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

Cf: Peirce's 1870 “Logic of Relatives” • Comment 12.2 (part 2 of 3) https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-c… Peirce's 1870 “Logic of Relatives” • Comment 12.2 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, It is instructive to examine the matrix representation of ℓ^w at this point, not the least because it effectively dispels the mystery of the name “involution”. First, we make the following observation. To say j is a lover of every woman is to say j loves k if k is a woman. This can be rendered in symbols as follows. • j loves k ⇐ k is a woman. Reading the formula ℓ^w as “j loves k if k is a woman” highlights the operation of converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name. The operations defined by the formulas x^y = z and (x ⇐ y) = z for x, y, z in the boolean domain B = {0, 1} are tabulated as follows. Table. Involution ≅ Implication https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-involution-… It is clear the two operations are isomorphic, being effectively the same operation of type B × B → B. All that remains is to see how operations like these on values in B induce the corresponding operations on sets and terms. To be continued ... Regards, Jon