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]