Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus

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

\tableofcontents

\section{Casual introduction}

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

\begin{figure}[h]\begin{centering}
\begin{footnotesize}\begin{verbatim}
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o-----------------------------------------------------------o
\end{verbatim}\end{footnotesize}
Figure 1.  Local Habitations, And Names
\end{centering}\end{figure}

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

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

\begin{figure}[h]\begin{centering}
\begin{footnotesize}\begin{verbatim}
o-----------------------------------------------------------o
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o-----------------------------------------------------------o
\end{verbatim}\end{footnotesize}
Figure 2.  Same Names, Different Habitations
\end{centering}\end{figure}

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

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

\begin{figure}[h]\begin{centering}
\begin{footnotesize}\begin{verbatim}
o-----------------------------------------------------------o
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| . . . | . . . . . Q . . | . . . | . . dQ. . . . . | . . . |
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\end{verbatim}\end{footnotesize}
Figure $1^\prime$.  Back, To The Future
\end{centering}\end{figure}

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

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

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

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

\begin{itemize}
\item
$\overline{q}\ \overline{\operatorname{d}q}$ describes $h$
\item
$\overline{q}\ \operatorname{d}q$ describes $k$
\item
$q\ \overline{\operatorname{d}q}$ describes $i$
\item
$q\ \operatorname{d}q$ describes $j$
\end{itemize}

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

\begin{center}\begin{tabular}{ccccccc}
\multicolumn{7}{c}{\textbf{Table 3.  Differential Inference Rules}} \\[12pt]
From & $\overline{q}$ & and & $\overline{\operatorname{d}q}$ & infer & $\overline{q}$ & next. \\[6pt]
From & $\overline{q}$ & and & $\operatorname{d}q$ & infer & $q$ & next. \\[6pt]
From & $q$ & and & $\overline{\operatorname{d}q}$ & infer & $q$ & next. \\[6pt]
From & $q$ & and & $\operatorname{d}q$ & infer & $\overline{q}$ & next. \\[6pt]
\end{tabular}\end{center}

$\ldots$

\section{Formal development}

Table 4

\begin{center}\begin{tabular}{|l|l|l|l|}
\multicolumn{4}{c}{\textbf{Table 4.  Propositional Calculus : Basic Notation}} \\
\hline
\textbf{Symbol} & \textbf{Notation} & \textbf{Description} & \textbf{Type} \\
\hline
$\mathcal{A}$ & $\{ a_1, \ldots, a_n \}$ & Alphabet & $[n] = \mathbf{n}$ \\
\hline
$A_i$ & $\{ \overline{a_i}, a_i \}$ & Dimension $i$ & $\mathbb{B}$ \\
\hline
$A$ & $\langle \mathcal{A} \rangle$      & Set of cells,      & $\mathbb{B}^n$ \\
    & $\langle a_1, \ldots, a_n \rangle$ & coordinate tuples, & \\
    & $\{ (a_1, \ldots, a_n) \}$         & points, or vectors & \\
    & $A_1 \times \ldots \times A_n$     & in the universe    & \\
    & $\textstyle \prod_{i=1}^n A_i$     & of discourse       & \\
\hline
$A^*$ & $(\operatorname{hom} : A \to \mathbb{B})$ & Linear functions &
$(\mathbb{B}^n)^* \cong \mathbb{B}^n$ \\
\hline
$A^\uparrow$ & $(A \to \mathbb{B})$ & Boolean functions &
$\mathbb{B}^n \to \mathbb{B}$ \\
\hline
$A^\circ$ & $[ \mathcal{A} ]$         & Universe of discourse    &
$(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))$                  \\
          & $(A, A^\uparrow)$         & based on the features    &
$(\mathbb{B}^n\ +\!\to \mathbb{B})$                              \\
          & $(A\ +\!\to \mathbb{B})$  & $\{ a_1, \ldots, a_n \}$ &
$[\mathbb{B}^n]$                                                 \\
          & $(A, (A \to \mathbb{B}))$ & & \\
          & $[ a_1, \ldots, a_n ]$    & & \\
\hline
\end{tabular}\end{center}

Table 5

\begin{center}\begin{tabular}{|l|l|l|l|}
\multicolumn{4}{c}{\textbf{Table 5.  Differential Extension : Basic Notation}} \\
\hline

\textbf{Symbol} &
\textbf{Notation} &
\textbf{Description} &
\textbf{Type} \\
\hline

$\operatorname{d}\mathcal{A}$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
Alphabet of differential features &
$[n] = \mathbf{n}$ \\
\hline

$\operatorname{d}A_i$ &
$\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}$ &
Differential dimension $i$ &
$\mathbb{D}$ \\
\hline

$\operatorname{d}A$ &
$\langle \operatorname{d}\mathcal{A} \rangle$ &
Tangent space at a point: &
$\mathbb{D}^n$
\\
&
$\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle$ &
Set of changes, &
\\
&
$\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}$ &
motions, steps, &
\\
&
$\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n$ &
tangent vectors &
\\
&
$\textstyle \prod_{i=1}^n \operatorname{d}A_i$ &
at a point &
\\
\hline

$\operatorname{d}A^*$ &
$(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})$ &
Linear functions on $\operatorname{d}A$ &
$(\mathbb{D}^n)^* \cong \mathbb{D}^n$ \\
\hline

$\operatorname{d}A^\uparrow$ &
$(\operatorname{d}A \to \mathbb{B})$ &
Boolean functions on $\operatorname{d}A$ &
$\mathbb{D}^n \to \mathbb{B}$ \\
\hline

$\operatorname{d}A^\circ$ &
$[ \operatorname{d}\mathcal{A} ]$ &
Tangent universe &
$(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))$
\\
&
$(\operatorname{d}A, \operatorname{d}A^\uparrow)$ &
at a point of $A^\circ,$ &
$(\mathbb{D}^n\ +\!\to \mathbb{B})$
\\
&
$(\operatorname{d}A\ +\!\to \mathbb{B})$ &
based on the &
$[\mathbb{D}^n]$
\\
&
$(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))$ &
tangent features &
\\
&
$[ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ]$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
\\
\hline
\end{tabular}\end{center}

$\ldots$

\section{Expository examples}

$\ldots$

Draft Conversion @ MyWikiBiz : Wiki Format

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

Casual introduction

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

o-----------------------------------------------------------o
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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o-----------------------------------------------------------o

Figure 1. Local Habitations, And Names

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

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

o-----------------------------------------------------------o
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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o-----------------------------------------------------------o

Figure 2. Same Names, Different Habitations

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

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

o-----------------------------------------------------------o
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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| . . . . . . o-------------o . o-------------o . . . . . . |
| . . h . . ./. . . . . . . .\./. . . . . . . .\. . . . . . |
| . . @ . . / . . . . . . . . o . . . . . . . . \ . . . . . |
| . . . . ./. . i . . . . . ./.\. . . . . . . . .\. . . . . |
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| . . . o . . . . . . . . o . j . o . . . . . . . . o . . . |
| . . . | . . . . . . . . | . @ . | . . . . . . . . | . . . |
| . . . | . . . . . . . . | . . . | . . . . . . . . | . . . |
| . . . | . . . . . Q . . | . . . | . . dQ. . . . . | . . . |
| . . . | . . . . . . . . | . . . | . . . . . k . . | . . . |
| . . . | . . . . . . . . | . . . | . . . . . @ . . | . . . |
| . . . o . . . . . . . . o . . . o . . . . . . . . o . . . |
| . . . .\. . . . . . . . .\. . ./. . . . . . . . ./. . . . |
| . . . . \ . . . . . . . . \ . / . . . . . . . . / . . . . |
| . . . . .\. . . . . . . . .\./. . . . . . . . ./. . . . . |
| . . . . . \ . . . . . . . . o . . . . . . . . / . . . . . |
| . . . . . .\. . . . . . . ./.\. . . . . . . ./. . . . . . |
| . . . . . . o-------------o . o-------------o . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
o-----------------------------------------------------------o

Figure 1′. Back, To The Future

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

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

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

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

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

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

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

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

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

Table 3. Differential Inference Rules

From \(\overline{q}\!\) and \(\overline{\operatorname{d}q}\!\) infer \(\overline{q}\!\) next.     From \(\overline{q}\!\) and \(\operatorname{d}q\!\) infer \(q\!\) next.     From \(q\!\) and \(\overline{\operatorname{d}q}\!\) infer \(q\!\) next.     From \(q\!\) and \(\operatorname{d}q\!\) infer \(\overline{q}\!\) next.

…

Formal development

…

Expository examples

…

Materials from "Dif Log Dyn Sys" for Reuse Here

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 {a₁, …, 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, …} 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 {x₁, …, 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 {a₁, …, a_n} provides a basis for generating an n-dimensional universe of discourse that I denote as [a₁, …, a_n]. It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points 〈a₁, …, a_n〉 and the set of propositions f : 〈a₁, …, a_n〉 → B that are implicit with the ordinary picture of a venn diagram on n features. Thus, we may regard the universe of discourse [a₁, …, a_n] as an ordered pair having the type (Bⁿ, (Bⁿ → B), and we may abbreviate this last type designation as Bⁿ +→ B, or even more succinctly as [Bⁿ]. (Used this way, the angle brackets 〈…〉 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 n to denote the data type of a finite set on n elements.

Table 2. Fundamental Notations for Propositional Calculus

Symbol	Notation	Description	Type
A	{a1, …, an}	Alphabet	[n] = n
Ai	{(ai), ai}	Dimension i	B
A	〈A〉 〈a1, …, an〉 {‹a1, …, an›} A1 × … × An ∏i Ai	Set of cells, coordinate tuples, points, or vectors in the universe of discourse	Bn
A*	(hom : A → B)	Linear functions	(Bn)* = Bn
A^	(A → B)	Boolean functions	Bn → B
A•	[A] (A, A^) (A +→ B) (A, (A → B)) [a1, …, an]	Universe of discourse based on the features {a1, …, an}	(Bn, (Bn → B)) (Bn +→ B) [Bn]

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.

Table 5. A Bridge Over Troubled Waters

Linear Space	Liminal Space	Logical Space
X {x1, …, xn} cardinality n	X {x1, …, xn} cardinality n	A {a1, …, an} cardinality n
Xi 〈xi〉 isomorphic to K	Xi {(xi), xi} isomorphic to B	Ai {(ai), ai} isomorphic to B
X 〈X〉 〈x1, …, xn〉 {‹x1, …, xn›} X1 × … × Xn ∏i Xi isomorphic to Kn	X 〈X〉 〈x1, …, xn〉 {‹x1, …, xn›} X1 × … × Xn ∏i Xi isomorphic to Bn	A 〈A〉 〈a1, …, an〉 {‹a1, …, an›} A1 × … × An ∏i Ai isomorphic to Bn
X* (hom : X → K) isomorphic to Kn	X* (hom : X → B) isomorphic to Bn	A* (hom : A → B) isomorphic to Bn
X^ (X → K) isomorphic to: (Kn → K)	X^ (X → B) isomorphic to: (Bn → B)	A^ (A → B) isomorphic to: (Bn → B)
X• [X] [x1, …, xn] (X, X^) (X +→ K) (X, (X → K)) isomorphic to: (Kn, (Kn → K)) (Kn +→ K) [Kn]	X• [X] [x1, …, xn] (X, X^) (X +→ B) (X, (X → B)) isomorphic to: (Bn, (Bn → B)) (Bn +→ B) [Bn]	A• [A] [a1, …, an] (A, A^) (A +→ B) (A, (A → B)) isomorphic to: (Bn, (Bn → B)) (Bn +→ B) [Bn]

The left side of the Table collects mostly standard notation for an n-dimensional vector space over a field 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 K, with a special interest in the continuous line R, to the qualitative and discrete situations that are instanced and typified by B.

I now proceed to explain these concepts in more detail. The two most important ideas developed in the table are:

• 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 = 〈x₁, …, x_n〉 \(\cong\) Rⁿ. The coordinate system X = {x_i} is a set of maps x_i : R_n → R, also known as the coordinate projections. Given a "dataset" of points x in 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 R, that is, a subset of R_n, and then we define the i^th threshold map, or limen x_i as follows:

x_i : R_n → B such that:

x_i(x) = 1 if x ∈ L_i,

x_i(x) = 0 if otherwise.

In other notations that are sometimes used, the operator \(\chi (\ )\) 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, given as elements of B. Finally, it is not uncommon to use the name of the relation itself as a predicate that maps n-tuples into truth values. In each of these notations, the above definition could be expressed as follows:

x_i(x) = \(\chi (x \in L_i)\) = \(\lceil x \in L_i \rceil\) = L_i(x).

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^th axis at a unique threshold value r_i ∈ 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^th axis, that is, points of the form ‹0, …, 0, r_i, 0, …, 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 = Rⁿ can now be expressed by taking the set X = {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^th threshold map. This can help to remind us that the threshold operator  )_i acts on x by setting up a kind of a "hurdle" for it. In this interpretation, the coordinate proposition x_i asserts that the representative point x resides above the i^th threshold.

Primitive assertions of the form x_i(x) can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state x of a contemplated system or a statistical ensemble of systems. Parentheses "( )" may be used to indicate negation. Eventually one discovers the usefulness of the k-ary just one false operators of the form "( , , , )", as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), X = 〈X〉 \(\cong\) Bⁿ, and a space of functions (regions, propositions), X^ \(\cong\) (Bⁿ → B). Together these form a new universe of discourse X ^• = [X] of the type (Bⁿ, (Bⁿ → B)), which we may abbreviate as Bⁿ +→ B, or most succinctly as [Bⁿ].

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, where we constantly think of the elementary cells x, the defining features x_i, and the potential shadings f : X → B, all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.

Finally, let X* denote the space of linear functions, (hom : X → K), which in the finite case has 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, which 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 : The Qualitative Analogues of Differential Equations

In order to define the differential extension of a universe of discourse [A], the initial alphabet A must be extended to include a collection of symbols for differential features, or basic changes that are capable of occurring in [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 [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 dA = {da₁, …, da_n}, in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet A = {a₁, …, 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 dA 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 A and dA.)

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 dA = 〈dA〉 = 〈da₁, …, da_n〉. 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:

dA = ∏_i dA_i = dA₁ × … × dA_n.

Here, dA_i is an alphabet of two symbols, dA_i = {(da_i), da_i}, where (da_i) is a symbol with the logical value of "not da_i". Each component dA_i has the type B, under the correspondence {(da_i), da_i} \(\cong\) {0, 1}. However, clarity is often served by acknowledging this differential usage with a superficially distinct type D, whose intension may be indicated as follows:

D = {(da_i), da_i} = {same, different} = {stay, change} = {stop, step}.

Viewed within a coordinate representation, spaces of type Bⁿ and Dⁿ 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]

It may help to get a sense of the relation between B and D by considering the path classifier (or equivalence class of curves) approach to tangent vectors. As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [X]. Given the boolean value system, a path in the space X = 〈X〉 is a map q : B → X. In this case, the set of paths (B → X) is isomorphic to the cartesian square X² = X × X, or the set of ordered pairs from X.

We may analyze X² = {‹u, v› : u, v ∈ X} into two parts, specifically, the pairs that lie on and off the diagonal:

X² = {‹u, v› : u = v} ∪ {‹u, v› : u ≠ v}

In symbolic terms, this partition may be expressed as:

X² \(\cong\) Diag(X) + 2 * Comb(X, 2),

where:

Diag(X) = {‹x, x› : x ∈ X},

and where:

Comb(X, k) = "X choose k" = {k-sets from X},

so that:

Comb(X, 2) = {{u, v} : u, v ∈ X}.

We can now use the features in dX = {dx_i} = {dx₁, …, dx_n} to classify the paths of (B → X) by way of the pairs in X². If X \(\cong\) Bⁿ then a path in X has the form q : (B → Bⁿ) \(\cong\) Bⁿ × Bⁿ \(\cong\) B²ⁿ \(\cong\) (B²)ⁿ. Intuitively, we want to map this (B²)ⁿ onto Dⁿ by mapping each component B² onto a copy of D. But in our current situation "D" is just a name we give, or an accidental quality we attribute, to coefficient values in B when they are attached to features in dX.

Therefore, define dx_i : X² → B such that:

dxi(‹u, v›)	=	( xi(u) , xi(v) )
	=	xi(u) + xi(v)
	=	xi(v) – xi(u).

In the above transcription, the operator bracket of the form "( … , … )" is a cactus lobe, signifying just one false, in this case among two boolean variables, while "+" is boolean addition in the proper sense of addition in GF(2), and is thus equivalent to "–", in the sense of adding the additive inverse.

The above definition is equivalent to defining dx_i : (B → X) → B such that:

dxi(q)	=	( xi(q0) , xi(q1) )
	=	xi(q0) + xi(q1)
	=	xi(q1) – xi(q0),

where q_b = q(b), for each b in B. Thus, the proposition dx_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.

Now we can use the language of features in 〈dX〉, indeed the whole calculus of propositions in [dX], to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions g : dX → B. For example, the paths corresponding to Diag(X) fall under the description (dx₁)…(dx_n), which says that nothing changes among the set of features {x₁, …, 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 which 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 so-called extended alphabet or bundled alphabet EA as:

EA = A ∪ dA = {a₁, …, a_n, da₁, …, da_n}

This supplies enough material to construct the differential extension EA, or the tangent bundle over the initial space A, in the following fashion:

EA	=	A × dA
	=	〈EA〉
	=	〈A ∪ dA〉
	=	〈a1, …, an, da1, …, dan〉,

thus giving EA the type Bⁿ × Dⁿ.

Finally, the tangent universe EA ^• = [EA] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features EA:

EA ^• = [EA] = [a₁, …, a_n, da₁, …, da_n],

thus giving the tangent universe EA ^• the type (Bⁿ × Dⁿ +→ B) = (Bⁿ × Dⁿ, (Bⁿ × Dⁿ → B)).

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
dA	{da1, …, dan}	Alphabet of differential features	[n] = n
dAi	{(dai), dai}	Differential dimension i	D
dA	〈dA〉 〈da1, …, dan〉 {‹da1, …, dan›} dA1 × … × dAn ∏i dAi	Tangent space at a point: Set of changes, motions, steps, tangent vectors at a point	Dn
dA*	(hom : dA → B)	Linear functions on dA	(Dn)* = Dn
dA^	(dA → B)	Boolean functions on dA	Dn → B
dA•	[dA] (dA, dA^) (dA +→ B) (dA, (dA → B)) [da1, …, dan]	Tangent universe at a point of A•, based on the tangent features {da1, …, dan}	(Dn, (Dn → B)) (Dn +→ B) [Dn]

The adjectives differential or tangent are systematically attached to every construct based on the differential alphabet dA, taken by itself. Strictly speaking, we probably ought to call dA the set of cotangent features derived from A, but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type (Bⁿ → B) → B from cotangent vectors as elements of type Dⁿ. In like fashion, having defined EA = A ∪ dA, we can systematically attach the adjective extended or the substantive bundle to all of the constructs associated with this full complement of 2n features.

Eventually we may want to extend our basic alphabet even further, to allow for discussion of higher order differential expressions. For those who want to run ahead, and would like to play through, I submit the following gamut of notation (Table 9).

Table 9. Higher Order Differential Features

A = d0A = {a1, …, an} dA = d1A = {da1, …, dan}     dkA = {dka1, …, dkan} d*A = {d0A, …, dkA, …}

E0A = d0A E1A = d0A ∪ d1A EkA = d0A ∪ … ∪ dkA E∞A = ∪ d*A

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, I produce the following scheme of notation, which extends any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators p^k and Q^k are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.

Table 10. A Realm of Intentional Features

p0A = A = {a1 , …, an } p1A = A′ = {a1′, …, an′} p2A = A″ = {a1″, …, an″} ...       ... pkA =     {pka1, …, pkan}

Q0A = A Q1A = A ∪ A′ Q2A = A ∪ A′ ∪ A″ ...   ... QkA = A ∪ A′ ∪ … ∪ pkA

The resulting augmentations of our logical basis found a series of discursive universes that may be called the intentional extension of propositional calculus. The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d^k and E^k, and there is an obvious and natural relation between these two extensions that falls within our purview to explore. In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain X through an indefinite number of higher reaches, I refer to a particular collection of domains based on X as a realm of X, and when the succession exhibits a temporal aspect, as a reign of X.

For the purposes of this discussion, let us define an intentional proposition as a proposition in the universe of discourse QX ^• = [QX], in other words, a map q : QX → B. The sense of this definition may be seen if we consider the following facts. First, the equivalence QX = X × X′ motivates the following chain of isomorphisms between spaces:

(QX → B)	\(\cong\)	(X × X′ → B)
	\(\cong\)	(X → (X′ → B))
	\(\cong\)	(X′ → (X → B)).

Viewed in this light, an intentional proposition q may be rephrased as a map q : X × X′ → 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 → (X′ → B) and as q : X′ → (X → 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, we may refer to a map of the type q : QⁱX → B as an "i^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 [EA] makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition q : EA → B is the set of models q^–1(1) in EA. Finding all of the models of q, the extended interpretations in EA that satisfy q, can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus 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 of [EA] 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, my 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 : Some 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 X = {x₁} = {A} be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "A" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions. At any rate, the basis element A = x₁ may be interpreted as a simple proposition or a coordinate projection A = x₁ : B₁ ¸> B. The space X = 〈A 〉 = {(A), A} of points (cells, vectors, interpretations) has cardinality 2ⁿ = 2¹ = 2 and is isomorphic to B = {0, 1}. Moreover, X may be identified with the set of singular propositions {x : B ××> B}. The space of linear propositions X* = {hom : B +> B} = {0, A} is algebraically dual to X and also has cardinality 2. Here, "0" is interpreted as denoting the constant function 0 : B → 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 {pos : B ¥> B} = {A, 1}, of rank 1 and 0, respectively, where "1" is understood as denoting the constant function 1 : B → B. In sum, there are \(2^{2^n} = 2^{2^1} = 4\) propositions altogether in the universe of discourse, comprising the set X^ = {f : X → B} = {0, (A), A, 1} \(\cong\) (B → B).

The first order differential extension of X is EX = {x₁, dx₁} = {A, dA}. If the feature "A" is understood as applying to some object or state, then the feature "dA" 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 dA are true at a given moment one may infer that (A) 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:

From (A) and (dA) infer (A) next.     From (A) and dA infer A next.     From A and (dA) infer A next.     From A and dA infer (A) next.

It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time. A time variable is actually a reference to a clock, that is, a canonical or a convenient process that is established or accepted 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 {(dA), dA} are preserved or changed in the next instance. In order to know this, we would have to determine d²A, 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 d^kA = 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 = dA and the law d²A = (A). Then, since "A = dA" ⇔ "A dA or (A)(dA)", we may infer two possible trajectories, as displayed in Table 11. In either of these cases, the state A(dA)(d²A) is a stable attractor or a terminal condition for both starting points.

Table 11. A Pair of Commodious Trajectories

Time	Trajectory 1	Trajectory 2
0 1 2 3 4	A dA (d2A) (A) dA d2A A (dA) (d2A) A (dA) (d2A) " " "	(A) (dA) d2A (A) dA d2A A (dA) (d2A) A (dA) (d2A) " " "

Because the initial space X = 〈A〉 is one-dimensional, we can easily fit the second order extension E²X = 〈A, dA, d²A〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.

Figure 12. The Anchor

If we eliminate from view the regions of E²X that are ruled out by the dynamic law d²A = (A), 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 d²A. 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 (A, d²A).

Figure 13. 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 I may continue with outlining the structure of the differential extension [EX] = [A, dA]. Over the extended alphabet EX = {x₁, dx₁} = {A, dA}, of cardinality 2ⁿ = 2, we generate the set of points, EX, of cardinality 2²ⁿ = 4, that bears the following chain of equivalent descriptions:

EX	=	〈A, dA〉
	=	{(A), A} × {(dA), dA}
	=	{(A)(dA), (A) dA, A (dA), A dA}.

The space EX may be assigned the mnemonic type B × D, which is really no different than B × B = B². An individual element of EX 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 EX ^• = [x₁, dx₁] = [A, dA], one must add the set of differential propositions EX^ = {g : EX → B} \(\cong\) (B × D → B) to the set of dispositions in EX. There are \(2^{2^{2n}}\) = 16 propositions in EX^, as detailed in Table 14.

Table 14. Differential Propositions

	A :	1 1 0 0
	dA :	1 0 1 0
f0	g0	0 0 0 0	( )	False	0
	g1 g2 g4 g8	0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0	(A)(dA) (A) dA A (dA) A dA	Neither A nor dA Not A but dA A but not dA A and dA	¬A ∧ ¬dA ¬A ∧ dA A ∧ ¬dA A ∧ dA
f1 f2	g3 g12	0 0 1 1 1 1 0 0	(A) A	Not A A	¬A A
	g6 g9	0 1 1 0 1 0 0 1	(A, dA) ((A, dA))	A not equal to dA A equal to dA	A ≠ dA A = dA
	g5 g10	0 1 0 1 1 0 1 0	(dA) dA	Not dA dA	¬dA dA
	g7 g11 g13 g14	0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0	(A dA) (A (dA)) ((A) dA) ((A)(dA))	Not both A and dA Not A without dA Not dA without A A or dA	¬A ∨ ¬dA A → dA A ← dA A ∨ dA
f3	g15	1 1 1 1	(( ))	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^. 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 → B and the corresponding function g_j : EX → B, even though they share the same logical expression. Being human, we insist on preserving all the aesthetic delights afforded by the abstractly unified form of the "cake" while giving up none of the diverse contents that its substantive consummation can provide. In short, 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 X is a subset of another alphabet Y, then we say that any proposition f : 〈X〉 → B has a tacit extension to a proposition \(\epsilon\)f : 〈Y〉 → B, and that the space (〈X〉 → B) has an automatic embedding within the space (〈Y〉 → B). The extension is defined in such a way that \(\epsilon\)f puts the same constraint on the variables of X that are contained in Y as the proposition f initially did, while it puts no constraint on the variables of Y outside of X, in effect, conjoining the two constraints.

If the variables in question are indexed as X = {x₁, …, x_n} and Y = {x₁, …, x_n, …, x_n+k}, then the definition of the tacit extension from X to Y may be expressed in the form of an equation:

\[\epsilon\]f(x₁, …, x_n, …, x_n+k) = f(x₁, …, x_n).

On formal occasions, such as the present context of definition, the tacit extension from X to Y is explicitly symbolized by the operator \(\epsilon\) : (〈X〉 → B) → (〈Y〉 → B), where the appropriate alphabets X and Y are understood from context, but normally one may leave the "\(\epsilon\)" silent.

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

Table 15. Tacit Extension of [A] to [A, dA]

0 = 0 · ((dA), dA) = 0     (A) = (A) · ((dA), dA) = (A)(dA) + (A) dA      A = A · ((dA), dA) =  A (dA) +  A  dA      1 = 1 · ((dA), dA) = 1

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 (A), 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 X = {x₁} = {A}. In the discussion that follows I will consider a class of trajectories having the property that d^kA = 0 for all k greater than some fixed m, and I 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 d^mA 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 d^mA the drive at that point. Curves of constant drive d^mA are then referred to as "m^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 1840's prefigured the Turing machines of the 1940's [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 particular Example that I take up here can be described as the family of 4^th gear curves through E⁴X = 〈A, dA, d²A, d³A, d⁴A〉. These are the trajectories generated subject to the dynamic law d⁴A = 1, where it is understood in such a statement that all higher order differences are equal to 0. Since d⁴A and all higher d^kA are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E³X = 〈A, dA, d²A, d³A〉. 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 E³X. It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.

Figure 16. 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 E^mX 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 = s/t whose denominator is the power of two t = 2^m and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next. The differential coefficients of the state q are just the values d^kA(q), for k = 0 to m, where d⁰A is defined as being identical to A. To form the binary index d₀.d₁…d_m of the state q the coefficient d^kA(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:

\(r(q)\!\) \(=\) \(\sum_k d_k \cdot 2^{-k}\) \(=\) \(\sum_k \mbox{d}^k A(q) \cdot 2^{-k}\) \(=\) \(\frac{s(q)}{t}\) \(=\) \(\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}\) \(=\) \(\frac{\sum_k \mbox{d}^k A(q) \cdot 2^{(m-k)}}{2^m}\)

Applied to the example of fourth 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 = 2⁴ = 16. Within 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.

Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1

Time	State	A	dA
pi	qj	d0A	d1A	d2A	d3A	d4A
p0 p1 p2 p3 p4 p5 p6 p7	q01 q03 q05 q15 q17 q19 q21 q31	0. 0 0 0 1 0. 0 0 1 1 0. 0 1 0 1 0. 1 1 1 1 1. 0 0 0 1 1. 0 0 1 1 1. 0 1 0 1 1. 1 1 1 1

Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2

Time	State	A	dA
pi	qj	d0A	d1A	d2A	d3A	d4A
p0 p1 p2 p3 p4 p5 p6 p7	q25 q11 q29 q07 q09 q27 q13 q23	1. 1 0 0 1 0. 1 0 1 1 1. 1 1 0 1 0. 0 1 1 1 0. 1 0 0 1 1. 1 0 1 1 0. 1 1 0 1 1. 0 1 1 1

Work Area

Table 3. Differential Inference Rules

From \(\overline{q}\!\) and \(\overline{\operatorname{d}q}\!\) infer \(\overline{q}\!\) next.     From \(\overline{q}\!\) and \(\operatorname{d}q\!\) infer \(q\!\) next.     From \(q\!\) and \(\overline{\operatorname{d}q}\!\) infer \(q\!\) next.     From \(q\!\) and \(\operatorname{d}q\!\) infer \(\overline{q}\!\) next.

Table 3. Differential Inference Rules

From \((q)\!\) and \((\operatorname{d}q)\!\) infer \((q)\!\) next.     From \((q)\!\) and \(\operatorname{d}q\!\) infer \(q\!\) next.     From \(q\!\) and \((\operatorname{d}q)\!\) infer \(q\!\) next.     From \(q\!\) and \(\operatorname{d}q\!\) 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
\(\mathcal{A}\)	\(\{ a_1, \ldots, a_n \}\)	Alphabet	\([n] = \mathbf{n}\)
\(A_i\!\)	\(\{ \overline{a_i}, a_i \}\!\)	Dimension \(i\!\)	\(\mathbb{B}\)
\(A\!\)	\(\langle \mathcal{A} \rangle\) \(\langle a_1, \ldots, a_n \rangle\) \(\{ (a_1, \ldots, a_n) \}\!\) \(A_1 \times \ldots \times A_n\) \(\textstyle \prod_i A_i\!\)	Set of cells, coordinate tuples, points, or vectors in the universe of discourse	\(\mathbb{B}^n\)
\(A^*\!\)	\((\operatorname{hom} : A \to \mathbb{B})\)	Linear functions	\((\mathbb{B}^n)^* \cong \mathbb{B}^n\)
\(A^\uparrow\)	\((A \to \mathbb{B})\)	Boolean functions	\(\mathbb{B}^n \to \mathbb{B}\)
\(A^\circ\)	\([ \mathcal{A} ]\) \((A, A^\uparrow)\) \((A\ +\!\to \mathbb{B})\) \((A, (A \to \mathbb{B}))\) \([ a_1, \ldots, a_n ]\)	Universe of discourse based on the features \(\{ a_1, \ldots, a_n \}\)	\((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))\) \((\mathbb{B}^n\ +\!\to \mathbb{B})\) \([\mathbb{B}^n]\)

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
\(\operatorname{d}\mathcal{A}\)	\(\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}\)	Alphabet of differential features	\([n] = \mathbf{n}\)
\(\operatorname{d}A_i\)	\(\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}\)	Differential dimension \(i\!\)	\(\mathbb{D}\)
\(\operatorname{d}A\)	\(\langle \operatorname{d}\mathcal{A} \rangle\) \(\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle\) \(\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}\) \(\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n\) \(\textstyle \prod_i \operatorname{d}A_i\)	Tangent space at a point: Set of changes, motions, steps, tangent vectors at a point	\(\mathbb{D}^n\)
\(\operatorname{d}A^*\)	\((\operatorname{hom} : \operatorname{d}A \to \mathbb{B})\)	Linear functions on \(\operatorname{d}A\)	\((\mathbb{D}^n)^* \cong \mathbb{D}^n\)
\(\operatorname{d}A^\uparrow\)	\((\operatorname{d}A \to \mathbb{B})\)	Boolean functions on \(\operatorname{d}A\)	\(\mathbb{D}^n \to \mathbb{B}\)
\(\operatorname{d}A^\circ\)	\([\operatorname{d}\mathcal{A}]\) \((\operatorname{d}A, \operatorname{d}A^\uparrow)\) \((\operatorname{d}A\ +\!\to \mathbb{B})\) \((\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))\) \([\operatorname{d}a_1, \ldots, \operatorname{d}a_n]\)	Tangent universe at a point of \(A^\circ,\) based on the tangent features \(\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}\)	\((\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))\) \((\mathbb{D}^n\ +\!\to \mathbb{B})\) \([\mathbb{D}^n]\)