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

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 11 https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-s… All, We continue with §3. Application of the Algebraic Signs to Logic. Peirce’s 1870 “Logic of Relatives” • Selection 11 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#S… <QUOTE CSP> The Signs for Multiplication (concl.) The conception of multiplication we have adopted is that of the application of one relation to another. So, a quaternion being the relation of one vector to another, the multiplication of quaternions is the application of one such relation to a second. Even ordinary numerical multiplication involves the same idea, for 2 × 3 is a pair of triplets, and 3 × 2 is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives. If we have an equation of the form xy = z, and there are just as many x’s per y as there are, per things, things of the universe, then we have also the arithmetical equation, [x][y] = [z]. For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then [t][f] = [tf] holds arithmetically. So if men are just as apt to be black as things in general, [m,][b] = [m,b], where the difference between [m] and [m,] must not be overlooked. It is to be observed that [_1_] = 1. Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation. Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76) </QUOTE> Regards, Jon