1. Re: Relevance of Peircean Semiotic to Computational Intelligence Augmentation
2. Re: Relevance of Peircean Semiotic to Computational Intelligence Augmentation
3. Re: Reductions Among Relations

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Subject: Re: Relevance of Peircean Semiotic to Computational Intelligence Augmentation
From: "Peter Skagestad" <Peter_Skagestad[…]uml.edu>
Date: Mon, 09 Dec 2002 08:49:36 -0800
X-Message-Number: 1

Armando,

I must confess that I am not familiar with Burch's notation. If you could
give me the citation for his paper, that would be helpful. Meanwhile, I
am actually planning to stop by the Brandeis University library today,
and will see if I can find it on my own.

Best regards,
Peter

Armando Sercovich wrote:
>
> Peter Skagestad,
>
> Initially I want to thank you your fineness of having sent me last year your important article 'Peirce's Inkstand as an External Embodiment of
Mind'.
>
> I apologize by my so long silence, but I assure to you that I have read it carefully. Particularly your two preliminary claims - as well as the
concept of "vi
>
> In your work you wrote:
>
> "My second claim is that Peirce's quotation is echoed in Karl Popper's later doctrine that human knowledge depends on the evolution of
exosomatic organs, such
>
> Being you an historian of ideas, I ask you:
>
> . Do you suppose that these "orality-literacy transformations" could be topologically represented (logical and geometrically) by means of Burch's
notation in h
> . That would allow us to understand different literacy processes in different cultures?
>
> and:
>
> . It would be that notation a exosomatic organ (or a machine maybe)?
>
> Thank you in advance.
>
> Best regards,
.



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Subject: Re: Relevance of Peircean Semiotic to Computational Intelligence Augmentation
From: "Armando Sercovich" <cispec[…]com4.com.ar>
Date: Mon, 9 Dec 2002 13:20:57 -0300
X-Message-Number: 2

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 Press, 1997, pp. 234/251, edited by Nathan Houser, =
Don D. Roberts and James Van Evra

Best regards,
Armando

Peter Skagestad wrote:

> Armando,
>=20
> I must confess that I am not familiar with Burch's notation. If you =
could=20
> give me the citation for his paper, that would be helpful. Meanwhile, =
I=20
> am actually planning to stop by the Brandeis University library today, =

> and will see if I can find it on my own.
>=20
> Best regards,
> Peter
.
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Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 09 Dec 2002 12:48:20 -0500
X-Message-Number: 3

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RAR. Note 18

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Relational Composition as Logical Matrix Multiplication

We have it within our reach to pick up another way of representing
2-adic relations, namely, the representation as logical matrices,
and also to grasp the analogy between relational composition and
ordinary matrix multiplication as it appears in linear algebra.

To begin, while we still have the data of a very simple concrete case
in mind, let us reflect on what we did in our last Example in order
to find the composition G o H of the 2-adic relations G and H.

Here is the setup that we had before:

X = {1, 2, 3, 4, 5, 6, 7},

G = 4:3 + 4:4 + 4:5 c X x X,

H = 3:4 + 4:4 + 5:4 c X x X.

Let us recall the rule for finding the relational composition of a pair
of 2-adic relations. Given the 2-adic relations P c X x Y, Q c Y x Z,
the "relational composition" of P and Q, in that order, is commonly
denoted as "P o Q" or more simply as "PQ" and obtained as follows:

To compute PQ, in general, where P and Q are 2-adic relations,
simply multiply out the two sums in the ordinary distributive
algebraic way, only subject to the following rule for finding
the product of two elementary relations of shapes a:b and c:d.

| (a:b)(c:d) = (a:d) if b = c,
|
| (a:b)(c:d) = 0 otherwise.

To find the relational composition G o H,
we write it as a quasi-algebraic product:

G o H = (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4).

Multiplying this out in accord with the applicable form
of distributive law one obtains the following expansion:

G o H = (4:3)(3:4) + (4:3)(4:4) + (4:3)(5:4) +
(4:4)(3:4) + (4:4)(4:4) + (4:4)(5:4) +
(4:5)(3:4) + (4:5)(4:4) + (4:5)(5:4)

Applying the rule that determines the product
of elementary relations, we obtain this array:

G o H = (4:4) + 0 + 0 +
0 + (4:4) + 0 +
0 + 0 + (4:4)

Since the plus sign in this context represents an operation of
logical disjunction or set-theoretic aggregation, all positive
multiplicites count as one, and this gives the ultimate result:

G o H = 4:4

To be continued ...

Jon Awbrey

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END OF DIGEST 12-10-02

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