## Peirce’s 1870 “Logic Of Relatives” • Comment 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, $(a^b)^c = a^{bc}.$

Then $(\mathit{s}^\mathit{l})^\mathrm{w}$ will denote whatever stands to every woman in the relation of servant of every lover of hers;  and $\mathit{s}^{(\mathit{l}\mathrm{w})}$ will denote whatever is a servant of everything that is lover of a woman.  So that $(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.$

(Peirce, CP 3.77)

Articulating the compound relative term $\mathit{s}^{(\mathit{l}\mathrm{w})}$ in set-theoretic terms is fairly immediate: $\displaystyle \mathit{s}^{(\mathit{l}\mathrm{w})} ~=~ \bigcap_{x \in LW} \mathrm{proj}_1 (S \star x) ~=~ \bigcap_{x \in LW} S \cdot x$

On the other hand, translating the compound relative term $(\mathit{s}^\mathit{l})^\mathrm{w}$ 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 dyadic relative term to the power of a dyadic 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 $\begin{array}{*{15}{c}} X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i\ & \} \\[6pt] L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} \\[6pt] S & = & \{ & b\!:\!a, & b\!:\!c, & d\!:\!c, & d\!:\!d, & d\!:\!e, & f\!:\!e, & f\!:\!f, & f\!:\!g, & h\!:\!g, & h\!:\!i\ & \} \end{array}$

There is a “servant of every lover of” link between $u$ and $v$ if and only if $u \cdot S ~\supseteq~ L \cdot v.$  But the vacuous inclusions, that is, the cases where $L \cdot v = \varnothing,$ have the effect of adding non-intuitive links to the mix.

The computational requirements are evidently met by the following formula: $\displaystyle (\mathsf{S}^\mathsf{L})_{xy} ~=~ \prod_{p \in X} \mathsf{S}_{xp}^{\mathsf{L}_{py}}$

In other words, $(\mathsf{S}^\mathsf{L})_{xy} = 0$ if and only if there exists a $p \in X$ such that $\mathsf{S}_{xp} = 0$ and $\mathsf{L}_{py} = 1.$

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