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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 21:18, 14 May 2008
, 21:18, 14 May 2008→Casual introduction: add text
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>
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>
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> Referring to the sample of points in Figure 1, <math>\operatorname{false}\!</math> holds of no points, <math>\lnot q\!</math> holds of <math>h\!</math> and <math>k,\!</math> <math>q\!</math> holds of <math>i\!</math> and <math>j,\!</math> and <math>\operatorname{true}\!</math> holds of all points in the sample.
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>
Figure 1′ maintains the same universe of discourse and extends the basis of discussion up to a set of two qualities, <math>\{ q, \operatorname{d}q \}.\!</math> In parallel fashion the initial propositional calculus is extended in the medium of a larger alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math> "<math>\operatorname{d}q\!</math>"<math>\}.\!</math>
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