Peirce’s 1870 “Logic of Relatives” • Comment 12.4

Peirce’s 1870 “Logic of Relatives”Comment 12.4

Peirce next considers a pair of compound involutions, stating an equation between them analogous to a law of exponents from 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

(s^ℓ)^w = s^(ℓw)

(Peirce, CP 3.77)

Articulating the compound relative term \mathit{s}^{(\mathit{l}\mathrm{w})} in set-theoretic terms is fairly immediate.

Denotation Equation s^(ℓw)

On the other hand, translating the compound relative term (\mathit{s}^\mathit{l})^\mathrm{w} into its set-theoretic equivalent is less immediate, the hang-up being we have yet to define the case of logical involution raising one dyadic relative term to the power of another.  As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.

Involution Example 2

Consider a universe of discourse X subject to the following data.

\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}

Bigraph Involution S^L
\text{Figure 56. Bigraph Involution}~ \mathsf{S}^\mathsf{L}

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.

Matrix Computation S^L

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.

Resources

cc: CyberneticsOntolog ForumStructural ModelingSystems Science
cc: FB | Peirce MattersLaws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

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5 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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