Jim, List,
The form x:y is just Peirce's notation for the ordered pair (x, y).
A 2-point universe like {I, J} provides us with another example of formal
degeneracy (loss of generality) since the number of “diagonal” terms (of the
form A:A) is equal to the number of “off-diagonal” terms (of the form A:B) and
so the case exhibits symmetries that will be broken as soon as one adds another
element to the universe. I am planning to take up a 3-point example in good
time but I wanted to essay the graph-theoretic representation of dyadic
relations first.
Regards,
Jon
http://inquiryintoinquiry.com
> On Apr 20, 2015, at 12:46 PM, Jim Willgoose <[email protected]> wrote:
>
> 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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