 $\mathrm{d}x$ $\mathrm{d}x$
 $\mathrm{d}x$ $\mathrm{d}x$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(\mathrm{d}x,\ \mathrm{d}y)$ $(\mathrm{d}x,\ \mathrm{d}y)$
 $(\mathrm{d}x,\ \mathrm{d}y)$ $(\mathrm{d}x,\ \mathrm{d}y)$
 $(\mathrm{d}x,\ \mathrm{d}y)$ $(\mathrm{d}x,\ \mathrm{d}y)$
 $(\mathrm{d}x,\ \mathrm{d}y)$ $(\mathrm{d}x,\ \mathrm{d}y)$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $\mathrm{d}y$ $\mathrm{d}y$
 $\mathrm{d}y$ $\mathrm{d}y$
 $\mathrm{d}y$ $\mathrm{d}y$
 $\mathrm{d}y$ $\mathrm{d}y$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((\mathrm{d}x)(\mathrm{d}y))$ $(\mathrm{d}x)\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$ $\mathrm{d}x\ \mathrm{d}y$
 $(\mathrm{d}x)\ \mathrm{d}y$ $((\mathrm{d}x)(\mathrm{d}y))$ $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$
 $\mathrm{d}x\ (\mathrm{d}y)$ $\mathrm{d}x\ \mathrm{d}y$ $((\mathrm{d}x)(\mathrm{d}y))$ $(\mathrm{d}x)\ \mathrm{d}y$
 $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x\ (\mathrm{d}y)$ $(\mathrm{d}x)\ \mathrm{d}y$ $((\mathrm{d}x)(\mathrm{d}y))$
$f_{15}\!$ $((~))\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$

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

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

So let us do just that.

I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way. The data for that account is contained in Table 3.

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

E : (UB) → (EUB),
E : f(x, y) → Ef(x, y, dx, dy),
Ef(x, y, dx, dy) = f(x + dx, y + dy).

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

Eij : (UB) → (UB),
Eij : f → : Eijf,
Eijf = Ef|<dx = i, dy = j> = f(x + i, y + j).

The notation is a little bit awkward, but the data of the Table should make the sense clear. The important thing to observe is that Eij has the effect of transforming each proposition f : U → B into some other proposition f´ : U → B. As it happens, the action is one-to-one and onto for each Eij, so the gang of four operators {Eij : i, j in B} is an example of what is called a transformation group on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as T00, T01, T10, T11, to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:

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

It happens that there are just two possible groups of 4 elements. One is the cyclic group Z4 (German Zyklus), which this is not. The other is Klein's four-group V4 (German Vier), which it is.

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called orbits. One says that the orbits are preserved by the action of the group. There is an Orbit Lemma of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, T00 operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:

Number of orbits = (4 + 4 + 4 + 16) ÷ 4 = 7.

Amazing!

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

We are still working well within the logical analogue of the classical finite difference calculus, taking in the novelties that the logical transmutation of familiar elements is able to bring to light. Soon we will take up several different notions of approximation relationships that may be seen to organize the space of propositions, and these will allow us to define several different forms of differential analysis applying to propositions. In time we will find reason to consider more general types of maps, having concrete types of the form X1 × … × Xk → Y1 × … × Yn and abstract types Bk → Bn. We will think of these mappings as transforming universes of discourse into themselves or into others, in short, as transformations of discourse.

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

For example, consider the proposition f of concrete type f : X × Y × Z → B and abstract type f : B3 → B that is written (x, y, z) in cactus syntax. Taken as an assertion in what Peirce called the existential interpretation, (x, y, z) says that just one of x, y, z is false. It is useful to consider this assertion in relation to the conjunction xyz of the features that are engaged as its arguments. A venn diagram of (x, y, z) looks like this:

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

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

The same form of boundary relationship is exhibited for any cell of origin that one might elect to indicate, say, by means of the conjunction of positive and negative basis features u1 … uk, where uj = xj or uj = (xj), for j = 1 to k. The proposition (u1, …, uk) indicates the disjunctive region consisting of the cells that are "just next door" to the cell u1 … uk.

#### The Pragmatic Maxim

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.

Charles Sanders Peirce, “The Maxim of Pragmatism”, CP 5.438.

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

Let us return to the example of the so-called four-group V4. We encountered this group in one of its concrete representations, namely, as a transformation group that acts on a set of objects, in this particular case a set of sixteen functions or propositions. Forgetting about the set of objects that the group transforms among themselves, we may take the abstract view of the group's operational structure, say, in the form of the group operation table copied here:

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

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

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

Reading "+" as a logical disjunction:

 G = e + f + g + h,

And so, by expanding effects, we get:

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

More on the pragmatic maxim as a representation principle later.

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.

Peirce, “Maxim of Pragmaticism”, Collected Papers, CP 5.438.

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

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

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

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

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

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

[Let] A:B be taken to denote the elementary relative which multiplied into B gives A. (Peirce, CP 3.123).

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

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

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

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

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

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

Also, let m be such that:

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

In sum:

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

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

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

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

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

If X, Y, Z denote the three rectangular components of a vector, and W' denote numerical unity (or a fourth rectangular component, involving space of four dimensions), and (Y:Z) denote the operation of converting the Y component of a vector into its Z component, then

 1 = (W:W) + (X:X) + (Y:Y) + (Z:Z) i = (X:W) - (W:X) - (Y:Z) + (Z:Y) j = (Y:W) - (W:Y) - (Z:X) + (X:Z) k = (Z:W) - (W:Z) - (X:Y) + (Y:X)

In the language of logic (Y:Z) is a relative term whose relate is a Y component, and whose correlate is a Z component. The law of multiplication is plainly (Y:Z)(Z:X) = (Y:X), (Y:Z)(X:W) = 0, and the application of these rules to the above values of 1, i, j, k gives the quaternion relations

 i^2 = j^2 = k^2 = -1, ijk = -1, etc.

The symbol a(Y:Z) denotes the changing of Y to Z and the multiplication of the result by a'. If the relatives be arranged in a block

 W:W W:X W:Y W:Z X:W X:X X:Y X:Z Y:W Y:X Y:Y Y:Z Z:W Z:X Z:Y Z:Z

then the quaternion w + xi + yj + zk is represented by the matrix of numbers

 w -x -y -z x w -z y y z w -x z -y x w

The multiplication of such matrices follows the same laws as the multiplication of quaternions. The determinant of the matrix = the fourth power of the tensor of the quaternion.

The imaginary x + y(-1)^(1/2) may likewise be represented by the matrix

 x y -y x

and the determinant of the matrix = the square of the modulus.

C.S. Peirce, Collected Papers, CP 3.323, (1882). Johns Hopkins University Circulars, No. 13, p. 179.

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

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

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

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

Also, let n be such that:

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

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

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

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

 n = [ 1 1 0 | | | | 0 1 1 | | | | 1 0 1 ]

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

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

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

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

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

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

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

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

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

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

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

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

#### Obstacles to Applying the Pragmatic Maxim

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

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

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

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

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

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

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

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

Here is is the operation table of V4 once again:

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

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

In the case of V4 = (G$\cdot$), where G is the "underlying set" {efgh}, we have the 3-adic relation L(V4) ⊆ G × G × G whose triples are listed below:



It is part of the definition of a group that the 3-adic relation L ⊆ G3 is actually a function L : G × G → G. It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type L : G × G → G, we can define a couple of substitution operators:

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

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

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

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

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

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because V4 is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group. This is a group of six elements, say, G = {efghij}, with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, X = {ABC}, usually notated as G = Sym(X) or more abstractly and briefly, as Sym(3) or S3. Here are the permutation (= substitution) operations in Sym(X):

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

Here is the operation table for S3, given in abstract fashion:

 Table 2. Symmetric Group S_3 ^ e / \ e / \ / e \ f / \ / \ f / \ / \ / f \ f \ g / \ / \ / \ g / \ / \ / \ / g \ g \ g \ h / \ / \ / \ / \ h / \ / \ / \ / \ / h \ e \ e \ h \ i / \ / \ / \ / \ / \ i / \ / \ / \ / \ / \ / i \ i \ f \ j \ i \ j / \ / \ / \ / \ / \ / \ j / \ / \ / \ / \ / \ / \ ( j \ j \ j \ i \ h \ j ) \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ h \ h \ e \ j \ i / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ i \ g \ f \ h / \ / \ / \ / \ / \ / \ / \ / \ / \ f \ e \ g / \ / \ / \ / \ / \ / \ / \ g \ f / \ / \ / \ / \ / \ e / \ / \ / v

By the way, we will meet with the symmetric group S3 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).

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group V4, let us write out as quickly as possible in "relative form" a minimal budget of representations for the symmetric group on three letters, Sym(3). After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.

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

Writing this table in relative form generates the following "natural representation" of S3.

 e = A:A + B:B + C:C f = A:C + B:A + C:B g = A:B + B:C + C:A h = A:A + B:C + C:B i = A:C + B:B + C:A j = A:B + B:A + C:C

I have without stopping to think about it written out this natural representation of S3 in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as X:Y constitutes the turning of X into Y. It is possible that the next time we check in with CSP that we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

To construct the regular representations of S3, we pick up from the data of its operation table:

 Table 1. Symmetric Group S_3 ^ e / \ e / \ / e \ f / \ / \ f / \ / \ / f \ f \ g / \ / \ / \ g / \ / \ / \ / g \ g \ g \ h / \ / \ / \ / \ h / \ / \ / \ / \ / h \ e \ e \ h \ i / \ / \ / \ / \ / \ i / \ / \ / \ / \ / \ / i \ i \ f \ j \ i \ j / \ / \ / \ / \ / \ / \ j / \ / \ / \ / \ / \ / \ ( j \ j \ j \ i \ h \ j ) \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ h \ h \ e \ j \ i / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ i \ g \ f \ h / \ / \ / \ / \ / \ / \ / \ / \ / \ f \ e \ g / \ / \ / \ / \ / \ / \ / \ g \ f / \ / \ / \ / \ / \ e / \ / \ / v

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

It is part of the definition of a group that the 3-adic relation L ⊆ G3 is actually a function L : G × G → G. It is from this functional perspective that we can see an easy way to derive the two regular representations.

Since we have a function of the type L : G × G → G, we can define a couple of substitution operators:

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

In (1), we consider the effects of each x in its practical bearing on contexts of the form «_, y», as y ranges over G, and the effects are such that x takes «_, y» into xy, for y in G, all of which is summarily notated as x = {(y : xy) : y in G}. The pairs (y : xy) can be found by picking an x from the left margin of the group operation table and considering its effects on each y in turn as these run along the right margin. This produces the regular ante-representation of S3, like so:

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

In (2), we consider the effects of each x in its practical bearing on contexts of the form «y, _», as y ranges over G, and the effects are such that x takes «y, _» into yx, for y in G, all of which is summarily notated as x = {(y : yx) : y in G}. The pairs (y : yx) can be found by picking an x on the right margin of the group operation table and considering its effects on each y in turn as these run along the left margin. This generates the regular post-representation of S3, like so:

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

If the ante-rep looks different from the post-rep, it is just as it should be, as S3 is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.

the way of heaven and earth
is to be long continued
in their operation
without stopping

i ching, hexagram 32

You may be wondering what happened to the announced subject of "Differential Logic". If you think that we have been taking a slight excursion my reply to the charge of a scenic rout would be both "yes and no". What happened was this. We chanced to make the observation that the shift operators Eij form a transformation group that acts on the set of propositions of the form f : B2 → B. Group theory is a very attractive subject, but it did not have the effect of drawing us so far off our initial course as one might at first think. For one thing, groups, in particular, the special family of groups that have come to be named after the Norwegian mathematician Marius Sophus Lie, turn out to be of critical importance in the solution of differential equations. For another thing, group operations afford us examples of 3-adic relations that have been extremely well-studied over the years, and thus they supply us with no small bit of guidance in the study of sign relations, another class of 3-adic relations that have significance for logical studies, in our brief acquaintance with which we have scarcely even begun to break the ice. Finally, I could not resist taking up the connection between group representations, which constitute a very generic class of logical models, and the all-important pragmatic maxim.

We've seen a couple of groups, V4 and S3, represented in various ways, and we've seen their representations presented in a variety of different manners. Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called "matrix representation" of a group.

Recalling the manner of our acquaintance with the symmetric group S3, we began with the "bigraph" (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set X = {ABC}.

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

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

 e = A:A + B:B + C:C f = A:C + B:A + C:B g = A:B + B:C + C:A h = A:A + B:C + C:B i = A:C + B:B + C:A j = A:B + B:A + C:C

These days one is much more likely to encounter the natural representation of S3 in the form of a "linear representation", that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these:

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

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

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

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

### Differential Logic : Brief Recap

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

We have been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, $X,\!$ to considering a larger universe of discourse, $\mathrm{E}X.\!$

Each of these operators, in general terms having the form $\mathrm{F} : X \to \mathrm{E}X,\!$ acts on each proposition $p : X \to \mathbb{B}\!$ of the source universe $X\!$ to produce a proposition $\mathrm{F}p : \mathrm{E}X \to \mathbb{B}\!$ of the target universe $\mathrm{E}X.\!$

The two main operators that we have worked with up to this point are the enlargement operator $\mathrm{E} : X \to \mathrm{E}X\!$ and the difference operator $\mathrm{D} : X \to \mathrm{E}X.\!$

$\mathrm{E}\!$ and $\mathrm{D}\!$ take a proposition in $X,\!$ that is, a proposition $p : X \mathbb{B}\!$ that is said to be about the subject matter of $X,\!$ and produce the extended propositions $\mathrm{E}p, \mathrm{D}p : \mathrm{E}X \to \mathbb{B},\!$ which may be interpreted as being about specified collections of changes that might occur in $X.\!$

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

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

In the field picture a proposition $p : X \to \mathbb{B}\!$ becomes a scalar field, that is, a field of values in $\mathbb{B},\!$ or a field of true-false indications.

Let us take a moment to view an old proposition in this new light, for example, the conjunction $uv : X \to \mathbb{B}\!$ that is depicted in Figure 1.

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

Each of the operators $\mathrm{E}, \mathrm{D} : X \to \mathrm{E}X~\!$ takes us from considering propositions $p : X \to \mathbb{B},\!$ here viewed as scalar fields over $X,\!$ to considering the corresponding differential fields over $X,\!$ analogous to what are usually called vector fields $X.\!$

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

A differential operator $\mathrm{F},\!$ of the first order sort that we have been considering, takes a proposition $p : X \to \mathbb{B}\!$ and gives back a differential proposition $\mathrm{F}p : \mathrm{E}X \to \mathbb{B}.\!$

In the field view, we see the proposition $p : X \to \mathbb{B}\!$ as a scalar field and we see the differential proposition $\mathrm{F}p : \mathrm{E}X \to \mathbb{B}\!$ as a vector field, specifically, a field of propositions about contemplated changes in $X.\!$

The field of changes produced by $\mathrm{E}\!$ on $uv\!$ is shown in Figure 2.

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

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

The field of changes produced by $\mathrm{D}\!$ on $uv\!$ is shown in Figure 3.

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

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

## Elegant Graveyard

### Transitional Remarks

Up to this point we have been treating the universe of discourse $X,\!$ the quality $q,\!$ and the symbol "$q\!$" as all of one piece, almost as if the entire context marked by $X\!$ and $q\!$ and "$q\!$" amounted to the only way of viewing $X.\!$ That is clearly not the case, but the fact is that people often use the term "universe of discourse" to cover a particular set of distinctions drawn in the space $X\!$ and even sometimes a particular calculus or language for discussing the elements of $X.\!$ If it were possible to coin a new phrase in this realm one might distinguish these latter components as the "discursive universe", but there is probably no escape from simply recognizing the equivocal senses of the terms already in use and trying to clarify the senses intended in context.

Suppose we discover or begin to suspect that something we had been treating as a simple quality, $q,\!$ is actually compounded of other qualities, $u\!$ and $v\!$, according to a propositional formula $q = q(u, v).\!$

## Work Area

### Formal Development

#### Differential Propositions

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

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

In the above terms, a typical tangent space of $A\!$ at a point $x,\!$ frequently denoted as $T_x(A),\!$ can be characterized as having the generic construction $\mathrm{d}A = \langle \mathrm{d}\mathcal{A} \rangle = \langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.$ Strictly speaking, the name cotangent space is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.

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

$\mathrm{d}A = \prod_{i=1}^n \mathrm{d}A_i = \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.$

Here, $\mathrm{d}\mathcal{A}_i$ is an alphabet of two symbols, $\mathrm{d}\mathcal{A}_i = \{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \},$ where $\overline{\mathrm{d}a_i}$ is a symbol with the logical value of $\mathrm{not}\ \mathrm{d}a_i.$ Each component $\mathrm{d}A_i$ has the type $\mathbb{B},$ under the correspondence $\{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.$ However, clarity is often served by acknowledging this differential usage with a superficially distinct type $\mathbb{D},$ whose intension may be indicated as follows:

$\mathbb{D} = \{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \} = \{ \mathrm{same}, \mathrm{different} \} = \{ \mathrm{stay}, \mathrm{change} \} = \{ \mathrm{stop}, \mathrm{step} \}.$

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

#### Extended Universe of Discourse

Next, we define the so-called extended alphabet or bundled alphabet $\mathrm{E}\mathcal{A}$ as:

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

This supplies enough material to construct the differential extension $\mathrm{E}A,$ or the tangent bundle over the initial space $A,\!$ in the following fashion:

 $\mathrm{E}A\!$ = $A \times \mathrm{d}A\!$ = $\langle \mathrm{E}\mathcal{A} \rangle\!$ = $\langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle\!$ = $\langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,\!$

thus giving $\mathrm{E}A\!$ the type $\mathbb{B}^n \times \mathbb{D}^n.\!$

Finally, the tangent universe $\mathrm{E}A^\circ = [\mathrm{E}\mathcal{A}]\!$ is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features $\mathrm{E}\mathcal{A}:\!$

$\mathrm{E}A^\circ = [\mathrm{E}\mathcal{A}] = [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n],\!$

thus giving the tangent universe $\mathrm{E}A^\circ\!$ the type:

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

A proposition in the tangent universe $[\mathrm{E}\mathcal{A}]\!$ is called a differential proposition and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.

With these constructions, to be specific, the differential extension $\mathrm{E}A\!$ and the differential proposition $f : \mathrm{E}A \to \mathbb{B},\!$ we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 5).

### Orbit Table Template

Orbit Table Template
$f\!$ $\mathrm{F}f|_{xy}$ $\mathrm{F}f|_{x(y)}$ $\mathrm{F}f|_{(x)y}$ $\mathrm{F}f|_{(x)(y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
$f_{15}\!$ $((~))\!$

### Differential Analytic Operators

#### Tacit Extension

$\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) = f(x_1, \ldots, x_n).$

## Tables : Various Old Versions

### Archive 1

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

Table 3. Differential Inference Rules
 From $(q)\!$ and $(\mathrm{d}q)\!$ infer $(q)\!$ next. From $(q)\!$ and $\mathrm{d}q\!$ infer $q\!$ next. From $q\!$ and $(\mathrm{d}q)\!$ infer $q\!$ next. From $q\!$ and $\mathrm{d}q\!$ infer $(q)\!$ next.

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

Table 2. Fundamental Notations for Propositional Calculus
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

$\mathbb{B}^n$
$A^*\!$ $(\mathrm{hom} : A \to \mathbb{B})$ 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
$\{ a_1, \ldots, a_n \}$

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

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

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

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

Table 8. Notation for the Differential Extension of Propositional Calculus
Symbol Notation Description Type
$\mathrm{d}\mathfrak{A}$ $\lbrace\!$ “$\mathrm{d}a_1\!$” $, \ldots,\!$ “$\mathrm{d}a_n$” $\rbrace\!$ Alphabet of

differential
symbols

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

differential
features

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

dimension $i\!$

$\mathbb{D}$
$\mathrm{d}A$ $\langle \mathrm{d}\mathcal{A} \rangle$

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

Tangent space

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

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

on $\mathrm{d}A$

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

on $\mathrm{d}A$

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

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

Tangent universe

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

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

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

### Archive 2

 $\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$ $x\!$ : 1 1 0 0 $y\!$ : 1 0 1 0 $f_{0}\!$ $f_{0000}\!$ 0 0 0 0 $(~)\!$ false $0\!$ $f_{1}\!$ $f_{0001}~\!$ 0 0 0 1 $(x)(y)\!$ neither x nor y $\lnot x \land \lnot y\!$ $f_{2}\!$ $f_{0010}\!$ 0 0 1 0 $(x)\ y\!$ y and not x $\lnot x \land y\!$ $f_{3}\!$ $f_{0011}\!$ 0 0 1 1 $(x)\!$ not x $\lnot x\!$ $f_{4}\!$ $f_{0100}\!$ 0 1 0 0 $x\ (y)\!$ x and not y $x \land \lnot y\!$ $f_{5}\!$ $f_{0101}\!$ 0 1 0 1 $(y)\!$ not y $\lnot y\!$ $f_{6}\!$ $f_{0110}\!$ 0 1 1 0 $(x,\ y)\!$ x not equal to y $x \ne y\!$ $f_{7}\!$ $f_{0111}\!$ 0 1 1 1 $(x\ y)\!$ not both x and y $\lnot x \lor \lnot y\!$ $f_{8}\!$ $f_{1000}\!$ 1 0 0 0 $x\ y\!$ x and y $x \land y\!$ $f_{9}\!$ $f_{1001}\!$ 1 0 0 1 $((x,\ y))\!$ x equal to y $x = y\!$ $f_{10}\!$ $f_{1010}\!$ 1 0 1 0 $y\!$ y $y\!$ $f_{11}\!$ $f_{1011}\!$ 1 0 1 1 $(x\ (y))\!$ not x without y $x \Rightarrow y\!$ $f_{12}\!$ $f_{1100}\!$ 1 1 0 0 $x\!$ x $x\!$ $f_{13}\!$ $f_{1101}\!$ 1 1 0 1 $((x)\ y)\!$ not y without x $x \Leftarrow y\!$ $f_{14}\!$ $f_{1110}\!$ 1 1 1 0 $((x)(y))\!$ x or y $x \lor y\!$ $f_{15}\!$ $f_{1111}\!$ 1 1 1 1 $((~))\!$ true $1\!$

### Archive 3

 $\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$ $x\!$ : 1 1 0 0 $y\!$ : 1 0 1 0 $f_{0}\!$ $f_{0000}\!$ 0 0 0 0 $(\!|~|\!)$ false $0\!$ $f_{1}\!$ $f_{0001}~\!$ 0 0 0 1 $(\!|x|\!)(\!|y|\!)$ neither x nor y $\lnot x \land \lnot y$ $f_{2}\!$ $f_{0010}\!$ 0 0 1 0 $(\!|x|\!)\ y$ y and not x $\lnot x \land y$ $f_{3}\!$ $f_{0011}\!$ 0 0 1 1 $(\!|x|\!)$ not x $\lnot x$ $f_{4}\!$ $f_{0100}\!$ 0 1 0 0 $x\ (\!|y|\!)$ x and not y $x \land \lnot y$ $f_{5}\!$ $f_{0101}\!$ 0 1 0 1 $(\!|y|\!)\!$ not y $\lnot y$ $f_{6}\!$ $f_{0110}\!$ 0 1 1 0 $(\!|x,\ y|\!)$ x not equal to y $x \ne y$ $f_{7}\!$ $f_{0111}\!$ 0 1 1 1 $(\!|x\ y|\!)$ not both x and y $\lnot x \lor \lnot y$ $f_{8}\!$ $f_{1000}\!$ 1 0 0 0 $x\ y$ x and y $x \land y$ $f_{9}\!$ $f_{1001}\!$ 1 0 0 1 $(\!|(\!|x,\ y|\!)|\!)$ x equal to y $x = y\!$ $f_{10}\!$ $f_{1010}\!$ 1 0 1 0 $y\!$ y $y\!$ $f_{11}\!$ $f_{1011}\!$ 1 0 1 1 $(\!|x\ (\!|y|\!)|\!)$ not x without y $x \Rightarrow y$ $f_{12}\!$ $f_{1100}\!$ 1 1 0 0 $x\!$ x $x\!$ $f_{13}\!$ $f_{1101}\!$ 1 1 0 1 $(\!|(\!|x|\!)\ y|\!)$ not y without x $x \Leftarrow y$ $f_{14}\!$ $f_{1110}\!$ 1 1 1 0 $(\!|(\!|x|\!)(\!|y|\!)|\!)$ x or y $x \lor y$ $f_{15}\!$ $f_{1111}\!$ 1 1 1 1 $(\!|(\!|~|\!)|\!)$ true $1\!$

 $f\!$ $\mathrm{E}f|_{xy}$ $\mathrm{E}f|_{x(\!|y|\!)}$ $\mathrm{E}f|_{(\!|x|\!)y}$ $\mathrm{E}f|_{(\!|x|\!)(\!|y|\!)}\!$ $f_{0}\!$ $(\!|~|\!)$ $(\!|~|\!)$ $(\!|~|\!)$ $(\!|~|\!)$ $(\!|~|\!)$ $f_{1}\!$ $(\!|x|\!)(\!|y|\!)$ $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x (\!|\mathrm{d}y|\!)$ $(\!|\mathrm{d}x|\!) \mathrm{d}y$ $(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)$ $f_{2}\!$ $(\!|x|\!) y$ $\mathrm{d}x (\!|\mathrm{d}y|\!)$ $\mathrm{d}x\ \mathrm{d}y$ $(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)$ $(\!|\mathrm{d}x|\!) \mathrm{d}y$ $f_{4}\!$ $x (\!|y|\!)$ $(\!|\mathrm{d}x|\!) \mathrm{d}y$ $(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)$ $\mathrm{d}x\ \mathrm{d}y$ $\mathrm{d}x (\!|\mathrm{d}y|\!)$ $f_{8}\!$ $x y\!$ $(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)$ $(\!|\mathrm{d}x|\!) \mathrm{d}y$ $\mathrm{d}x (\!|\mathrm{d}y|\!)$ $\mathrm{d}x\ \mathrm{d}y$ $f_{3}\!$ $(\!|x|\!)$ $\mathrm{d}x$ $\mathrm{d}x$ $(\!|\mathrm{d}x|\!)$ $(\!|\mathrm{d}x|\!)$ $f_{12}\!$ $x\!$ $(\!|\mathrm{d}x|\!)$ $(\!|\mathrm{d}x|\!)$ $\mathrm{d}x$ $\mathrm{d}x$ $f_{6}\!$ $(\!|x, y|\!)$ $(\!|\mathrm{d}x, \mathrm{d}y|\!)\!$ $(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)$ $(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)$ $(\!|\mathrm{d}x, \mathrm{d}y|\!)\!$ $f_{9}\!$ $(\!|(\!|x, y|\!)|\!)$ $(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)$ $(\!|\mathrm{d}x, \mathrm{d}y|\!)\!$ $(\!|\mathrm{d}x, \mathrm{d}y|\!)\!$ $(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)$ $f_{5}\!$ $(\!|y|\!)\!$ $\mathrm{d}y$ $(\!|\mathrm{d}y|\!)$ $\mathrm{d}y$ $(\!|\mathrm{d}y|\!)$ $f_{10}\!$ $y\!$ $(\!|\mathrm{d}y|\!)$ $\mathrm{d}y$ $(\!|\mathrm{d}y|\!)$ $\mathrm{d}y$ $f_{7}\!$ $(\!|x y|\!)$ $(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)$ $(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)$ $(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)$ $(\!|\mathrm{d}x\ \mathrm{d}y|\!)$ $f_{11}\!$ $(\!|x (\!|y|\!)|\!)$ $(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)$ $(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)$ $(\!|\mathrm{d}x\ \mathrm{d}y|\!)$ $(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)$ $f_{13}\!$ $(\!|(\!|x|\!) y|\!)$ $(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)$ $(\!|\mathrm{d}x\ \mathrm{d}y|\!)$ $(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)$ $(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)$ $f_{14}\!$ $(\!|(\!|x|\!)(\!|y|\!)|\!)$ $(\!|\mathrm{d}x\ \mathrm{d}y|\!)$ $(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)$ $(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)$ $(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)$ $f_{15}\!$ $(\!|(\!|~|\!)|\!)$ $(\!|(\!|~|\!)|\!)$ $(\!|(\!|~|\!)|\!)$ $(\!|(\!|~|\!)|\!)$ $(\!|(\!|~|\!)|\!)$

### Archive 4

 $\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$ $x\!$ : 1 1 0 0 $y\!$ : 1 0 1 0 $f_{0}\!$ $f_{0000}\!$ 0 0 0 0 $(~)\!$ $\mathrm{false}$ $0\!$ $f_{1}\!$ $f_{0001}~\!$ 0 0 0 1 $(x)(y)\!$ $\mathrm{neither}\ x\ \mathrm{nor}\ y$ $\lnot x \land \lnot y\!$ $f_{2}\!$ $f_{0010}\!$ 0 0 1 0 $(x)\ y\!$ $y\ \mathrm{without}\ x$ $\lnot x \land y\!$ $f_{3}\!$ $f_{0011}\!$ 0 0 1 1 $(x)\!$ $\mathrm{not}\ x$ $\lnot x\!$ $f_{4}\!$ $f_{0100}\!$ 0 1 0 0 $x\ (y)\!$ $x\ \mathrm{without}\ y$ $x \land \lnot y\!$ $f_{5}\!$ $f_{0101}\!$ 0 1 0 1 $(y)\!$ $\mathrm{not}\ y$ $\lnot y\!$ $f_{6}\!$ $f_{0110}\!$ 0 1 1 0 $(x,\ y)\!$ $x\ \mathrm{not~equal~to}\ y$ $x \ne y\!$ $f_{7}\!$ $f_{0111}\!$ 0 1 1 1 $(x\ y)\!$ $\mathrm{not~both}\ x\ \mathrm{and}\ y$ $\lnot x \lor \lnot y\!$ $f_{8}\!$ $f_{1000}\!$ 1 0 0 0 $x\ y\!$ $x\ \mathrm{and}\ y$ $x \land y\!$ $f_{9}\!$ $f_{1001}\!$ 1 0 0 1 $((x,\ y))\!$ $x\ \mathrm{equal~to}\ y$ $x = y\!$ $f_{10}\!$ $f_{1010}\!$ 1 0 1 0 $y\!$ $y\!$ $y\!$ $f_{11}\!$ $f_{1011}\!$ 1 0 1 1 $(x\ (y))\!$ $\mathrm{not}\ x\ \mathrm{without}\ y$ $x \Rightarrow y\!$ $f_{12}\!$ $f_{1100}\!$ 1 1 0 0 $x\!$ $x\!$ $x\!$ $f_{13}\!$ $f_{1101}\!$ 1 1 0 1 $((x)\ y)\!$ $\mathrm{not}\ y\ \mathrm{without}\ x$ $x \Leftarrow y\!$ $f_{14}\!$ $f_{1110}\!$ 1 1 1 0 $((x)(y))\!$ $x\ \mathrm{or}\ y$ $x \lor y\!$ $f_{15}\!$ $f_{1111}\!$ 1 1 1 1 $((~))\!$ $\mathrm{true}$ $1\!$

Table 2. Propositional Forms on Two Variables
$\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$

$x\!$ :

1 1 0 0

$y\!$ :

1 0 1 0

$f_{0}\!$

$f_{0000}\!$

0 0 0 0

$(~)\!$

$\mathrm{false}$

$0\!$

 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $f_{0001}~\!$ $f_{0010}\!$ $f_{0100}\!$ $f_{1000}\!$
 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $\mathrm{neither}\ x\ \mathrm{nor}\ y$ $y\ \mathrm{without}\ x$ $x\ \mathrm{without}\ y$ $x\ \mathrm{and}\ y$
 $\lnot x \land \lnot y$ $\lnot x \land y$ $x \land \lnot y$ $x \land y$
 $f_{3}\!$ $f_{12}\!$
 $f_{0011}\!$ $f_{1100}\!$
 0 0 1 1 1 1 0 0
 $(x)\!$ $x\!$
 $\mathrm{not}\ x$ $x\!$
 $\lnot x$ $x\!$
 $f_{6}\!$ $f_{9}\!$
 $f_{0110}\!$ $f_{1001}\!$
 0 1 1 0 1 0 0 1
 $(x,\ y)\!$ $((x,\ y))\!$
 $x\ \mathrm{not~equal~to}\ y$ $x\ \mathrm{equal~to}\ y$
 $x \ne y$ $x = y\!$
 $f_{5}\!$ $f_{10}\!$
 $f_{0101}\!$ $f_{1010}\!$
 0 1 0 1 1 0 1 0
 $(y)\!$ $y\!$
 $\mathrm{not}\ y$ $y\!$
 $\lnot y$ $y\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $f_{0111}\!$ $f_{1011}\!$ $f_{1101}\!$ $f_{1110}\!$
 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $\mathrm{not~both}\ x\ \mathrm{and}\ y$ $\mathrm{not}\ x\ \mathrm{without}\ y$ $\mathrm{not}\ y\ \mathrm{without}\ x$ $x\ \mathrm{or}\ y$
 $\lnot x \lor \lnot y$ $x \Rightarrow y$ $x \Leftarrow y$ $x \lor y$

$f_{15}\!$

$f_{1111}\!$

1 1 1 1

$((~))\!$

$\mathrm{true}$

$1\!$

Table 2. Propositional Forms on Two Variables
$\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$

$x\!$ :

1 1 0 0

$y\!$ :

1 0 1 0

$f_{0}\!$

$f_{0000}\!$

0 0 0 0

$(~)\!$

$\mathrm{false}$

$0\!$

 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $f_{0001}~\!$ $f_{0010}\!$ $f_{0100}\!$ $f_{1000}\!$
 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $\mathrm{neither}\ x\ \mathrm{nor}\ y$ $y\ \mathrm{without}\ x$ $x\ \mathrm{without}\ y$ $x\ \mathrm{and}\ y$
 $\lnot x \land \lnot y$ $\lnot x \land y$ $x \land \lnot y$ $x \land y$
 $f_{3}\!$ $f_{12}\!$
 $f_{0011}\!$ $f_{1100}\!$
 0 0 1 1 1 1 0 0
 $(x)\!$ $x\!$
 $\mathrm{not}\ x$ $x\!$
 $\lnot x$ $x\!$
 $f_{6}\!$ $f_{9}\!$
 $f_{0110}\!$ $f_{1001}\!$
 0 1 1 0 1 0 0 1
 $(x,\ y)\!$ $((x,\ y))\!$
 $x\ \mathrm{not~equal~to}\ y$ $x\ \mathrm{equal~to}\ y$
 $x \ne y$ $x = y\!$
 $f_{5}\!$ $f_{10}\!$
 $f_{0101}\!$ $f_{1010}\!$
 0 1 0 1 1 0 1 0
 $(y)\!$ $y\!$
 $\mathrm{not}\ y$ $y\!$
 $\lnot y$ $y\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $f_{0111}\!$ $f_{1011}\!$ $f_{1101}\!$ $f_{1110}\!$
 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $\mathrm{not~both}\ x\ \mathrm{and}\ y$ $\mathrm{not}\ x\ \mathrm{without}\ y$ $\mathrm{not}\ y\ \mathrm{without}\ x$ $x\ \mathrm{or}\ y$
 $\lnot x \lor \lnot y$ $x \Rightarrow y$ $x \Leftarrow y$ $x \lor y$

$f_{15}\!$

$f_{1111}\!$

1 1 1 1

$((~))\!$

$\mathrm{true}$

$1\!$

 $f\!$ $\mathrm{E}f|_{xy}$ $\mathrm{E}f|_{x(y)}$ $\mathrm{E}f|_{(x)y}$ $\mathrm{E}f|_{(x)(y)}$ $f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $f_{1}\!$ $(x)(y)\!$ $\mathrm{d}x\ \mathrm{d}y\!$ $\mathrm{d}x (\mathrm{d}y)\!$ $(\mathrm{d}x) \mathrm{d}y\!$ $(\mathrm{d}x)(\mathrm{d}y)\!$ $f_{2}\!$ $(x) y\!$ $\mathrm{d}x (\mathrm{d}y)\!$ $\mathrm{d}x\ \mathrm{d}y\!$ $(\mathrm{d}x)(\mathrm{d}y)\!$ $(\mathrm{d}x) \mathrm{d}y\!$ $f_{4}\!$ $x (y)\!$ $(\mathrm{d}x) \mathrm{d}y\!$ $(\mathrm{d}x)(\mathrm{d}y)\!$ $\mathrm{d}x\ \mathrm{d}y\!$ $\mathrm{d}x (\mathrm{d}y)\!$ $f_{8}\!$ $x y\!$ $(\mathrm{d}x)(\mathrm{d}y)\!$ $(\mathrm{d}x) \mathrm{d}y\!$ $\mathrm{d}x (\mathrm{d}y)\!$ $\mathrm{d}x\ \mathrm{d}y\!$ $f_{3}\!$ $(x)\!$ $\mathrm{d}x\!$ $\mathrm{d}x\!$ $(\mathrm{d}x)\!$ $(\mathrm{d}x)\!$ $f_{12}\!$ $x\!$ $(\mathrm{d}x)\!$ $(\mathrm{d}x)\!$ $\mathrm{d}x\!$ $\mathrm{d}x\!$ $f_{6}\!$ $(x, y)\!$ $(\mathrm{d}x, \mathrm{d}y)\!$ $((\mathrm{d}x, \mathrm{d}y))\!$ $((\mathrm{d}x, \mathrm{d}y))\!$ $(\mathrm{d}x, \mathrm{d}y)\!$ $f_{9}\!$ $((x, y))\!$ $((\mathrm{d}x, \mathrm{d}y))\!$ $(\mathrm{d}x, \mathrm{d}y)\!$ $(\mathrm{d}x, \mathrm{d}y)\!$ $((\mathrm{d}x, \mathrm{d}y))\!$ $f_{5}\!$ $(y)\!$ $\mathrm{d}y\!$ $(\mathrm{d}y)\!$ $\mathrm{d}y\!$ $(\mathrm{d}y)\!$ $f_{10}\!$ $y\!$ $(\mathrm{d}y)\!$ $\mathrm{d}y\!$ $(\mathrm{d}y)\!$ $\mathrm{d}y\!$ $f_{7}\!$ $(x y)\!$ $((\mathrm{d}x)(\mathrm{d}y))\!$ $((\mathrm{d}x) \mathrm{d}y)\!$ $(\mathrm{d}x (\mathrm{d}y))\!$ $(\mathrm{d}x\ \mathrm{d}y)\!$ $f_{11}\!$ $(x (y))\!$ $((\mathrm{d}x) \mathrm{d}y)\!$ $((\mathrm{d}x)(\mathrm{d}y))\!$ $(\mathrm{d}x\ \mathrm{d}y)\!$ $(\mathrm{d}x (\mathrm{d}y))\!$ $f_{13}\!$ $((x) y)\!$ $(\mathrm{d}x (\mathrm{d}y))\!$ $(\mathrm{d}x\ \mathrm{d}y)\!$ $((\mathrm{d}x)(\mathrm{d}y))\!$ $((\mathrm{d}x) \mathrm{d}y)\!$ $f_{14}\!$ $((x)(y))\!$ $(\mathrm{d}x\ \mathrm{d}y)\!$ $(\mathrm{d}x (\mathrm{d}y))\!$ $((\mathrm{d}x) \mathrm{d}y)\!$ $((\mathrm{d}x)(\mathrm{d}y))\!$ $f_{15}\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$