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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 20:00, 4 May 2009
, 20:00, 4 May 2009→Commentary Note 12.3
===Commentary Note 12.3===
===Commentary Note 12.3===
Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to one of the usual laws of exponents in ordinary arithmetic, namely, <math>(a^b)^c = a^{bc}.\!</math>
Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to one of the laws of exponents in ordinary arithmetic, namely, the law that states <math>(a^b)^c = a^{bc}.\!</math>
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Articulating the compound relative term <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math> in set-theoretic terms is fairly immediate:
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| <math>\mathit{s}^{(\mathit{l}\mathrm{w})} ~=~ \bigcap_{x \in LW} \operatorname{proj}_1 (S \star x) ~=~ \bigcap_{x \in LW} (S \cdot x)</math>
| <math>\mathit{s}^{(\mathit{l}\mathrm{w})} ~=~ \bigcap_{x \in LW} \operatorname{proj}_1 (S \star x) ~=~ \bigcap_{x \in LW} (S \cdot x)</math>
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But doing the same thing for the compound relative term <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math> 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 teram.
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