PEIRCE-L Digest for Wednesday, December 11, 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: Relevance of Peircean Semiotic to Computational Intelligence Augmentation
2. Re: Reductions Among Relations
3. Peirce

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Subject: Re: Relevance of Peircean Semiotic to Computational Intelligence Augmentation
Date: Wed, 11 Dec 2002 10:41:24 -0800
X-Message-Number: 1

Armando,

OK - I should easily be able to lay my hands on that volume.

Peter

Armando Sercovich wrote:
>
> Peter,

>
> You can find a reduced version of Burch's original paper (without a detailed mathematical exposition) in STUDIES IN THE LOGIC OF CHARLES
SANDERS PEIRCE, IU Pre
>
> Best regards,
> Armando

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Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Wed, 11 Dec 2002 15:40:12 -0500
X-Message-Number: 2

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RAR. Note 21
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Relational Composition as Logical Matrix Multiplication (cont.)

All that remains in order to obtain a computational formula for
the relational composite G o H of the 2-adic relations G and H
is to collect its coefficients (G o H)_ij over the appropriate
basis of elementary relations i:j, as i and j range through X.

G o H = Sum_ij (G o H)_ij (i:j) = Sum_ij (Sum_k (G_ik H_kj)) (i:j).

This is the logical analogue of matrix multiplication in linear algebra,
the difference in the logical setting being that all of the operations
performed on coefficients take place in a system of boolean arithmetic
where summation corresponds to logical disjunction and multiplication
corresponds to logical conjunction.

By way of disentangling this formula, we notice that the form
Sum_k (G_ik H_kj) is what is usually called a "scalar product".
In this case it is the scalar product of the i^th row of G with
the j^th column of H.

To make this statement more concrete, let us go back to
the particular examples of G and H that we came in with:

G =

0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 1 1 1 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0

H =

0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 1 0 0 0
0 0 0 1 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0

The formula for computing G o H tells us this:

(G o H)_ij

= the ij^th entry in the matrix representation for G o H

= the entry in the i^th row and the j^th column of G o H

= the scalar product of the i^th row of G with the j^th column of H

= Sum_k (G_ik H_kj)

As it happens, we are enabled to make exceedingly light work of this example,
since there is only one row of G and one column of H that are not all zeroes.
Taking the scalar product, in a logical way, of the fourth row of G with the
fourth column of H produces the sole non-zero entry for the matrix of G o H.

G o H =

0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0

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

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Subject: Peirce
From: Søren_Lund <lund.s[…]mail.tele.dk>
Date: Wed, 11 Dec 2002 23:20:33 +0100
X-Message-Number: 3

Dear Peirce-l,

Two questions:

(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?

(2) Can someone address me to Peirce commentators who have
written about Peirce's concept of 'Form'?

Yours,
Søren Lund

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END OF DIGEST 12-11-02

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