## Differential Propositional Calculus • 3

### Formal Development

The preceding discussion outlined the ideas leading to the differential extension of propositional logic.  The next task is to lay out the concepts and terminology needed to describe various orders of differential propositional calculi.

#### Elementary Notions

Logical description of a universe of discourse begins with a collection of logical signs.  For simplicity in a first approach, we may assume these logical signs are collected in the form of a finite alphabet, $\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}.$  Each of these signs is interpreted as denoting a logical feature, for example, a property that objects of the universe of discourse may have or a proposition about objects in the universe of discourse.  There is then corresponding to the alphabet $\mathfrak{A}$ a set of logical features, $\mathcal{A} = \{ a_1, \ldots, a_n \}.$

A set of logical features, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ affords a basis for generating an $n$-dimensional universe of discourse, written $A^\bullet = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].$  It is useful to consider a universe of discourse as a categorical object incorporating both the set of points $A = \langle a_1, \ldots, a_n \rangle$ and the set of propositions $A^\uparrow = \{ f : A \to \mathbb{B} \}$ implicit with the ordinary picture of a venn diagram on $n$ features.  Accordingly, the universe of discourse $A^\bullet$ may be regarded as an ordered pair $(A, A^\uparrow)$ having the type $(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),$ and this last type designation may be abbreviated as $\mathbb{B}^n\ +\!\!\to \mathbb{B},$ or even more succinctly as $[ \mathbb{B}^n ].$  For convenience, the data type of a finite set on $n$ elements may be indicated by either one of the equivalent notations, $[n]$ or $\mathbf{n}.$

Table 7 summarizes the notations needed to describe ordinary propositional calculi in a systematic fashion.

Table 7.  Propositional Calculus : Basic Notation

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