Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus

Current Version @ PlanetMath : TeX Format

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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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\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}
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\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}
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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}

$\ldots$

\section{Formal development}

$\ldots$

\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 . . . . . . . . \ . . . . . |
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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 \(j\!\)

…

Formal development

…

Expository examples

…

Materials from "Dif Log Dyn Sys" for Reuse Here

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.