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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 11:59, 6 May 2009
, 11:59, 6 May 2009→Selection 12: reorganize subsections
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====Example 6====
===Commentary Note 12.3===
We now have two ways of computing a logical involution that raises a 2-adic relative term to the power of a 1-adic absolute term, for example, <math>\mathit{l}^\mathrm{w}\!</math> for "lover of every woman".
We now have two ways of computing a logical involution that raises a 2-adic relative term to the power of a 1-adic absolute term, for example, <math>\mathit{l}^\mathrm{w}\!</math> for "lover of every woman".
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Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved. The Figure below represents a universe of discourse <math>X\!</math> that is subject to the following data:
Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved.
====Example 6====
The Figure below represents a universe of discourse <math>X\!</math> that is subject to the following data:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
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===Commentary Note 12.3===
===Commentary Note 12.4===
Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to a law of exponents in ordinary arithmetic, namely, the law that states <math>(a^b)^c = a^{bc}.\!</math>
Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to a law of exponents in ordinary arithmetic, namely, the law that states <math>(a^b)^c = a^{bc}.\!</math>
On the other hand, translating the compound relative term <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math> into a set-theoretic equivalent is less immediate, the hang-up being that we have yet to define the case of logical involution that raises a 2-adic relative term to the power of a 2-adic relative term. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.
On the other hand, translating the compound relative term <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math> into a set-theoretic equivalent is less immediate, the hang-up being that we have yet to define the case of logical involution that raises a 2-adic relative term to the power of a 2-adic relative term. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.
====Example 7====
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"