Changes

There are a number of issues, that arise especially in establishing the proper use of STRs, that are appropriate to discuss at this juncture.  The notation <math>\downharpoonleft s \downharpoonright</math> is intended to represent ''the proposition denoted by the sentence <math>s.\!</math>''  There is only one problem with the use of this form.  There is, in general, no such thing as "the" proposition denoted by <math>s.\!</math>  Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote.  But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind.  Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form.  This requires a closer examination of the very notion of ''context'', and it involves engaging in enough reflection on the ''contextual evaluation'' of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms.

There are a number of issues, that arise especially in establishing the proper use of STRs, that are appropriate to discuss at this juncture.  The notation <math>\downharpoonleft s \downharpoonright</math> is intended to represent ''the proposition denoted by the sentence <math>s.\!</math>''  There is only one problem with the use of this form.  There is, in general, no such thing as "the" proposition denoted by <math>s.\!</math>  Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote.  But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind.  Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form.  This requires a closer examination of the very notion of ''context'', and it involves engaging in enough reflection on the ''contextual evaluation'' of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms.

A sentence that is written in a context where it represents a value of <math>\underline{1}</math> or <math>\underline{0}</math> as a function of things in the universe <math>X,\!</math> where it stands for a value of <math>\operatorname{truth}</math> or <math>\operatorname{falsehood},</math> depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in <math>X,\!</math> in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in <math>X,\!</math> is a sentence that might as well be written in the context <math>\downharpoonleft \ldots \downharpoonright,</math> whether this frame is explicitly marked around it or not.

A sentence that is written in a context where it represents a value of 1 or 0 as a function of things in the universe U, where it stands for a value of "true" or "false", depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in U, in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in U, is a sentence that might as well be written in the context "[ ... ]", whether or not this frame is explicitly marked around it.

More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations.  Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.

More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations.  Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.

[u C X]  = Lambda (u, C, X).(u C X).

{| align="center" cellpadding="8" width="90%"

| <math>\downharpoonleft x \in Q \downharpoonright ~=~ \lambda (x, \in, Q).(x \in Q).</math>

As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms.  But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged.  Consider the sense of equivalence among sentences that is recorded in Rule 4.

As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms.  But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged.  Consider the sense of equivalence among sentences that is recorded in Rule 4.