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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

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Suppose we discover or begin to suspect that something we had been treating as a sumple quality, <math>q,\!</math> is actually compounded of other qualities, <math>u\!</math> and <math>v,\!</math> according to a propositional formula <math>q = q(u, v).\!</math>

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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.

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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, <math>\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace.\!</math> 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 <math>\mathfrak{A}</math> there is then a set of properties or propositions, <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.</math>

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Table 4 summarizes the basic notations that are needed to describe ordinary propositional calculi in a parametric fashion.

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Jon Awbrey

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