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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
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This new quality, <math>\operatorname{d}q,\!</math> 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 <math>X\!</math> outside and inside the region <math>\operatorname{d}Q.\!</math>
This new quality, <math>\operatorname{d}q,\!</math> 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 <math>X\!</math> outside and inside the region <math>\operatorname{d}Q.\!</math>
Figure 1 represents a universe of discourse, <math>X,\!</math> together with a basis of discussion, <math>\{ q \},\!</math> for expressing propositions about the contents of that universe. Once the quality <math>q\!</math> is given a name, say, the symbol "<math>q\!</math>", we have the basis for a formal language that is specifically cut out for discussing <math>X\!</math> in terms of <math>q,\!</math> and this formal language is more formally known as the ''propositional calculus'' with alphabet <math>\{\!</math>"<math>q\!</math>"<math>\}.\!</math>
Within the pale of <math>X\!</math> and <math>\{ q \}\!</math> there are but four different pieces of information that can be given expression in the corresponding propositional calculus, namely, the propositions: <math>\operatorname{false},\!</math> <math>\lnot q,\!</math> <math>q,\!</math> <math>\operatorname{true}.\!</math>
Figure 1′ maintains the same universe of discourse and extends the basis of discussion to a set of two qualities, <math>\{ q, \operatorname{d}q \}.\!</math> In corresponding fashion the initial propositional calculus is extended in the medium of the new alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math> "<math>\operatorname{d}q\!</math>"<math>\}.\!</math>
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