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

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.5 https://inquiryintoinquiry.com/2014/06/15/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 12.5 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, The equation (s^ℓ)^w = s^(ℓw) can be verified by establishing the corresponding equation in matrices. • (S^L)^W = S^(LW) If A and B are two 1-dimensional matrices over the same index set X then A = B if and only if A_x = B_x for every x in X. Thus, a routine way to check the validity of (S^L)^W = S^(LW) is to check whether the following equation holds for arbitrary x in X. • ((S^L)^W)_x = (S^(LW))_x Taking both ends toward the middle, we proceed as follows. Matrix Equation ((S^L)^W)_x = (S^(LW))_x https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-s5el5ew_x-s… The products commute, so the equation holds. In essence, the matrix identity turns on the fact that the law of exponents (a^b)^c = a^(bc) in ordinary arithmetic holds when the values a, b, c are restricted to the boolean domain B = {0, 1}. Interpreted as a logical statement, the law of exponents (a^b)^c = a^(bc) amounts to a theorem of propositional calculus otherwise expressed in the following ways. • ((a ⇐ b) ⇐ c) = (a ⇐ b ∧ c) • (c ⇒ (b ⇒ a)) = (c ∧ b ⇒ a) Regards, Jon