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===Commentary Note 12===

===Commentary Note 12===

Let us make a few preliminary observations about the operation of ''logical involution", as Peirce introduces it here:

Let us make a few preliminary observations about the operation of ''logical involution'', as Peirce introduces it here:

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In arithmetic, the "involution" ''x''^''y'', 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 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, 'l'^w is the iterated multiplication of 'l', repeated as many times as there are individuals under the term w.

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>

For example, suppose that the universe of discourse has, among other things, just the three women, W₁, W₂, W₃. This could be expressed in Peirce's notation by writing:

For example, suppose that the universe of discourse has, among other things, just the three women, <math>\mathrm{W}_1, \mathrm{W}_2, \mathrm{W}_3.\!</math>  This could be expressed in Peirce's notation by writing:

: w = W₁ +, W₂ +, W₃.

| <math>\mathrm{w} ~=~ \mathrm{W}_1 ~+\!\!,~ \mathrm{W}_2 ~+\!\!,~ \mathrm{W}_3</math>

In this setting, we would have:

In this setting, we would have: