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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 19:12, 11 August 2009
, 19:12, 11 August 2009→Analysis of contingent propositions: TeX formats
Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact a couple of different ways to execute the picture.
Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact a couple of different ways to execute the picture.
Figure 1 indicates the points of the universe of discourse ''X'' for which the proposition ''f'' : ''X'' → '''B''' has the value 1 (= true). In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe ''X'', going according to some previously adopted convention, for instance: Let the cells that get the value 0 under ''f'' remain untinted, and let the cells that get the value 1 under ''f'' be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the 0, 1 in '''B''', but in the pattern of regions that they indicate. NB. In this Ascii version, I use [ ] for 0 and [ ` ` ` ] for 1.
Figure 1 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1 (<math>= \operatorname{true}</math>). In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance: Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate. (Note. In this Ascii version, I use [ ] for 0 and [ ` ` ` ] for 1.)
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