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
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