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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 02:42, 15 May 2008
, 02:42, 15 May 2008→Casual introduction: del dupe lines
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.
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′ preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, <math>\{ q, \operatorname{d}q \}.\!</math> In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math> "<math>\operatorname{d}q\!</math>"<math>\}.\!</math> Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows:
Figure 1′ preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, <math>\{ q, \operatorname{d}q \}.\!</math> In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math> "<math>\operatorname{d}q\!</math>"<math>\}.\!</math> Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows: