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Directory talk:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 15:04, 30 April 2009
, 15:04, 30 April 2009→Commentary Work Area
==Commentary Work Area==
==Commentary Work Area==
===Commentary Note 12.2===
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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<p>Then</p>
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| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math>
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<p>will denote whatever stands to every woman in the relation of servant of every lover of hers;</p>
<p>and</p>
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| align="center" | <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math>
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<p>will denote whatever is a servant of everything that is lover of a woman.</p>
<p>So that</p>
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| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.</math>
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<p>(Peirce, CP 3.77).</p>
|}
Proceeding as before, assume the following definitions:
{| align="center" cellspacing="6" width="90%"
| height="40" | <math>X\!</math> is the universe of discourse,
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| height="40" | <math>W \subseteq X</math> is the denotation of the absolute term <math>\mathrm{w} = \text{woman},\!</math>
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| height="40" | <math>L \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{l} = \text{lover of}\,\underline{~~~~},</math>
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| height="40" | <math>S \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{s} = \text{servant of}\,\underline{~~~~}.</math>
|}
Then we have the following results:
{| align="left" cellspacing="6" width="100%"
|- style="height:60px"
| width="5%" |
| width="10%" | <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math>
| width="5%" | <math>=\!</math>
| width="20%" | <math>\bigcap_{x \in LW} \operatorname{proj}_1 (S \star x)</math>
| width="60%" |
|- style="height:60px"
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| <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math>
| <math>=\!</math>
| <math>\bigcap_{x \in W} \operatorname{proj}_1 (S^L \star x)?</math>
|
|}
But what is <math>S^L?\!</math>
Suppose we try this:
{| align="center" cellspacing="6" width="90%"
| <math>S^L ~=~ \bigcap_{x \in \operatorname{proj}_1 L} \operatorname{proj}_1 (S \star x)</math>
|}
No, need to think about it some more …
===Commentary on Selection 12===
===Commentary on Selection 12===