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=====1.3.10.5.  Articulate Sentences=====

=====1.3.10.5.  Articulate Sentences=====

A sentence is ''articulate'' (1) if it has a significant form, a compound constitution, or a non-trivial structure as a sign, and (2) if there is an informative relationship that exists between its structure as a sign and the proposition that it happens to denote.  A sentence of this kind is typically given in the form of a ''description'', an ''expression'', or a ''formula'', in other words, as an articulated sign or a well-structured element of a formal language.  As a general rule, the class of sentences that one is willing to contemplate is compiled from a particular brand of complex signs and syntactic strings, those that are put together from the basic building blocks of a formal language and held in a special esteem for the roles that they play within its grammar.  However, even if a typical sentence is a sign that is generated by a formal regimen, having its form, its meaning, and its use governed by the principles of a comprehensive grammar, the class of sentences that one has a mind to contemplate can also include among its number many other signs of an arbitrary nature.

A sentence is "articulate" (1) if it has a significant form, a compound constitution, or a non�trivial structure as a sign, and (2) if there is an informative relationship that exists between its structure as a sign and the proposition that it happens to denote.  A sentence of this kind is typically given in the form of a "description", an "expression", or a "formula", in other words, as an articulated sign or a well�structured element of a formal language.  As a general rule, the class of sentences that one is willing to contemplate is compiled from a particular brand of complex signs and syntactic strings, those that are put together from the basic building blocks of a formal language and held in a special esteem for the roles that they play within its grammar.  However, even if a typical sentence is a sign that is generated by a formal regimen, having its form, its meaning, and its use governed by the principles of a comprehensive grammar, the class of sentences that one has a mind to contemplate can also include among its number many other signs of an arbitrary nature.

Frequently this "formula" has a "variable" in it that "ranges over" the universe U.  A "variable" is an ambiguous or equivocal sign that can be interpreted as denoting any element of the set that it "ranges over".

Frequently this formula has a ''variable'' in it that ''ranges over'' the universe <math>X.\!</math> A variable is an ambiguous or equivocal sign that can be interpreted as denoting any element of the set that it ranges over.

If a sentence denotes a proposition f : U �> B, then the "value" of the sentence with regard to u C U is the value f(u) of the proposition at u, where "0" is interpreted as "false" and "1" is interpreted as "true".

If a sentence denotes a proposition <math>f : X \to \underline\mathbb{B},</math> then the ''value'' of the sentence with regard to <math>x \in X</math> is the value <math>f(x)\!</math> of the proposition at <math>x,\!</math> where <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> is interpreted as ''false'' and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> is interpreted as ''true''.

Since the value of a sentence or a proposition depends on the universe of discourse to which it is "referred", and since it also depends on the element of the universe with regard to which it is evaluated, it is usual to say that a sentence or a proposition "refers" to a universe and to its elements, though perhaps in a variety of different senses.  Furthermore, a proposition, acting in the role of as an indicator function, "refers" to the elements that it "indicates", namely, the elements on which it takes a positive value.  In order to sort out the possible confusions that are capable of arising here, I need to examine how these various notions of reference are related to the notion of denotation that is used in the pragmatic theory of sign relations.

Since the value of a sentence or a proposition depends on the universe of discourse to which it is "referred", and since it also depends on the element of the universe with regard to which it is evaluated, it is usual to say that a sentence or a proposition "refers" to a universe and to its elements, though perhaps in a variety of different senses.  Furthermore, a proposition, acting in the role of as an indicator function, "refers" to the elements that it "indicates", namely, the elements on which it takes a positive value.  In order to sort out the possible confusions that are capable of arising here, I need to examine how these various notions of reference are related to the notion of denotation that is used in the pragmatic theory of sign relations.