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===Commentary Note 12===
===Commentary Note 12===
Let us make a few preliminary observations about the "logical sign of involution", as Peirce uses it here:
Let us make a few preliminary observations about the
"logical sign of involution", as Peirce uses it here:
<blockquote>
<p>'''The Sign of Involution'''</p>
<p>I shall take involution in such a sense that ''x''<sup>''y''</sup> will denote everything which is an ''x'' for every individual of ''y''.</p>
<p>Thus</p>
: <p>'l'<sup>w</sup></p>
w = W_1 +, W_2 +, W_3.
<p>will be a lover of every woman.</p>
<p>(C.S. Peirce, CP 3.77).</p>
</blockquote>
In arithmetic, the "involution" ''x''<sup>''y''</sup>, or the "exponentiation" of ''x'' to the power of ''y'', is the iterated multiplication of the factor ''x'', repeated as many times as there are ones making up the exponent ''y''.
In analogous fashion, 'l'<sup>w</sup> is the iterated multiplication of 'l', repeated as many times as there are individuals under the term w.
For example, suppose that the universe of discourse has, among other things, just the three women, W<sub>1</sub>, W<sub>2</sub>, W<sub>3</sub>. This could be expressed in Peirce's notation by writing:
: w = W<sub>1</sub> +, W<sub>2</sub> +, W<sub>3</sub>.
In this setting, we would have:
In this setting, we would have:
'l'^w = 'l'^(W_1 +, W_2 +, W_3) = 'l'W_1 , 'l'W_2 , 'l'W_3.
: 'l'<sup>w</sup> = 'l'<sup>(W<sub>1</sub> +, W<sub>2</sub> +, W<sub>3</sub>)</sup> = 'l'W<sub>1</sub> , 'l'W<sub>2</sub> , 'l'W<sub>3</sub>.
That is, a lover of every woman in the universe of discourse
That is, a lover of every woman in the universe of discourse would be a lover of W<sub>1</sub> and a lover of W<sub>2</sub> and lover of W<sub>3</sub>.
would be a lover of W_1 and a lover of W_2 and lover of W_3.
</pre>
==References==
==References==