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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 15:28, 4 May 2009
, 15:28, 4 May 2009→Selection 12: reorganize subsections
===Commentary Note 12.1===
===Commentary Note 12.1===
To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions:
To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions:
\end{array}</math>
\end{array}</math>
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===Commentary Note 12.2===
Let us make a few preliminary observations about the operation of ''logical involution'', as Peirce introduces it here:
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<p>I shall take involution in such a sense that <math>x^y\!</math> will denote everything which is an <math>x\!</math> for every individual of <math>y.\!</math></p>
<p>Thus</p>
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| align="center" | <math>\mathit{l}^\mathrm{w}\!</math>
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<p>will be a lover of every woman.</p>
<p>(Peirce, CP 3.77).</p>
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In ordinary arithmetic the ''involution'' <math>x^y,\!</math> or the ''exponentiation'' of <math>x\!</math> to the power of <math>y,\!</math> is the repeated application of the multiplier <math>x\!</math> for as many times as there are ones making up the exponent <math>y.\!</math>
In analogous fashion, the logical involution <math>\mathit{l}^\mathrm{w}\!</math> is the repeated application of the term <math>\mathit{l}\!</math> for as many times as there are individuals under the term <math>\mathrm{w}.\!</math> According to Peirce's interpretive rules, the repeated applications of the base term <math>\mathit{l}\!</math> are distributed across the individuals of the exponent term <math>\mathrm{w}.\!</math> In particular, the base term <math>\mathit{l}\!</math> is not applied successively in the manner that would give something like "a lover of a lover of … a lover of a woman".
For example, suppose that a universe of discourse numbers among its contents just three women, <math>\mathrm{W}^{\prime}, \mathrm{W}^{\prime\prime}, \mathrm{W}^{\prime\prime\prime}.</math> This could be expressed in Peirce's notation by writing:
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| <math>\mathrm{w} ~=~ \mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime}</math>
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Under these circumstances the following equation would hold:
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| <math>\mathit{l}^\mathrm{w} ~=~ \mathit{l}^{(\mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime})} ~=~ (\mathit{l}\mathrm{W}^{\prime}), (\mathit{l}\mathrm{W}^{\prime\prime}), (\mathit{l}\mathrm{W}^{\prime\prime\prime}).</math>
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This says that a lover of every woman in the given universe of discourse is a lover of <math>\mathrm{W}^{\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime\prime}.</math> In other words, a lover of every woman in this context is a lover of <math>\mathrm{W}^{\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime\prime}.</math>
The denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> is a subset of <math>X\!</math> that can be obtained as follows: For each flag of the form <math>L \star x</math> with <math>x \in W,</math> collect the elements <math>\operatorname{proj}_1 (L \star x)</math> that appear as the first components of these ordered pairs, and then take the intersection of all these subsets. Putting it all together:
The denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> is a subset of <math>X\!</math> that can be obtained as follows: For each flag of the form <math>L \star x</math> with <math>x \in W,</math> collect the elements <math>\operatorname{proj}_1 (L \star x)</math> that appear as the first components of these ordered pairs, and then take the intersection of all these subsets. Putting it all together:
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===Commentary Note 12.2===
===Commentary Note 12.3===
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->