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}

LOR 1870 Figure 56

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.

This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

3 Responses to Peirce’s 1870 “Logic Of Relatives” • Comment 12.4

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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