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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
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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 two 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 two different ways to execute the picture.
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, here interpreted as the logical value <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.)
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, here interpreted as the logical value <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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{| align="center" cellpadding="10" style="text-align:center; width:90%"
While we go through each of these ways let us keep one eye out for the character and the conduct of each type of proceeding as a semiotic process, that is, as an orbit, in this case discrete, through a locus of signs, in this case propositional expressions, and as it happens in this case, a sequence of transformations that perseveres in the denotative objective of each expression, that is, in the abstract proposition that it expresses, while it preserves the informed constraint on the universe of discourse that gives us one viable candidate for the informational content of each expression in the interpretive chain of sign metamorphoses.
While we go through each of these ways let us keep one eye out for the character and the conduct of each type of proceeding as a semiotic process, that is, as an orbit, in this case discrete, through a locus of signs, in this case propositional expressions, and as it happens in this case, a sequence of transformations that perseveres in the denotative objective of each expression, that is, in the abstract proposition that it expresses, while it preserves the informed constraint on the universe of discourse that gives us one viable candidate for the informational content of each expression in the interpretive chain of sign metamorphoses.
A ''sign relation'' ''L'' is a subset of a cartesian product ''O'' × ''S'' × ''I'', where ''O'', ''S'', ''I'' are sets known as the ''object'', ''sign'', and ''interpretant sign'' domains, respectively. One writes ''L'' ⊆ ''O'' × ''S'' × ''I'', where the symbol "⊆"
A ''sign relation'' <math>L\!</math> is a subset of a cartesian product <math>O \times S \times I,</math> where <math>O, S, I\!</math> are sets known as the ''object'', ''sign'', and ''interpretant sign'' domains, respectively. One writes <math>L \subseteq O \times S \times I.</math> Accordingly, a sign relation <math>L\!</math> consists of ordered triples of the form <math>(o, s, i),\!</math> where <math>o, s, i\!</math> belong to the domains <math>O, S, I,\!</math> respectively.
The ''syntactic domain'' of a sign relation ''L'' ⊆ ''O'' × ''S'' × ''I'' is just the set-theoretic union ''S'' ∪ ''I'' of its sign domain ''S'' and its interpretant domain ''I''.
The ''syntactic domain'' of a sign relation <math>L \subseteq O \times S \times I</math> is defined as the set-theoretic union <math>S \cup I</math> of its sign domain <math>S\!</math> and its interpretant domain <math>I.\!</math>
It is not uncommon, especially in formal examples, for the sign domain and the interpretant domain to be equal as sets, in short, to have ''S'' = ''I''.
It is not uncommon, especially in formal examples, for the sign domain and the interpretant domain to be equal as sets, in short, to have <math>S = I.\!</math>
Elsewhere I have discussed examples of sign relations that consist of a finite set of triples of the form (''o'', ''s'', ''i''), where ''o'', ''s'', ''i'' are the ''object'', ''sign'', ''interpretant sign'', respectively, of what is called the ''sign triple'' or the ''elementary sign relation'' (''o'', ''s'', ''i'').
Elsewhere I have discussed examples of sign relations that consist of a finite set of triples of the form (''o'', ''s'', ''i''), where ''o'', ''s'', ''i'' are the ''object'', ''sign'', ''interpretant sign'', respectively, of what is called the ''sign triple'' or the ''elementary sign relation'' (''o'', ''s'', ''i'').