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

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.3 (part 2 of 2) https://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 12.3 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, Last time we looked at two ways of computing a logical involution which raises a dyadic relative term to the power of a monadic absolute term, for example, ℓ^w for “lover of every woman”. The first method computes the denotation of the term ℓ^w by intersecting the family of sets produced by applying ℓ to the elements of w, as given by the following equation. Equation 1. Denotation Equation for L^W https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-denotation-… The second method operates in the matrix representation, raising the matrix for ℓ to the power of the matrix for w, as given by the following equation. Equation 2. Matrix Computation for L^W https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-comp… To gain a more intuitive grasp of what these formulas mean, let's cook up a concrete example and draw a picture of the relations involved. Involution Example ================== Consider a universe of discourse X subject to the following data. • X = {a, b, c, d, e, f, g, h, i} • W = {d, f} • L = {b:a, b:c, c:b, c:d, e:d, e:e, e:f, g:f, g:h, h:g, h:i} Figure 55 shows the placement of W within X and the placement of L within X × X. Figure 55. Bigraph for the Involution L^W https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-bigraph-inv… To highlight the role of W more clearly, the Figure represents the absolute term “w” by means of the relative term “w,” which conveys the same information. Computing the denotation of ℓ^w by way of the class intersection formula, we can show our work as follows. Equation 3. Class Intersection Formula for L^W https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-class-inter… With Figure 55 in mind we can visualize the computation of (L^W)_u as follows. 1. Pick a specific u in the bottom row of the Figure. 2. Pan across the elements v in the middle row of the Figure. 3. If u links to v then L_uv = 1, otherwise L_uv = 0. 4. If v in the middle row links to v in the top row then W_v = 1, otherwise W_v = 0. 5. Compute the value of (L_uv)^W_v, which equals the value of the converse implication L_uv ⇐ W_v, for each v in the middle row. 6. If any of the values (L_uv)^W_v is 0 then the product ∏_v (L_uv)^W_v is 0, otherwise it is 1. As a general observation, we know the value of (L_uv)^W_v goes to 0 just as soon as we find a v in X such that L_uv = 0 and W_v = 1, in other words, such that (u, v) is not L but v is in W. If there is no such v then (L_uv)^W_v = 1. Running through the program for each u in X, the only case producing a non-zero result is ((L_uv)^W_v)_e = 1. That portion of the work can be summarized as follows. Equation 4. Matrix Coefficient Formula for L^W https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-coef… Regards, Jon