[CG] Re: Peirce's 1870 “Logic of Relatives”

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.4 https://inquiryintoinquiry.com/2014/06/14/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 12.4 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, 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}. Law of Exponents (a^b)^c = a^(bc) https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-a5eb5ec-a5e… <QUOTE CSP:> Then (s^ℓ)^w will denote whatever stands to every woman in the relation of servant of every lover of hers; and s^(ℓw) will denote whatever is a servant of everything that is lover of a woman. So that (s^ℓ)^w = s^(ℓw). https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-s5el5ew-s5e… (Peirce, CP 3.77) https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-s… </QUOTE> Articulating the compound relative term s^(ℓw) in set-theoretic terms is fairly immediate. Denotation Equation s^(ℓw) https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-denotation-… On the other hand, translating the compound relative term (s^ℓ)^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. • X = {a, b, c, d, e, f, g, h, i} • L = {b:a, b:c, c:b, c:d, e:d, e:e, e:f, g:f, g:h, h:g, h:i} • S = {b:a, b:c, d:c, d:d, d:e, f:e, f:f, f:g, h:g, h:i} Figure 56. Bigraph for the Involution S^L https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-bigraph-inv… There is a “servant of every lover of” link between u and v if and only if u∙S ⊇ L∙v. But the vacuous inclusions, that is, the cases where L∙v = ∅, have the effect of adding non‑intuitive links to the mix. The computational requirements are evidently met by the following formula. Matrix Computation for S^L https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-matrix-comp… In other words, (S^L)_xy = 0 if and only if there exists a p in X such that S_xp = 0 and L_py = 1. Regards, Jon