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Subject: Re: Logic Of Relatives
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Sun, 15 Dec 2002 10:50:58 -0500
X-Message-Number: 1

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LOR. Note 3

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| Numbers Corresponding to Letters
|
| I propose to use the term "universe" to denote that class of individuals
| 'about' which alone the whole discourse is understood to run. The universe,
| therefore, in this sense, as in Mr. De Morgan's, is different on different
| occasions. In this sense, moreover, discourse may run upon something which
| is not a subjective part of the universe; for instance, upon the qualities
| or collections of the individuals it contains.
|
| I propose to assign to all logical terms, numbers; to an absolute term,
| the number of individuals it denotes; to a relative term, the average
| number of things so related to one individual. Thus in a universe of
| perfect men ('men'), the number of "tooth of" would be 32. The number
| of a relative with two correlates would be the average number of things
| so related to a pair of individuals; and so on for relatives of higher
| numbers of correlates. I propose to denote the number of a logical term
| by enclosing the term in square brackets, thus ['t'].
|
| C.S. Peirce, CP 3.65
|
| Charles Sanders Peirce,
|"Description of a Notation for the Logic of Relatives,
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).

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Subject: Re: [Arisbe] Re: Logic Of Relatives
From: Gary Richmond <garyrichmond[…]rcn.com>
Date: Sun, 15 Dec 2002 11:13:25 -0500
X-Message-Number: 2

Jon,

I wonder if the necessary "elementary triad" spoken of below isn't
somehow implicated in those discussions
"invoking a 'closure principle'."

> CP 1.292 It can further be said in advance, not, indeed, purely a
> priori but with the degree of apriority that is proper to logic,
> namely, as a necessary deduction from the fact that there are signs,
> that there must be an elementary triad. For were every element of the
> phaneron a monad or a dyad, without the relative of teridentity
> (which is, of course, a triad), it is evident that no triad could ever
> be built up. Now the relation of every sign to its object and
> interpretant is plainly a triad. A triad might be built up of pentads
> or of any higher perissad elements in many ways. But it can be proved
> -- and really with extreme simplicity, though the statement of the
> general proof is confusing -- that no element can have a higher
> valency than three.

(Of course this passage also directly relates to the recent thread on
Identity and Teridentity.)

Gary

PS I came upon the above passage last night reading through the Peirce
selections in John J. Stuhr's
Classical American Philosophy: Essential Readings and Interpretive
Essays , Oxford University,
1987 (the passage above is found on pp 61-62), readily available in
paperback in a new edition, I
believe.

An aside: These excerpts in Sturh include versions of a fascinating
"Intellectual Autobiography," Peirce's summary
of his scientific, especially, philosophic accomplishments. I've seen
them published nowhere else.

Jon Awbrey wrote:

>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
>LOR. Note 2
>
>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
>I am going to experiment with an interlacing commentary
>on Peirce's 1870 "Logic of Relatives" paper, revisiting
>some critical transitions from several different angles
>and calling attention to a variety of puzzles, problems,
>and potentials that are not so often remarked or tapped.
>
>What strikes me about the initial installment this time around is its
>use of a certain pattern of argument that I can recognize as invoking
>a "closure principle", and this is a figure of reasoning that Peirce
>uses in three other places, his discussion of "continuous relations",
>his definition of sign relations, and the pragmatic maxim itself.
>
>Jon Awbrey
>
>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>_______________________________________________
>Arisbe mailing list
>Arisbe[…]stderr.org
>http://stderr.org/cgi-bin/mailman/listinfo/arisbe
>

.


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Subject: Re: Logic Of Relatives
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Sun, 15 Dec 2002 12:30:24 -0500
X-Message-Number: 3

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LOR. Discussion Note 1

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GR = Gary Richmond

GR: I wonder if the necessary "elementary triad" spoken of below isn't somehow
implicated in those discussions "invoking a 'closure principle'".

GR, quoting CSP:

| CP 1.292. It can further be said in advance, not, indeed,
| purely a priori but with the degree of apriority that is
| proper to logic, namely, as a necessary deduction from
| the fact that there are signs, that there must be an
| elementary triad. For were every element of the
| phaneron a monad or a dyad, without the relative
| of teridentity (which is, of course, a triad),
| it is evident that no triad could ever be
| built up. Now the relation of every sign
| to its object and interpretant is plainly
| a triad. A triad might be built up of
| pentads or of any higher perissad
| elements in many ways. But it
| can be proved -- and really
| with extreme simplicity,
| though the statement of
| the general proof is
| confusing -- that no
| element can have
| a higher valency
| than three.

GR: (Of course this passage also directly relates
to the recent thread on Identity and Teridentity.)

Yes, generally speaking, I think that there are deep formal principles here
that manifest themselves in these various guises: the levels of intention
or the orders of reflection, the sign relation, pragmatic conceivability,
the generative sufficiency of 3-adic relations for all practical intents,
and the irreducibility of continuous relations. I have run into themes
in combinatorics, group theory, and Lie algebras that are tantalizingly
reminiscent of the things that Peirce says here, but it will take me
some time to investigate them far enough to see what's going on.

GR: PS. I came upon the above passage last night reading through
the Peirce selections in John J. Stuhr's 'Classical American
Philosophy: Essential Readings and Interpretive Essays',
Oxford University, 1987 (the passage above is found on
pp 61-62), readily available in paperback in a new
edition, I believe.

GR: An aside: These excerpts in Sturh include versions of a fascinating
"Intellectual Autobiography", Peirce's summary of his scientific,
especially, philosophic accomplishments. I've seen them published
nowhere else.

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

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Subject: Re: Logic Of Relatives
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Sun, 15 Dec 2002 16:44:23 -0500
X-Message-Number: 4

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LOR. Note 4

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Peirce's remarks at CP 3.65 are so replete with remarkable ideas,
some of them so taken for granted in mathematical discourse that
they usually escape explicit mention, and others so suggestive
of things to come in a future remote from his time of writing,
and yet so smoothly introduced in passing that it is all too
easy to overlook their consequential significance, that I
can do no better here than to highlight these ideas in
other words, whose main advantage is to be a little
more jarring to the mind's sensibilities.

| Numbers Corresponding to Letters
|
| I propose to use the term "universe" to denote that class of individuals
| 'about' which alone the whole discourse is understood to run. The universe,
| therefore, in this sense, as in Mr. De Morgan's, is different on different
| occasions. In this sense, moreover, discourse may run upon something which
| is not a subjective part of the universe; for instance, upon the qualities
| or collections of the individuals it contains.
|
| I propose to assign to all logical terms, numbers; to an absolute term,
| the number of individuals it denotes; to a relative term, the average
| number of things so related to one individual. Thus in a universe of
| perfect men ('men'), the number of "tooth of" would be 32. The number
| of a relative with two correlates would be the average number of things
| so related to a pair of individuals; and so on for relatives of higher
| numbers of correlates. I propose to denote the number of a logical term
| by enclosing the term in square brackets, thus ['t'].
|
| C.S. Peirce, 'Collected Papers', CP 3.65

1. This mapping of letters to numbers, or logical terms to mathematical quantities,
is the very core of what "quantification theory" is all about, and definitely
more to the point that the mere "innovation" of using distinctive symbols
for the so-called "quantifiers". We will speak of this more later on.

2. The mapping of logical terms to numerical measures,
to express it in current language, would probably be
recognizable as some kind of "morphism" or "functor"
from a logical domain to a quantitative co-domain.

3. Notice that Peirce follows the mathematician's usual practice,
then and now, of making the status of being an "individual" or
a "universal" relative to a discourse in progress. I have come
to appreciate more and more of late how radically different this
"patchwork" or "piecewise" approach to things is from the way of
some philosophers who seem to be content with nothing less than
many worlds domination, which means that they are never content
and rarely get started toward the solution of any real problem.
Just my observation, I hope you understand.

4. It is worth noting that Peirce takes the "plural denotation"
of terms for granted, or what's the number of a term for,
if it could not vary apart from being one or nil?

5. I also observe that Peirce takes the individual objects of a particular
universe of discourse in a "generative" way, not a "totalizing" way,
and thus they afford us with the basis for talking freely about
collections, constructions, properties, qualities, subsets,
and "higher types", as the phrase is mint.

Jon Awbrey

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Subject: Re: TPM Online Information: Posting No. 49
From: Gary Richmond <garyrichmond[…]rcn.com>
Date: Sun, 15 Dec 2002 20:05:01 -0500
X-Message-Number: 5

.

Dr Jeremy Stangroom wrote:

>It's Sunday afternoon, and I've been thinking about Christmas. What I want
>to know is why it isn't possible to mention Christmas without someone opining
>that it has become 'much too commercial'? What on earth are they talking
>about? It's always been commercial. After all, didn't some wise men bring the
>baby Jesus presents as it lay in its crib. Gold and Frank's incense, if I
>remember correctly. Anyway, the whole point of Christmas is to run up a huge
>credit card bill. Unless, of course, you've maxed it out buying an oil tanker.
>
>
I think the point of the critics of the commercialism of Christmas is
that to the extent that the original message is
obscured--that message being, that the world is redeemed exactly
through love--commercialism remains an
impediment to the growth of the Law of Love in the world, to the health
and healing of the world through generous,
and even selfless, acts of loving each other as we would be loved.

The symbol of the rich gifts given by the wisemen to the christchild
does not in any way support commercialism;
perhaps a more faithful interpretation is that is is possible for even
the wealthiest and most powerful to come to see,
that we ought give of what is ours in the world to that which is worthy,
to that which is really and truly of value.

From exercising this (which is, nothing more than the fundamental
expression of our deepest humanity) follows the power
of "the Gospel of Love" (as opposed to what the scientist and
philosopher Charles S. Peirce called "the Gospel of Greed"), and
even the intellectual summum bonum, the reasonable in itself (Peirce)

Best regards and Merry Christmas,

Ga;ry Richmond
City University of New York

PS


JS: That's it for now. Got to go and heckle some people who insist on calling
Christmas a winter solstice celebration...

GR: Now this I can go along with!


>
>It's Sunday afternoon, and I've been thinking about Christmas. What I want
>to know is why it isn't possible to mention Christmas without someone opining
>that it has become 'much too commercial'? What on earth are they talking
>about? It's always been commercial. After all, didn't some wise men bring the
>baby Jesus presents as it lay in its crib. Gold and Frank's incense, if I
>remember correctly. Anyway, the whole point of Christmas is to run up a huge
>credit card bill. Unless, of course, you've maxed it out buying an oil tanker.
>
>
>


.


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END OF DIGEST 12-15-02

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