Changes

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'' <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.

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.  These facts are symbolized by writing <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.  An ordered triple of the form <math>(o, s, i) \in L</math> is referred to as a ''sign triple'' or an ''elementary sign relation''.

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>

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 <math>S = 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 <math>S = I.\!</math>

Sign relations may contain any number of sign triples, finite or infinite.  Finite sign relations do arise in applications and can be very instructive as expository examples, but most of the sign relations of significance in logic have infinite sign and interpretant domains, and usually infinite object domains, in the long run, at least, though one frequently works up to infinite domains by a series of finite approximations and gradual stages.

With that preamble behind us, let us turn to consider the case of semiosis, or sign transformation process, that is generated by our first proof of the propositional equation <math>E_1.\!</math>

 

 

With that preamble behind us, let us turn to consider the case of semiosis, or sign transformation process, that is generated by our first proof of the propositional equation ''E''₁.

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