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 $\operatorname{true}$ to the region where $q$ is $\operatorname{false}$ while $k$ has moved from the region where $q$ is $\operatorname{false}$ to the region where $q$ is $\operatorname{true}.$

Figure $1^\prime$ reprises the situation shown in Figure 1, but configures a new quality designed to explain the sequel to come in 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 $\{ ``q" \}.$

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

Figure $1^\prime$ preserves the same universe of discourse and extends the basis of discussion up 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 . . . . . . . . . . . . . . . |
| . . h . . ./. . . . . . . .\. . . . . . . . . . . . . . . |
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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 \(\operatorname{true}\!\) to the region where \(q\!\) is \(\operatorname{false}\!\) while \(k\!\) has moved from the region where \(q\!\) is \(\operatorname{false}\!\) to the region where \(q\!\) is \(\operatorname{true}.\!\)

Figure 1′ reprises the situation shown in Figure 1, but configures a new quality designed to explain the sequel to come in Figure 2.

o-----------------------------------------------------------o
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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| . . . . . . o-------------o . o-------------o . . . . . . |
| . . h . . ./. . . . . . . .\./. . . . . . . .\. . . . . . |
| . . @ . . / . . . . . . . . o . . . . . . . . \ . . . . . |
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| . . . | . . . . . Q . . | . . . | . . dQ. . . . . | . . . |
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| . . . . . . o-------------o . o-------------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 \(\{\!\)"\(q\!\)"\(\}.\!\)

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

…