jon list. Very nice work! I got stuck on the large number of objects that could be attached to a relative and drifted off into prime numbers, factors, subgroups and the 'conversion' formula(s). It is good to see a valuation for interpreting some of these things. Given the arrays of zeroes and ones, I can begin to see how to replace the ":" sign with various operations that are less general. Jim W > Date: Sun, 19 Apr 2015 14:40:33 -0400 > From: [email protected] > To: [email protected] > Subject: [PEIRCE-L] Peirce's 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2 > > Post : Peirce's 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2 > http://inquiryintoinquiry.com/2015/04/19/peirces-1880-algebra-of-logic-chapter-3-%e2%80%a2-comment-7-2/ > Date : April 19, 2015 at 1:00 pm > > Peircers, > > Note. This post has a lot of math formatting, > so please follow the link above for a more > readable text. > > Because it can sometimes be difficult to reconnect abstractions with > their concrete instances, especially after the abstract types have > become autonomous and taken on a life of their own, let us resort > to a simple concrete case and examine the implications of what > Peirce is saying about the relation between general relatives > and individual relatives. > > Suppose our initial universe of discourse has > exactly two individuals, I and J. Then there > are exactly four individual dual relatives or > ordered pairs of universe elements: > > • I:I, I:J, J:I, J:J. > > It is convenient arrange these in a square array: > > ⎛ I:I I:J ⎞ > ⎝ J:I J:J ⎠ > > There are 2^4 = 16 dual relatives in general over this universe of discourse, > since each one is formed by choosing a subset of the four ordered pairs and > then “aggregating” them, forming their logical sum, or simply regarding them > as a subset. Taking the square array of ordered pairs as a backdrop, any one > of the 16 dual relatives may be represented by a square matrix of binary > values, > a value of 1 occupying the place of each ordered pair that belongs to the > subset > and a value of 0 occupying the place of each ordered pair that does not belong > to the subset in question. The matrix representations of the 16 dual > relatives > or dyadic relations over the universe {I, J} are displayed below: > > ⎛ 0 0 ⎞ ⎛ 1 0 ⎞ ⎛ 0 0 ⎞ ⎛ 1 0 ⎞ > ⎝ 0 0 ⎠ ⎝ 0 0 ⎠ ⎝ 0 1 ⎠ ⎝ 0 1 ⎠ > > ⎛ 0 1 ⎞ ⎛ 1 1 ⎞ ⎛ 0 1 ⎞ ⎛ 1 1 ⎞ > ⎝ 0 0 ⎠ ⎝ 0 0 ⎠ ⎝ 0 1 ⎠ ⎝ 0 1 ⎠ > > ⎛ 0 0 ⎞ ⎛ 1 0 ⎞ ⎛ 0 0 ⎞ ⎛ 1 0 ⎞ > ⎝ 1 0 ⎠ ⎝ 1 0 ⎠ ⎝ 1 1 ⎠ ⎝ 1 1 ⎠ > > ⎛ 0 1 ⎞ ⎛ 1 1 ⎞ ⎛ 0 1 ⎞ ⎛ 1 1 ⎞ > ⎝ 1 0 ⎠ ⎝ 1 0 ⎠ ⎝ 1 1 ⎠ ⎝ 1 1 ⎠ > > Relative to the universe {I, J}, the individual dual relatives of > the form A:A are I:I and J:J while the individual dual relatives of > the form A:B are I:J and J:I. > > Peirce assigns the name ‘concurrents’ to dual relatives all whose > individual aggregants are of the form A:A. There are exactly 4 of > these and their matrices are shown in the top row of the above display. > All the rest are called ‘opponents’ and their matrices are listed in > the bottom three rows. > > Peirce gives the name ‘alio-relatives’ to dual relatives all whose > individual aggregants are of the form A:B. There are exactly 4 of > these and their matrices are shown in the first column of the above > display. All the rest are called ‘self-relatives’ and their matrices > are listed in the right hand three columns. > > Notice that the relative 0, represented by a matrix with all 0 entries, > falls under the definitions of both a concurrent and an alio-relative. > > References > > • Peirce, C.S. (1880), “On the Algebra of Logic”, > American Journal of Mathematics 3, 15–57. > Collected Papers (CP 3.154–251), > Chronological Edition (CE 4, 163–209). > > • Peirce, C.S., Collected Papers of Charles Sanders Peirce, > vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), > vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, > Cambridge, MA, 1931–1935, 1958. Volume 3 : Exact Logic, 1933. > > • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, > Peirce Edition Project (eds.), Indiana University Press, Bloomington > and Indianapolis, IN, 1981–. Volume 4 (1879–1884), 1986. > > Resources > > • Peirce’s 1870 Logic Of Relatives > http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_1870_Logic_Of_Relatives > > -- > > academia: http://independent.academia.edu/JonAwbrey > my word press blog: http://inquiryintoinquiry.com/ > inquiry list: http://stderr.org/pipermail/inquiry/ > isw: http://intersci.ss.uci.edu/wiki/index.php/JLA > oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey > facebook page: https://www.facebook.com/JonnyCache
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