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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 that are needed to describe various orders of differential propositional calculi.

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 that are needed to describe various orders of differential propositional calculi.

Logical description of a universe of discourse begins with a set of logical signs.  For the sake of simplicity in a first approach, assume that these form a finite alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \}.$  Each of these signs is interpreted as denoting either a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse.  Corresponding to the alphabet $\mathfrak{A}$ there is then a set of properties or propositions, $\mathcal{A} = \{ a_1, \ldots, a_n \}.$

Logical description of a universe of discourse begins with a set of logical signs.  For the sake of simplicity in a first approach, assume that these form a finite alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \}.$  Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse.  Corresponding to the alphabet $\mathfrak{A}$ there is then 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 $A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].$  It is useful to consider a universe of discourse as a unified categorical object that incorporates both the set of points $A = \langle a_1, \ldots, a_n \rangle$ and the set of propositions $A^\uparrow = \{ f : A \to \mathbb{B} \}$ that are implicit with the ordinary picture of a venn diagram on $n$ features.  Accordingly, the universe of discourse $A^\circ$ 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 $[ \mathbb{B}^n ].$

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