Changes

For example, the annnotation <math>X_1 : A_1 :: X_2 : A_2\!</math> may be read to say that <math>X_1\!</math> is to <math>A_1\!</math> as <math>X_2\!</math> is to <math>A_2,\!</math> where the step from <math>A_1\!</math> to <math>A_2\!</math> is permitted by a previously accepted rule.

For example, the annnotation <math>X_1 : A_1 :: X_2 : A_2\!</math> may be read to say that <math>X_1\!</math> is to <math>A_1\!</math> as <math>X_2\!</math> is to <math>A_2,\!</math> where the step from <math>A_1\!</math> to <math>A_2\!</math> is permitted by a previously accepted rule.

This can be illustrated by considering the derivation of Rule&nbsp;3 in the following augmented form:

This can be illustrated by considering the derivation of Rule&nbsp;3 in the augmented form that follows:

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

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

For example, extracting the expressions <math>\text{R3a}\!</math> and <math>\text{R3c}\!</math> that given as equivalents in Rule&nbsp;3 and explictly stating their equivalence produces the equation recorded in Corollary&nbsp;1.

For example, extracting the expressions <math>\text{R3a}\!</math> and <math>\text{R3c}\!</math> that given as equivalents in Rule&nbsp;3 and explictly stating their equivalence produces the equation recorded in Corollary&nbsp;1.

<pre>

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If X c U

| style="border-top:1px solid black" | <math>\text{If}\!</math>

| style="border-top:1px solid black" | <math>Q \subseteq X</math>

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C1a. u C X  <=> {X}(u) = 1. R3a=R3c

There are a number of issues, that arise especially in establishing the proper use of STR's, that are appropriate to discuss at this juncture.  The notation "[S]" is intended to represent "the proposition denoted by the sentence S".  There is only one problem with the use of this form.  There is, in general, no such thing as "the" proposition denoted by S.  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 STR's, that are appropriate to discuss at this juncture.  The notation "[S]" is intended to represent "the proposition denoted by the sentence S".  There is only one problem with the use of this form.  There is, in general, no such thing as "the" proposition denoted by S.  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.