## Differential Propositional Calculus • 1

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

### Casual Introduction

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

Figure 1.  Local Habitations, And Names

The area of the rectangle represents a universe of discourse, $X.$  The universe under discussion may be a population of individuals having various additional properties or it may be a collection of locations occupied by various individuals.  The area of the “circle” represents the individuals having the property $q$ or the locations in the corresponding region $Q.$  Four individuals, $a, b, c, d,$ are singled out by name.  It happens that $b$ and $c$ currently reside in region $Q$ while $a$ and $d$ do not.

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

Figure 2.  Same Names, Different Habitations

Figure 2 differs from Figure 1 solely in the circumstance that the object $c$ is outside the region $Q$ while the object $d$ is inside the region $Q.$  So far, nothing says our encountering these Figures in this order is other than purely accidental but if we interpret this sequence of frames as a “moving picture” representation of their natural order in a temporal process then it would be natural to suppose $a$ and $b$ have remained as they were with regard to the quality $q$ while $c$ and $d$ have changed their standings in that respect.  In particular, $c$ has moved from the region where $q$ is true to the region where $q$ is false while $d$ has moved from the region where $q$ is false to the region where $q$ is true.

Figure 3 returns to the situation in Figure 1, but this time interpolates a new quality specifically tailored to account for the relation between Figure 1 and Figure 2.

Figure 3.  Back, To The Future

This new quality, $\mathrm{d}q,$ 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” distinguishing two halves of the universe of discourse, in this case, the portions of $X$ outside and inside the region $\mathrm{d}Q.$

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 ${}^{\backprime\backprime} q {}^{\prime\prime},$ we have the basis for a formal language specifically cut out for discussing $X$ in terms of $q.$  This language is more formally known as the propositional calculus with alphabet $\{ {}^{\backprime\backprime} q {}^{\prime\prime} \}.$

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:  $\mathrm{false}, ~ \lnot q, ~ q, ~ \mathrm{true}.$  Referring to the sample of points in Figure 1, the constant proposition $\mathrm{false}$ holds of no points, the proposition $\lnot q$ holds of $a$ and $d,$ the proposition $q$ holds of $b$ and $c,$ and the constant proposition $\mathrm{true}$ holds of all points in the sample.

Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, $\{ q, \mathrm{d}q \}.$  In corresponding fashion, the initial propositional calculus is extended by means of the enlarged alphabet, $\{ {}^{\backprime\backprime} q {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{d}q {}^{\prime\prime} \}.$  Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together.  Just by way of salient examples, we can pick out the most informative proposition applying to each sample point at the initial moment of observation.  Table 4 shows these initial state descriptions, using overlines to express logical negations.

Table 4.  Initial State Descriptions

Table 5 shows the rules of inference responsible for giving the differential quality $\mathrm{d}q$ its meaning in practice.

Table 5.  Differential Inference Rules

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