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Revision as of 16:04, 18 January 2009
, 16:04, 18 January 2009→1.3.10.7. Stretching Operations: mathematical markup mystery tour
Roughly sketched, the relations of denotation and indication that exist among sets, propositions, sentences, and values can be diagrammed as in Table 11.
Roughly sketched, the relations of denotation and indication that exist among sets, propositions, sentences, and values can be diagrammed as in Table 11.
<pre>
<br>
Table 11. Levels of Indication
Object
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
Sign
|+ '''Table 11. Levels of Indication'''
Higher Order Sign
|- style="background:whitesmoke"
Set
| width="33%" | <math>\operatorname{Object}</math>
Proposition
| width="33%" | <math>\operatorname{Sign}</math>
Sentence
| width="34%" | <math>\operatorname{Higher~Order~Sign}</math>
f-1(v)
|- style="background:whitesmoke"
f
| width="33%" | <math>\operatorname{Set}</math>
| width="33%" | <math>\operatorname{Proposition}</math>
| width="34%" | <math>\operatorname{Sentence}</math>
1
|-
| <math>f^{-1} (y)\!</math>
| <math>f\!</math>
0
| <math>^{\backprime\backprime} f \, ^{\prime\prime}</math>
|-
| <math>Q\!</math>
| <math>\underline{1}</math>
| <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math>
|-
| <math>{}^{_\sim} Q</math>
| <math>\underline{0}</math>
| <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math>
|}
<br>
<pre>
Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 11 have to be taken with the indicated grains of salt. Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning. Therefore, it needs to be understood that a proposition f can be said to "indicate" a set X only insofar as the values of 1 and 0 that it assigns to the elements of the universe U are positive and negative indications, respectively, of the elements in X, and thus indications of the set X and of its complement ~X = U - X, respectively. It is actually these values, when rendered by a concrete implementation of the indicator function f, that are the actual signs of the objects that are inside the set X and the objects that are outside the set X, respectively.
Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 11 have to be taken with the indicated grains of salt. Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning. Therefore, it needs to be understood that a proposition f can be said to "indicate" a set X only insofar as the values of 1 and 0 that it assigns to the elements of the universe U are positive and negative indications, respectively, of the elements in X, and thus indications of the set X and of its complement ~X = U - X, respectively. It is actually these values, when rendered by a concrete implementation of the indicator function f, that are the actual signs of the objects that are inside the set X and the objects that are outside the set X, respectively.