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

Cf: Peirce's 1870 “Logic of Relatives” • Comment 12.2 [part 1 of 2] 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, Let us make a few preliminary observations about the operation of “logical involution” which Peirce introduces in the following words. <QUOTE CSP:> I shall take involution in such a sense that x^y will denote everything which is an x for every individual of y. Thus ℓ^w will be a lover of every woman. (Peirce, CP 3.77) https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-s… </QUOTE> In ordinary arithmetic the involution x^y, or the exponentiation of x to the power y, is the repeated application of the multiplier x for as many times as there are ones making up the exponent y. In analogous fashion, the logical involution ℓ^w is the repeated application of the term ℓ for as many times as there are individuals under the term w. On Peirce’s interpretive rules, the repeated applications of the base term ℓ are distributed across the individuals of the exponent term w. In particular, the base term ℓ is not applied successively in the manner that would give something on the order of “a lover of a lover of ... a lover of a woman”. By way of example, suppose a universe of discourse numbers among its elements just three women, W′, W″, W‴. In Peirce's notation the fact may be written as follows. • w = W′ +, W″ +, W‴ In that case the following equation would hold. • ℓ^w = ℓ^(W′ +, W″ +, W‴) = (ℓW′),(ℓW″),(ℓW‴) The equation says a lover of every woman in the aggregate W′ +, W″ +, W‴ is a lover of W′ that is a lover of W″ that is a lover of W‴. In other words, a lover of every woman in the universe at hand is a lover of W′ and a lover of W″ and a lover of W‴. The denotation of the term ℓ^w is a subset of X which may be obtained by the following procedure. For each flag of the form L ∗ x with x in W collect the subset proj₁(L ∗ x) of elements which appear as the first components of the pairs in L ∗ x and then take the intersection of all those subsets. Putting it all together, we have the following equation. Denotation Equation ℓ^w https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-denotation-… Regards, Jon