PEIRCE-L Digest for Friday, December 13, 2002.

NOTE: This record of what has been posted to PEIRCE-L
has been nodified by omission of redundant quotations in
the messages. both for legibility and to save space.
-- Joseph Ransdell, PEIRCE-L manager/owner]

1. RE: Peirce
2. Re: Reductions Among Relations
3. Re: Reductions Among Relations

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Subject: RE: Peirce
From: "Mats Bergman" <mats.bergman[…]helsinki.fi>
Date: Fri, 13 Dec 2002 12:58:01 +0200
X-Message-Number: 1

Søren,

(1) I have heart that T. L. Short has in an article compared Peirce's
and Jakobson's doctrines of sign. In which journal (if any) is it
published.

I think you are referring to Short's "Jakobson's Problematic
Appropriation of Peirce", published in the third volume of the Peirce
Seminar Papers (edited by Michael Shapiro; 1998).

I will get back to the questions you asked in your private mail soon.

Best,

Mats

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```Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Fri, 13 Dec 2002 10:00:28 -0500
X-Message-Number: 2

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

RAR.  Note 23

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Relational Composition as Logical Matrix Multiplication (cont.)

We have now seen three different representations of 2-adic relations.
If one has a strong preference for letters, or numbers, or pictures,
then one may be tempted to take one or the other as being canonical,
but each of them will be found to have its peculiar advantages and
is therefore approached by keeping all three of them in mind.

To see the promised utility of the bigraph picture of 2-adic relations,
let us devise a slightly more complex example of a composition problem,
and use it to illustrate the logic of the matrix multiplication formula.

Keeping to the same space X = {1, 2, 3, 4, 5, 6, 7},
define the 2-adic relations M, N c X x X as follows:

M  =  2:1 + 2:2 + 2:3 + 4:3 + 4:4 + 4:5 + 6:5 + 6:6 + 6:7

N  =  1:2 + 2:2 + 3:3 + 4:3    +    4:5 + 5:5 + 6:7 + 7:7

Here are the bigraph pictures:

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
/|\     /|\     /|\
M     / | \   / | \   / | \
/  |  \ /  |  \ /  |  \
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
|  /    |  / \  |    \  |
N   | /     | /   \ |     \ |
|/      |/     \|      \|
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

To form the composite relation M o N, we simply follow the bigraph for M
by the bigraph for N, here arranging the bigraphs in order down the page,
and then we proceed to "edge out the middle person", that is, we call any
non-empty set of paths of length two between two nodes as being equivalent
to a single directed edge between them in the composite bigraph for M o N.

Here is how it looks in pictures:

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
/|\     /|\     /|\
M     / | \   / | \   / | \
/  |  \ /  |  \ /  |  \
X   o   o   o   o   o   o   o
|  /    |  / \  |    \  |
N   | /     | /   \ |     \ |
|/      |/     \|      \|
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

Figure 15.  M Followed By N

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
/ \     / \     / \
MoN   /   \   /   \   /   \
/     \ /     \ /     \
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

Figure 16.  M Composed With N

Let us hark back to that mysterious matrix multiplication formula,
and see how it appears in the light of the bigraph representation.

The coefficient of the composition M o N
between i and j in X is given as follows:

(M o N)_ij  =  Sum_k (M_ik N_kj)

Graphically interpreted, this is a "sum over paths".
Starting at the node i, M_ik being 1 indicates that
there is an edge in the bigraph of M from node i to
node k, and N_kj being 1 indicates that there is an
edge in the bigraph of N from node k to node j.  So
the Sum_k ranges over all possible intermediaries k,
ascending from 0 to 1 just as soon as there happens
to be some path of length two between nodes i and j.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

----------------------------------------------------------------------

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Fri, 13 Dec 2002 16:30:23 -0500
X-Message-Number: 3

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

RAR.  Note 23

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Relational Composition as Logical Matrix Multiplication (cont.)

We have now seen three different representations of 2-adic relations.
If one has a strong preference for letters, or numbers, or pictures,
then one may be tempted to take one or the other as being canonical,
but each of them will be found to have its peculiar advantages and
is therefore approached by keeping all three of them in mind.

To see the promised utility of the bigraph picture of 2-adic relations,
let us devise a slightly more complex example of a composition problem,
and use it to illustrate the logic of the matrix multiplication formula.

Keeping to the same space X = {1, 2, 3, 4, 5, 6, 7},
define the 2-adic relations M, N c X x X as follows:

M  =  2:1 + 2:2 + 2:3 + 4:3 + 4:4 + 4:5 + 6:5 + 6:6 + 6:7

N  =  1:2 + 2:2 + 3:3 + 4:3    +    4:5 + 5:5 + 6:7 + 7:7

Here are the bigraph pictures:

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
/|\     /|\     /|\
M     / | \   / | \   / | \
/  |  \ /  |  \ /  |  \
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
|  /    |  / \  |    \  |
N   | /     | /   \ |     \ |
|/      |/     \|      \|
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

To form the composite relation M o N, we simply follow the bigraph for M
by the bigraph for N, here arranging the bigraphs in order down the page,
and then we proceed to "edge out the middle person", that is, we call any
non-empty set of paths of length two between two nodes as being equivalent
to a single directed edge between them in the composite bigraph for M o N.

Here is how it looks in pictures:

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
/|\     /|\     /|\
M     / | \   / | \   / | \
/  |  \ /  |  \ /  |  \
X   o   o   o   o   o   o   o
|  /    |  / \  |    \  |
N   | /     | /   \ |     \ |
|/      |/     \|      \|
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

Figure 15.  M Followed By N

1   2   3   4   5   6   7
X   o   o   o   o   o   o   o
/ \     / \     / \
MoN   /   \   /   \   /   \
/     \ /     \ /     \
X   o   o   o   o   o   o   o
1   2   3   4   5   6   7

Figure 16.  M Composed With N

Let us hark back to that mysterious matrix multiplication formula,
and see how it appears in the light of the bigraph representation.

The coefficient of the composition M o N
between i and j in X is given as follows:

(M o N)_ij  =  Sum_k (M_ik N_kj)

Graphically interpreted, this is a "sum over paths".
Starting at the node i, M_ik being 1 indicates that
there is an edge in the bigraph of M from node i to
node k, and N_kj being 1 indicates that there is an
edge in the bigraph of N from node k to node j.  So
the Sum_k ranges over all possible intermediaries k,
ascending from 0 to 1 just as soon as there happens
to be some path of length two between nodes i and j.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o```

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END OF DIGEST 12-13-02

Page last modified by B.U. April 28, 2012, earliest in summer 2011 — B.U.