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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 01:18:17 EST
X-Message-Number: 1

PB wrote:

----quote------
Jon would you please point to where John shows he "does not know what the
dif-ference between a relation and one of its instances is" and "does not
know what decomposition or reduction is"? Thanks.
----quote------

Good question, Peter, I was wondering where Jon might have got this idea
myself.
There are some interesting questions suggested, of course. But the
suppositions which Jon adds seem simply so unsympathetic as to be doubtful
and contentious --that is, they require some argument or support, as you
rightly point out.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)



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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 01:50:35 EST
X-Message-Number: 2

You wrote, Charles,

----quote---------
I have not been following this thread closely, but I haven't seen it pointed
out that someplace (I don't have access to Peirce texts just now) Peirce
explicitly discusses the example of 'between' in terms of the relation of
Philadelphia as between Washing-ton and New York, arguing that 'between' is
not a relation of genuine thirdness, but is rather degenerate thirdness. In
other words, as I understand Peirce's thinking, 'between' is not a relation
of thirdness, but rather a relation of compounded secondnesses.
----end quote-----

I recall something like this. Yet it seems rather misleading to say here
(whether this is true or not) that "between is not a relation of genuine
thirdness." After all, the prior question was whether triadic relations can
be analyzed --not what to count as "genuine thirdness." Recalling my
analysis of the passage from Peirce, recently discussed; it seemed there that
the argument was from examples of genuinely triadic relations to thirdness.
If I've got this right, then the question is not, to this point, whether
"between" is a relation of genuine thirdness, but whether it is a triadic
relation open to analysis.

In terms of the example offered, it is one thing to say that Philadelphia is
between New York and Washington, though we might perhaps analyze this, for
some pur-poses, this by saying that New York in North of Philadelphia and
Philadelphia is North of Washington; it is quite something different to say,
perhaps, that Philadel-phia mediates between New York and Washington. Whether
or not Philadelphia has ever or could mediate between New York and Washington
in some way or other (consider that Pennsylvania calls itself the "keystone
state"), it is certainly located between New York and Washington. So, being a
triadic relation, even a genuine triadic relation, so far at least, seems
quite different from being "a relation of genuine thirdness."

Much more needs to be said. It would be rather a difficulty for Peirce
studies, to find, after long consideration and discussions, that every
triadic or three-placed relation for which a plausible analysis can be given
or found, will then turn out to be not a "genuine triadic relation," because
not exemplifying thirdness. It seems to me that we have a clearer and more
definite idea of what a three-placed relation is than we do of what thirdness
is. That seems to be Peirce's view, too. Beyond that, if we find regarding
some three-placed relations, that we have no plausible analysis, then this
may just be to say that we haven't looked hard enough or that the related
subject-matter stands in need of development. While I do not think that that
is an inevitable conclusion, it is a kind of conclusion witrh some stadning
as a kind of hypothesis.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 02:06:47 EST
X-Message-Number: 3

Joe,

You wrote, in reply to John,

----quote----------
I thought the basic claim is that representation is irreducibly triadic,
John. Peirce recogmizes any degree of adicity you like, if it is only a
matter of verbal form. So I understand you to be saying that you don't
regard representation as triadic, other than verbally so. I don't know what
you would have in mind in saying that Peirce convinced you of its
triadicity, though, since I don't recall him trying to make any claim for it
that wasn't a claim to its irreducibility to a complex of dyads.
----end quote-----------

Here I would just like to interject the observation that we may need to
consider "reducibility" and "open to analysis" as importantly distinct. For,
suppose that representation is, as you say, "irreducibly triadic," if we give
an analysis of representation, say in terms of other relations, it would not
follow that we have reduced representation to something else. I recall in
this connection, a recent news story, according to which there is some work
going on to attempt to create a new (simple) biological organism, by removing
the DNA from a very simple organism and replacing it with a synthetic DNA.
The plan is to analyze this simple organism in terms of its bio-chemical
constituents and processes. Still, I think that if such an ambitious plan
where actually accomplished, it would not follow that the simple biological
organism, capable of all life functions usual in such an organism, is not a
biological organism at all --but merely, say, a collection of chemicals
interacting in various specified ways. If people come to understand this
simple organism on the plan of analysis and intervention discussed in the
news story, it would not follow that the successful analysis reduces the
living organism to something else.

In an similar way, it seems clear that an analysis of representation is not
quite the same as a reduction of representation to something else. We have to
ask whether the aim of the analysis is to eliminate all reference to
representation. Likewise, if some relation is "irreducibly tiadic," this does
not suggests to me that it is incapable of analysis or definition making use
of less-than-triadic relations.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 02:25:57 EST
X-Message-Number: 4

John & list,

In amongst a long reply of much interest, you said something that struck my
eye and which I am inclined to question.

----quote----------
So, returning to the issue of what Peirce meant, my best guess is that he
means that interpretation is non-linear and does not have a synthetic model.
He was mis-taken in thinking that there are logically irreducible triadic
relations.
---end quote------

As I said to Joe, there is some danger in equating analysis of a relation
with reduction of a relation. So, I would rather say that the question of
whether a given relation is open to some interesting analysis is a question
for inquiry, depending in degree on how the subject-matter may develop --it
is not simply a matter of set theory. I think that Quine merely offers a
mathematical model of reduction to a dyadic relation, which model will remain
without broad non-mathematical interest in many cases. Whether we will justly
regard some analysis offered as a "reduction" is a distinct question. In
consequence, I see no sufficient grounds for thinking, as of yet, at least,
that Peirce "was mistaken in thinking that there are logically irreducible
triadic relations." This, as it seems to me, is a rather strong claim in the
present context.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 02:40:44 EST
X-Message-Number: 5

Jon,

You wrote:

----quote-----------
Again, Jon seems to be assuming the irreducibility of teridentity, which was,
as I recall from the subject line, exactly the point that was at issue.
---end quote------

I don't want to address the issue of what Jon said, or didn't say. But I
think you usefully focus on a point at issue in this thread. Let us represent
teridentity for present purposes, in this fashion:

Ixyz,

which we regard as true of every ordered triple, such that x=y and y=z. It
seems plausible to say that we can analyze "Ixyz" in standard notation by
reference to "x=y & y=z." There seems to be some advantage, or possible
advantage, to the analysis, since we have axioms for identity theory,
including e.g., "(x) (x=x)" and the substitutivity principle --if x=y, then
whatever is true or x is true of y and vice versa.

Now, I am less certain that this is a reduction than I am that it is a pretty
good analysis. To say that it is a reduction, it seems, is just to say that
once we under-stand the analysis, we see no further need of teridentity. That
result I also regard as plausible, but I think there is an extra step to it,
which is not always so plausible, regarding other relations. For example,
suppose that representation is triadic and that we are able to give some
analysis of it which does not depend on something equally triadic. It would
not follow, as I see the matter, that we have no need of a triadic conception
of representation. Some additional argument would seem to be needed there.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 03:01:01 EST
X-Message-Number: 6

Gary John & list,

You wrote, Gary, commenting on Peirce,

---quote--------------
So, identity in the way it's been traditionally seen in logic is hardly
denied, but added to this is the concept of teridenity, which "is not mere
identity. It is identity and identity" in the very sense of the man seen on
consecutive days.
---end quote---------

It is important to see that Peirce does not identity as traditionally
employed in logic.
That helps sharpen the question of whether we need teridentity in addition,
and whether and how it might be analyzed or reduced to (bi-)identity. See my
last reply to John, which formulates just this kind of question of the
relationship between analysis and reduction of triadic relations. In the case
of eridentity, the analysis seems clear and the reduction plausible --though
I have been insisting that these are two separate types of questions with
regard to their general application.

You go on to quote a further passage from Peirce:

> Peirce: CP 1.346
> The other premiss of the argument that genuine triadic relations can
> never be built of dyadic relations and of qualities is easily shown.
> In existential graphs, a spot with one tail -- X represents a quality,
> a spot with two tails -- R -- a dyadic relation.+1 Joining the ends of
> two tails is also a dyadic relation. But you can never by such joining
> make a graph with three tails. You may think that a node connecting
> three lines of identity Y is not a triadic idea. But analysis will
> show that it is so. I see a man on Monday. On Tuesday I see a man, and
> I exclaim, "Why, that is the very man I saw on Monday." We may say,
> with sufficient accuracy, that I directly experienced the identity. On
> Wednesday I see a man and I say, "That is the same man I saw on
> Tuesday, and consequently is the same I saw on Monday." There is a
> recognition of triadic identity; but it is only brought about as a
> conclusion from two premisses, which is itself a triadic relation. If
> I see two men at once, I cannot by any such direct experience identify
> both of them with a man I saw before. I can only identify them if I
> regard them, not as the very same, but as two different manifestations
> of the same man. But the idea of manifestation is the idea of a sign.
> Now a sign is something, A, which denotes some fact or object, B, to
> some interpretant thought, C.

You comment:

---quote-------
I am no expert in the EG's, but connecting this example with the discussion
of identity/teridenity above ought at least to provide some basis for
discussion.
----end quote-----

Agreed. You've got another interesting quotation here. It may help us to
answer the question of why Peirce thought teridentity important though the
analysis seems so obvious: ("x=y & y=z" for "Ixyz"). Let's see if we get the
attention of some others on the list for this passage from Peirce. It is an
interesting passage. My argument has been that it is some particularity of
Peirce system of graphs which requires the notion of teridentity, and I think
we are yet to isolate or identify what exactly this comes to, or what
assumptions or suppositions are built into the system of graphs, so as to
suggest the need of teridentity.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 03:10:38 EST
X-Message-Number: 7

( I have changed the salutation on this posting, please excuse the typo, John,
---HGC)

John,

You wrote:

----quote-----------
Again, Jon seems to be assuming the irreducibility of teridentity, which
was,
as I recall from the subject line, exactly the point that was at issue.
---end quote------

I don't want to address the issue of what Jon said, or didn't say. But I
think you usefully focus on a point at issue in this thread. Let us
represent
teridentity for present purposes, in this fashion:

Ixyz,

which we regard as true of every ordered triple, such that x=y and y=z. It
seems plausible to say that we can analyze "Ixyz" in standard notation by
reference to "x=y & y=z." There seems to be some advantage, or possible
advantage, to the analysis, since we have axioms for identity theory,
including e.g., "(x) (x=x)" and the substitutivity principle --if x=y, then
whatever is true or x is true of y and vice versa.

Now, I am less certain that this is a reduction than I am that it is a
pretty
good analysis. To say that it is a reduction, it seems, is just to say that
once we under-stand the analysis, we see no further need of teridentity.
That
result I also regard as plausible, but I think there is an extra step to it,
which is not always so plausible, regarding other relations. For example,
suppose that representation is triadic and that we are able to give some
analysis of it which does not depend on something equally triadic. It would
not follow, as I see the matter, that we have no need of a triadic
conception
of representation. Some additional argument would seem to be needed there.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)
>>

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

Subject: Late Gothic Architecture
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 08:44:11 EST
X-Message-Number: 8

Peirce-l,

I want to recommend an article which I recently read from the London Review
of Books, Vol. 24, No. 20 dated 17 October 2002, by Andrew Saint, titled
"The Danger of Giving In:" A fascinating and curious review of _An Architect
of Promise: George Gilbert Scott Jr (1839-97) and the Late Gothic Revival_ by
Gavin Stamp, Shaun Tyas, 427 pp, 349.50.

I have often wondered about 19th century Gothic and the meaning it had to
those who adopted that style. This review suggested some answers, however
indirectly, though I cannot say that that was the objective of the review.
Some of what you find in the review may strike you a bit strange, but it is worth reading through.

I was reminded of the article last night, when I watched a recent movie about
a haunted palace and noticed how closely we tend to associate such ghost
stories with Victorian gothic arhitecture and style. Of course, one may
wonder about the significance of ghost stories more generally, but I suppose
that we all understand something of their cultural significance. But why are
ghost stories associated with gothic style? Again, is it any accident that
gothic style arose in the 19th century, continuing almost up to WWI? Keep in
mind that this was the dominant style of the time when Peirce lived and
wrote.

To get the review, go to the London Review of Books webpages, and search for

"architecture."

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Identity & Teridentity
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 08:48:49 -0500
X-Message-Number: 9

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

I&T. Note 24

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

Peter,

I still can't get into the Conceptual Graphs site you cited,
but I have read Sowa's 1984 to present work fairly closely,
so a minimal excerpt of the example that you have in mind
will probably be enough -- I'm guessing something of the
"blocks world" variety?

Meawhile, scanning trough the Standard Upper Ontology archives
has reminded me of many more of the most common fallacies that
pass for "conventional wisdom" of the non-Peircean variety due
to the decline in logical literacy brought on by the Principian
mould of reasoning -- I do not exaggerate -- Russell expressly
advised students of logic not to read classical sources, which
appears to be one of the reasons that so many of his students
fail to understand even the statements of the classic problems.

So I will attach an updated list of relation-theoretic fallacies and
downright infelicities. Not all of these are necessarily accounted
to John Collier -- after all, one person can only do so much!

You Can't Tell the Topics from the Fallacies without a Programme:

1. Topic. What is a relation?

1.1. Fallacy. The "I lied about the set" (ILATS) indirection.

2. Topic. What is a relation instance?
AKA: elementary relation, individual relation, tuple.

2.1. Fallacy. The "direct and full access to beings in themselves"
(DAFATBITS) fallacy.

2.2. Fallacy. The "change of variables" (COV) switcheroo.

3. Topic. What is the difference between a relation and one of its instances?

3.1. Fallacy. The "one row database" (ORD) illusion.

4. Topic. What are composition and decomposition, production and reduction?

4.1. Fallacy. The "2-ped dog" (2PD) dogma.
4.2. Fallacy. The "alchemist's dodge" (AD).
4.3. Fallacy. The "stone soup" (SS) parable.
4.4. Fallacy. The "who's on third" (WOT) joke.
AKA. "Truth-functions compute themselves".

Also in the mean time, here is
a helpful diagnostic hint for
the ILATS malady:

A relation is a set.

An immediate consequence of this familiar jingle
is that the analysis of a 3-adic relation into
a couple of 2-adic relations is the analysis
of a set into a couple of sets.

So what, exactly, in this context, is the analysis of set into a number of sets?
Well, it's not a question that many of our wannabe analysts often get as far as.
But all we need for the present's sake is the following:

Dx. Diagnostic Criterion.

A person who does not present the decomposition of a set into sets
is not presenting the decomposition of a relation into relations.

So far, John Collier has not presented the analysis of even a single relation.

To be continued ...

Jon Awbrey

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

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

Subject: Re: Late Gothic Architecture
From: William Thomas Sherman <gunjones1[…]earthlink.net>
Date: Mon, 25 Nov 2002 06:44:26 -0800
X-Message-Number: 10

The Gothic style largely dominated Christian art through most of the history of
Christianity, and such architectural and cultural locations became places for
spirits of the departed to congregate: either as a way of associating themselves
with or antagonizing the church or church related society.

Strange as this may sound, I have seen real ghosts and spoken to real spirit
people (in literal conversation), but as I have been previously forbidden by the
list to speak on the subject I can say no more. If, however, Howard you are
curious I would be happy to send you (as it presently stands) my treatise on
Hell.

William



HGCALLAWAY[…]aol.com wrote:

> Peirce-l,
>
> I want to recommend an article which I recently read from the London Review
> of Books, Vol. 24, No. 20 dated 17 October 2002, by Andrew Saint, titled
> "The Danger of Giving In:" A fascinating and curious review of _An Architect
> of Promise: George Gilbert Scott Jr (1839-97) and the Late Gothic Revival_ by
> Gavin Stamp, Shaun Tyas, 427 pp, #49.50.
>
> I have often wondered about 19th century Gothic and the meaning it had to
> those who adopted that style. This review suggested some answers, however
> indirectly, though I cannot say that that was the objective of the review.
> Some of what you find in the review may strike you a bit strange, but it is
> worth reading through.
>
> I was reminded of the article last night, when I watched a recent movie about
> a haunted palace and noticed how closely we tend to associate such ghost
> stories with Victorian gothic arhitecture and style. Of course, one may
> wonder about the significance of ghost stories more generally, but I suppose
> that we all understand something of their cultural significance. But why are
> ghost stories associated with gothic style? Again, is it any accident that
> gothic style arose in the 19th century, continuing almost up to WWI? Keep in
> mind that this was the dominant style of the time when Peirce lived and
> wrote.
>
> To get the review, go to the London Review of Books webpages, and search for
> "architecture."
>

--
William Thomas Sherman
1604 NW 70th St.
Seattle, WA 98117
206-784-1132
gunjones1[…]earthlink.net

http://www.angelfire.com/d20/htfh

"In the senses is deception and illusion, in the understanding is the source of
truth." ~ Xenophanes



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

Subject: Re: Identity & Teridentity
From: "Charles Pyle" <pyle[…]modempool.com>
Date: Mon, 25 Nov 2002 09:57:44 -0500
X-Message-Number: 11

Howard, and List

I hesitate to wade deeper into this discussion, because I don't understand
the fine points, e.g. what is at stake in distinguishing between analysis
and reduction, but I believe Peirce's view is precisely that to call
something a degenerate third is to say that it can be analyzed into seconds,
and to say that it is a genuine third is to say that it can't. In any case,
I would like to cite some quotes from Peirce that I think are relevant to
this discussion and might contribute usefully.

First, here is the passage about Philadelphia I was remembering, from
Collected Works:

3.367. We now come to thirds degenerate in the second degree. The dramatist
Marlowe had something of that character of diction in which Shakespeare and
Bacon agree. This is a trivial example; but the mode of relation is
important. In natural history, intermediate types serve to bring out the
resemblance between forms whose similarity might otherwise escape attention,
or not be duly appreciated. In portraiture, photographs mediate between the
original and the likeness. In science, a diagram or analogue of the observed
fact leads on to a further analogy. The relations of reason which go to the
formation of such a triple relation need not be all resemblances. Washington
was eminently free from the faults in which most great soldiers resemble one
another. A centaur is a mixture of a man and a horse. Philadelphia lies
between New York and Washington. Such thirds may be called intermediate
thirds or thirds of comparison.


And in the following paragraph note the relation between "a genuine three"
and "a triad cannot be analyzed into dyads"


3.363. But it will be asked, why stop at three? Why not go on to find a new
conception in four, five, and so on indefinitely? The reason is that while
it is impossible to form a genuine three by any modification of the pair,
without introducing something of a different nature from the unit and the
pair, four, five, and every higher number can be formed by mere
complications of threes. To make this clear, I will first show it in an
example. The fact that A presents B with a gift C, is a triple relation, and
as such cannot possibly be resolved into any combination of dual relations.
Indeed, the very idea of a combination involves that of thirdness, for a
combination is something which is what it is owing to the parts which it
brings into mutual relationship. But we may waive that consideration, and
still we cannot build up the fact that A presents C to B by any aggregate of
dual relations between A and B, B and C, and C and A. A may enrich B, B may
receive C, and A may part with C, and yet A need not necessarily give C to
B. For that, it would be necessary that these three dual relations should
not only coexist, but be welded into one fact. Thus we see that a triad
cannot be analyzed into dyads.

366. Among thirds, there are two degrees of degeneracy. The first is where
there is in the fact itself no Thirdness or mediation, but where there is
true duality; the second degree is where there is not even true Secondness
in the fact itself. Consider, first, the thirds degenerate in the first
degree. A pin fastens two things together by sticking through one and also
through the other: either might be annihilated, and the pin would continue
to stick through the one which remained. A mixture brings its ingredients
together by containing each. We may term these accidental thirds. "How did I
slay thy son?" asked the merchant, and the jinnee replied, "When thou
threwest away the date-stone, it smote my son, who was passing at the time,
on the breast, and he died forthright." Here there were two independent
facts, first that the merchant threw away the date-stone, and second that
the date-stone struck and killed the jinnee's son. Had it been aimed at him,
the case would have been different; for then there would have been a
relation of aiming which would have connected together the aimer, the thing
aimed, and the object aimed at, in one fact. What monstrous injustice and
inhumanity on the part of that jinnee to hold that poor merchant responsible
for such an accident! I remember how I wept at it, as I lay in my father's
arms and he first told me the story. It is certainly just that a man, even
though he had no evil intention, should be held responsible for the
immediate effects of his actions; but not for such as might result from them
in a sporadic case here and there, but only for such as might have been
guarded against by a reasonable rule of prudence. Nature herself often
supplies the place of the intention of a rational agent in making a
Thirdness genuine and not merely accidental; as when a spark, as third,
falling into a barrel of gunpowder, as first, causes an explosion, as
second. But how does nature do this? By virtue of an intelligible law
according to which she acts. If two forces are combined according to the
parallelogram of forces, their resultant is a real third. Yet any force may,
by the parallelogram of forces, be mathematically resolved into the sum of
two others, in an infinity of different ways. Such components, however, are
mere creations of the mind. What is the difference? As far as one isolated
event goes, there is none; the real forces are no more present in the
resultant than any components that the mathematician may imagine. But what
makes the real forces really there is the general law of nature which calls
for them, and not for any other components of the resultant. Thus,
intelligibility, or reason objectified, is what makes Thirdness genuine.

3.371. Let us now consider a triple character, say that A gives B to C.
This is not a mere congeries of dual characters. It is not enough to say
that A parts with C, and that B receives C. A synthesis of these two facts
must be made to bring them into a single fact; we must express that C, in
being parted with by A, is received by B.

Charles Pyle



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

Subject: Re: Late Gothic Architecture
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 10:22:22 EST
X-Message-Number: 12

William,

You wrote:

---quote----------
Strange as this may sound, I have seen real ghosts and spoken to real spirit
people (in literal conversation), but as I have been previously forbidden by
the list to speak on the subject I can say no more. If, however, Howard you
are curious I would be happy to send you (as it presently stands) my
treatise on Hell.
---end quote-------

Thanks but no thanks, William. I must say, however, that you have never
stopped amazing me.

Best wishes,

Howard

H.G. Callaway
(hgcallaway[…]aol.com)



(

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

Subject: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 10:24:23 -0500
X-Message-Number: 13

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

Subj: Reductions Among Relations
Date: Sat, 14 Apr 2001 19:48:55 -0400
From: Jon Awbrey <jawbrey[…]oakland.edu>
To: Stand Up Ontology <standard-upper-ontology[…]ieee.org>

One of the things that makes the general problem of RAR seem
just a little bit ill-defined is that you would have to survey
all of the conceivable ways of "getting new relations from old"
just to say what you mean by "L is reducible to {L_j : j in J}",
in other words, that if you had a set of "simpler" relations L_j,
for indices j in some set J, that this data would somehow fix
the original relation L that you are seeking to analyze,
to determine, to specify, to synthesize, or whatever.

In my experience, however, most people will eventually settle on
either one of two different notions of reducibility as capturing
what they have in mind, namely:

1. Compositive Reduction
2. Projective Reduction

As it happens, there is an interesting relationship between these
two notions of reducibility, the implications of which I am still
in the middle of studying, so I will try to treat them in tandem.

Jon Awbrey

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

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

Subject: Re: Jamesian Impasse
From: "Axel Schlotzhauer"
<axel.schlotzhauer[…]philosophie.uni-freiburg.de>
Date: Mon, 25 Nov 2002 16:28:00 +0100
X-Message-Number: 14

On Fri, 22 Nov 2002 12:05:41 -0800
"R. Jeffrey Grace" <rjgrace[…]yahoo.com> wrote:
> Axel,
>
> I'm leaving our exchange intact at the end of this
> message and am clipping the sections I respond to. As I
>read your responses, I'm beginning to think this is
>probably an issue which shows why Peirce renamed his
>philosophy "pragmaticism" (wasn't this before
> 1908?) when he saw what pragmaticism (correction:
>pragmatism) was becoming in the hands of James and Dewey.

Surely yes. As James approached the position of Dewey,
lauding his Logics (1898) although Peirce was the man in
these fields, whereas Dewey saught help from Bentley (1938)
for his logics, he drew a distinction by the word pragma-
ticism, although Dewey (1922) sees in Peirce the inventor
of the word pragmatism and such movement. Philosophers
believing in quarrels promoting the common end.

> While it may be true to say that James strove to arrive
> at a compromise position which would have made the
>various schools of pragmatism more uniform, did Peirce go
>along with it?
>

James and presumably also Peirce were against making
pragmatism uniform in a totalitarian younghegelian way
using only in the anarchistical Bakunist manner the dyadic,
separating form instead of the middling triadic form. For
them such operations were degenerate forms of dialectics,
as I remember well on this list as a position of Peirce.
As James published his "A pluralistic universe" (1908)
Dewey
and Mead were against such liberal views and also the
peaceful nonconflicting "socialistic" manners of James in
"The equivalent of war" (1906, 1910) giving the program to
the later Peace Corps as a deviation from a coming
american-
japanese conflict promoted by American generals already at
that time. Dewey and Mead saught the conflicting way for
a later pax americana won by economic means and on the
battlefield by superior technical means but also tried like
James to promote a socialism in America. James and presuma-
bly also Peirce were to liberal and appeasing for them.

> Axel wrote:
> ---
> No, Cunga claims the Aquinas having made already such
> fault of separating active and contemplative christendom
>in his theology and oikunomeia forgetting the christian
>unity of both. It's not James having made such separation.
>I put in my posting a certain difference seen by
> the prostestant- catholic schism accusing the catholic
> monks and priests of contemplative laziness separated
from >and not working in this world.<snip>
> ---
>
> Jeff responds:
> Well, I think the same accusation arises amongst Catholic
> theologians these days as well, so it's not a
> protestant/catholic issue. LaCunga was Catholic, for >
instance.

I agree, but the ideological arguments were of such content
as schism arose. Similarily the pragmatist schism for more
direct political and social engagement on the Mead / Dewey
wing.

>I think the point of Levering's article was
> that her claim was based upon a Jamesian influence.
> Levering supports it in the following quote (from pp.
>410-411 of the article):
>
>
> The Jamesian impulse in LaCugna's Trinitarian theology is
> striking. This is not to say that LaCugna's work is at >
> >odds with the majority of recent European Trinitarian
>theologies. As we have seen, both William James and
> Emerson drew heavily upon European intellectual
> movements, largely Schleiermachian and Hegelian, in
>constructing their American account of
> religious pragmatism. Yet, LaCugna's work echoes James in
> profound, if not always uniquely American, ways. Like
>James, she finds the roots of arid Christian faith in the
>"metaphysical monster" (to recall James's
> phrase) set up by those who sought to identify
>
>
> God's attributes by the steps of metaphysical reasoning.
> Like James, she critiques the entire tradition of
>Christian intellectual argumentation about the triune God.
>Like James, her central question is what practical
> import the doctrine of the triune God can be shown to
> have in the lives of Christians. Theological expression
>about the God of Jesus Christ should have for its goal
>exposing "God for us," the God we experience in
> and through salvation history, thereby impressing upon us
> the religious feelings and practical actions that flow
>from a proper encounter with the relational "God for us."
>As in James--a point emphasized in regard to James's work
>by Stanley Hauerwas in his Gifford Lectures--these
> feelings and practical actions bear a striking
> resemblance to the liberal democratic norms prevalent in
>mainstream Western intellectual culture today. (Matthew
>Levering, "Beyond the Jamesian Impasse in
> Trinitarian Theology", The Thomist, Vol. 66 No. 3 July
> 2002, pp. 410-411.)
>
Well, the "Dio lo vult" (God wishes it) of the crusades
and Spanish inquisition has similar traits like the prote-
stant "God for us" blessing the weapons in his German
edition. American patriotism which James tried to civilize
from brute militarist forms in his "The moral equivalent
of war (1906)" by socially working for example in normal
common projects like working in mines as a possibility of
such task bringing the classes together instead of
separating them undermined in a certain way such "God for
us." I don't think "God for us" was the real position of
James as he rather openly opposed militarist American
patriotism and general Shea in his article. The position
of Peirce I don't know.

> Axel wrote:
> ---
> <snip>
> But similarily such accusation is made by
> young-hegelianism as a basis
> of American pragmatism against the Schleiermachian wing
> of German
> idealist philosophy being socially lazy and not engaged
> in the real
> world of the working class by the pietistically only
> contemplative
> church services outside reality based solely on pious
> feelings. Bruno
> Bauer and Ludwig Feuerbach already accused the prote-
> stantism of 1840
> of such attitudes and Karl Marx especially together with
> Friedrich
> Engels accused the new pietist Prussian king together
> with Bruno Bruno
> being a "Holy family" only contemplating on Christ and
> not looking for
> the social necessities of their time. You see this way
> Bruno Bauer was
> made dialectically against truth a sort of Schleiermacher
> although Bauer
> critisized Schleiermacher.
>
> But such dialectical operations were not made by James
> but in the schism
> of pragmatism 1908 by Dewey and Mead although they kept"
> dialectically
> the Jamesian heritage and later accepted on a logical
> basis the Peircean
> one although Peirce also was severed in the said
> pragmatist schism from
> the Dewey and Mead line as an "objective, absolute
> idealist".
> <snip>
> ---
>
> Jeff responds:
> I think Levering is tracing the difference back to it's
> roots, namely the rejction by James of the metaphysical
>foundations of contemplation.

Did he really? I think his main contribution was to
research
religious experience also by the laboratory method and
the physiological components. Metaphysics were not rejected
also by Peirce.

> By the time you see these various schools you mention,
> they have
> replaced those foundations with a metaphysic that is
> antithetical to the
> goal of contemplation.

I agree, with the exception of the esthetics of Peirce and
James. If the musing of Peirce is contemplation is the
question, more amusing by times as I said in the snipped
parts. The "sharing" of experience and feelings in groups
following Lewin I think by Dewey (Art as experience 1925
The nature of esthetical experience (Mead 1925) is not
equivalent especially for the philosophical operations of
metaphysics basing on Aristoteles etc.


What they have in common with
> James is a
> rejection of the metaphysics that serves as a foundation
> for
> contemplation. Levering argues that Schleirmacher
> understood this
> metaphysic as "secondary products" which are at best
> "attempts to
> express religious feelings" (pg. 402) and that James
> accepts this theory
> and even expands it beyond metaphysical claims to
> encompass "..the whole
> variety of religious expression" (pg. 402).
>
Really? Seems correct claiming an Jamesian impasse in this
case.

> In other words, argues Levering, James doesn't accept the
> intellectual
> seriousness of theology, since he sees metaphysical
> accounts as
>
> "...nothing more than meaningless words, quite cut off
> from anything
> relevant to a religious person. These abstractions,
> James suggests, are
> even demonic- 'they have the trail of the serpent over
> them' insofar as
> they serve as substitutes for anything worthy of worship
> and religious
> feeling. He concludes 'So much for the metaphysical
> aspects of God!
> From the point of view of practical religion, the
> metaphysical monster
> which they offer to our worship is an absolutely
> worthless invention of
> the scholarly mind!' (This quote from James is from The
> Varieties of
> Religious Experience) (Levering, pg. 403).
>
> It seems to me that Levering has a pretty good argument!
> Anyway, I
> thought this topic would be of interest on the Peirce
> list because I'm
> convinced that this is a criticial difference between
> Perice and James.
>
>
> Pax...
>
> ---
> R. Jeffrey Grace
> rjgrace[…]pobox.com
> http://www.rjgrace.com <http://www.rjgrace.com/>
>
Well, it seems Levering made some good points although the
nearer context of his citations should be researched and
gives perhaps a different color. I don't think, that James
was really against theologian metaphysics in a pure form.
As Christian Science with its openness for scientific
criticism was a sound form of christianity for him, he was
not against metaphysical kernels like this form much
influenced by theosophy and its hinduist chakras.

Axel

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

Subject: Re: Morality and Courage in Science
From: William Thomas Sherman <gunjones1[…]earthlink.net>
Date: Mon, 25 Nov 2002 07:42:06 -0800
X-Message-Number: 15

Howard,

As you prefer. But this brings to mind my deeply held belief that true science
requires dispassioned objectivity, honesty and courage. Science without these is
feeble science or no science at all. It is easy enough to talk of the desire for
scientific truth, and some are even capable of rationally understanding
phenomena. Yet some scientific matters are such that without courage, such as
that of a soldier in battle willing to give his life, no progress can be made at
all. It comes as no surprise then that the timid should say that such inquiry is
unnecessary, or to deride such inquiry as superfluous. History is replete with
instances of such.

William



HGCALLAWAY[…]aol.com wrote:

> William,
>
> You wrote:
>
> ---quote----------
> Strange as this may sound, I have seen real ghosts and spoken to real spirit
> people (in literal conversation), but as I have been previously forbidden by
> the list to speak on the subject I can say no more. If, however, Howard you
> are curious I would be happy to send you (as it presently stands) my
> treatise on Hell.
> ---end quote-------
>
> Thanks but no thanks, William. I must say, however, that you have never
> stopped amazing me.

--
William Thomas Sherman
1604 NW 70th St.
Seattle, WA 98117
206-784-1132
gunjones1[…]earthlink.net

http://www.angelfire.com/d20/htfh

"In the senses is deception and illusion, in the understanding is the source of
truth." ~ Xenophanes



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

Subject: Re: Identity & Teridentity
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 10:40:01 -0500
X-Message-Number: 16

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

I found a few caches of ideas that may be pertinent here,
notes that I posted to the OCA and SUO Lists last year.
I haven't even looked at all of these yet, as I intend
to expand, revise, and rewrite them more clearly, and
with a more especial reference to Peirce -- on the
SUO List one could not even mention Peirce's name
without being subject to robo-filtering and
a general hue and cry. Luckily that only
occasionally happens on the Peirce List.

Jon Awbrey

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

Detached Ideas On Virally Important Topics

01. http://suo.ieee.org/ontology/msg01716.html
02. http://suo.ieee.org/ontology/msg01722.html

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

Inquiry Into Irreducibility

01. http://suo.ieee.org/ontology/msg03163.html
02. http://suo.ieee.org/ontology/msg03164.html
03. http://suo.ieee.org/ontology/msg03165.html
04. http://suo.ieee.org/ontology/msg03166.html
05. http://suo.ieee.org/ontology/msg03167.html
06. http://suo.ieee.org/ontology/msg03168.html

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

Reductions Among Relations

01. http://suo.ieee.org/ontology/msg01727.html
02. http://suo.ieee.org/ontology/msg01738.html
03. http://suo.ieee.org/ontology/msg01747.html
04. http://suo.ieee.org/ontology/msg01766.html
05. http://suo.ieee.org/ontology/msg01818.html
06. http://suo.ieee.org/ontology/msg01821.html
07. http://suo.ieee.org/ontology/msg02167.html
08. http://suo.ieee.org/ontology/msg02475.html

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

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

Subject: Re: Identity & Teridentity
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 10:55:39 EST
X-Message-Number: 17

Charles & list,

You wrote, Charles,

----quote------------
I hesitate to wade deeper into this discussion, because I don't understand
the fine points, e.g. what is at stake in distinguishing between analysis and
reduction, but I believe Peirce's view is precisely that to call something a
degenerate third is to say that it can be analyzed into seconds, and to say
that it is a genuine third is to say that it can't. In any case, I would like
to cite some quotes from Peirce that I think are relevant to this discussion
and might contribute usefully.
----end quote--------

Thanks for the valuable quotations, Charles. I have not reproduced them here,
but I want to recommend them to the readers of the list. Of particular
interest, I think is Peirce's paradigm of "A gives B to C." That one would
possibly be worth some attention, since if someone could give a plausible
analysis, then we might see more clearly what is at stake in the Peircean
irreducibility of genuine triadic relations. Part of the advntage is, of
course, that it seems a relatively simple example, and we all seem to
understand it adequately. This is an advantage particularly in contrast with
something like "representation."

Interestingly, perhaps, I have sometimes observed some differences in
connection with related practices and customs. Some folks make a much smaller
distinction between "giving" and "lending" or even perhaps "transfering
possession." If C gets B from A, that is to say, the intention of the letting
have may be regarded as relatively unimportant --perhaps because it is
thought that the awkawrdness of asking to have B back is the deciding
consideration in whether C can keep B. Again, having to ask to have B back
may be thought to be a very lowly sort of thing to do, so that what is lent
is not regarded as something one will likely ever have back.

Possession, we hear, is nine tenth of the law. So, if there is a lack of
concern for intentions, or intentions are regarded as something finally
subject to social media-tion, then it may be felt that those with better
social connection will always be able to have such things decided in their
own interest. Again, the distinction between giving and transfering
possession may then be considerably diminished. I recall in this connection
the story about Peirce, one time under some considerable social duress,
offering someone all of his books. He no doubt thought to evoke some sense of
guilt, though nothing similar happened. Did he give away his books? Or was he
merely attempting to produce particular feelings and more reasonable actions?
The role of intention is pretty central in our ordinary conception of giving,
but the intentions of people are sometimes simply ignored. Can we imagine a
society in which there is no such thing as giving? Or, can we perhaps imagine
a society in which giving is only possible for the powerful? If so, then we
might have no need for a conception of giving, in such a society, which can
not be reduced to something less refined. It will perhaps be thought that I
have simply changed the example.

These reflections do not answer your questions, I'm afriad. It is more that I
am reflecting on the plausibilities of reductionism. I do think it is an
excellent example.
Regarding the distinction between analysis and reduction, I think I stated
this most clearly in connection with identity and teridentity --the example I
explained to Joe, connected with the analysis of a newly created simple
organism.

Thanks again for the quotations.

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 11:01:01 -0500
X-Message-Number: 18

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

RAR. Note 2

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

Projective Reduction of Relations in General

I will start out with the notion of "projective reduction"
of relations, in part because it is easier and much more
intuitive (in the visual sense), but also because there
are a number of tools that we need for the other brand
of reduction that arise quite naturally as a part of
the projective setting.

Before we get into the operational machinery and the
vocational vocabulary of it all, let me just suggest
that you keep in mind the following style of picture,
which pretty much says it all, in that reducing to a
unity the "motley of the ten thousand terms" (MOT^4)
manner that the aptest genres and the fittest motifs
of representations are genreally found to immanifest.

Picture a k-adic relation L as a body
that resides in k-dimensional space X.
If the dimensions are X_1, ..., X_k,
then the "extension" of L, an object
that I will, for the whole time that
I am working this "extensional" vein,
regard as tantamount to the relation
itself, is a subset of the cartesian
product space X = X_1 x ... x X_k.

If you pick out your favorite family F of domains among these
dimensions, say, X_F = {X_j : j in F}, then the "projection" of
a point x of X on the subspace that gets "generated" along these
dimensions of X_F can be denoted by the notation "Proj_F (x)".

By extension, the projection of any relation L on that subspace
is denoted by "Proj_F (L)", or still more simply, by "Proj_F L".

The question of "projective reduction" for k-adic relations
can be stated with moderate generality in the following way:

| Given a set of k-place relations in the same space X and
| a set of projections from X to the associated subspaces,
| do the projections afford sufficient data to tell the
| different relations apart?

Next time, in order to make this discussion more concrete,
I will focus on some examples of triadic relations. In fact,
to start within the bounds of no doubt familiar examples by now,
I will begin with the examples of sign relations that I used before.

http://suo.ieee.org/email/msg00729.html
http://suo.ieee.org/email/msg01224.html

Jon Awbrey

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

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

Subject: Re: Morality and Courage in Science
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 11:04:06 EST
X-Message-Number: 19

William,

I must say, that I entirely agree with what you say in the quotation below:

----quote--------
As you prefer. But this brings to mind my deeply held belief that true
science re-quires dispassioned objectivity, honesty and courage. Science
without these is
feeble science or no science at all. It is easy enough to talk of the desire
for
scientific truth, and some are even capable of rationally understanding
phenomena. Yet some scientific matters are such that without courage, such as
that of a soldier in battle willing to give his life, no progress can be made
at all. It comes as no sur-prise then that the timid should say that such
inquiry is unnecessary, or to deride such inquiry as superfluous. History is
replete with instances of such.
----end quote------

Again, I have no dispute with any of this. Still, all of this may leave
considerable room for differences concerning particular cases. Right? That is
why it may strike some as pretty much a matter of empty words. (Not that I
think so --it is worth saying.) Thanks.

Best wishes,

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 11:11:02 -0500
X-Message-Number: 20

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

RAR. Note 3

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

Projective Reduction of Triadic Relations

We are ready to take up the question of whether
3-adic relations, in general, and in particular
cases, are "determined by", "reducible to", or
"reconstructible from" their 2-adic projections.

Suppose that L contained in XxYxZ is an arbitrary 3-adic relation,
and consider the three 2-adic relations that are gotten by taking
its projections, its "shadows", if you will, on each of the three
planes XY, XZ, YZ. Using the notation that I introduced before,
and compressing it just a bit or two in passing, one can write
these projections in each of the following ways, depending on
which turns out to be most convenient in a given context:

1. Proj_{X,Y} (L) = Proj_{1,2} (L) = Proj_XY L = Proj_12 L = L_XY = L_12.

2. Proj_{X,Z} (L) = Proj_{1,3} (L) = Proj_XZ L = Proj_13 L = L_XZ = L_13.

3. Proj_{Y,Z} (L) = Proj_{2,3} (L) = Proj_YZ L = Proj_23 L = L_YZ = L_23.

If you picture the relation L as a body in the 3-space XYZ, then
the issue of whether L is "reducible to" or "reconstuctible from"
its 2-adic projections is just the question of whether these three
projections, "shadows", or "2-faces" determine the body L uniquely.

Stating the matter the other way around, L is "not reducible to"
or "not reconstructible from" its 2-dim projections if & only if
there are two distinct relations L and L' which have exactly the
same projections on exactly the same planes.

The next series of Tables illustrates the projection operations
by means of their actions on the sign relations L(A) and L(B)
that I introduced earlier on, in the "Sign Relations" thread.
Recall that we had the following set-up:

| L(A) and L(B) are "contained in" or "subsets of" OxSxI:
|
| O = {A, B},
|
| S = {"A", "B", "i", "u"},
|
| I = {"A", "B", "i", "u"}.

| L(A) has the following eight triples
| of the form <o, s, i> in OxSxI:
|
| <A, "A", "A">
| <A, "A", "i">
| <A, "i", "A">
| <A, "i", "i">
| <B, "B", "B">
| <B, "B", "u">
| <B, "u", "B">
| <B, "u", "u">

| L(B) has the following eight triples
| of the form <o, s, i> in OxSxI:
|
| <A, "A", "A">
| <A, "A", "u">
| <A, "u", "A">
| <A, "u", "u">
| <B, "B", "B">
| <B, "B", "i">
| <B, "i", "B">
| <B, "i", "i">

Taking the 2-adic projections of L(A)
we obtain the following set of data:

| L(A)_OS has these four pairs
| of the form <o, s> in OxS:
|
| <A, "A">
| <A, "i">
| <B, "B">
| <B, "u">

| L(A)_OI has these four pairs
| of the form <o, i> in OxI:
|
| <A, "A">
| <A, "i">
| <B, "B">
| <B, "u">

| L(A)_SI has these eight pairs
| of the form <s, i> in SxI:
|
| <"A", "A">
| <"A", "i">
| <"i", "A">
| <"i", "i">
| <"B", "B">
| <"B", "u">
| <"u", "B">
| <"u", "u">

Taking the dyadic projections of L(B)
we obtain the following set of data:

| L(B)_OS has these four pairs
| of the form <o, s> in OxS:
|
| <A, "A">
| <A, "u">
| <B, "B">
| <B, "i">

| L(B)_OI has these four pairs
| of the form <o, i> in OxI:
|
| <A, "A">
| <A, "u">
| <B, "B">
| <B, "i">

| L(B)_SI has these eight pairs
| of the form <s, i> in SxI:
|
| <"A", "A">
| <"A", "u">
| <"u", "A">
| <"u", "u">
| <"B", "B">
| <"B", "i">
| <"i", "B">
| <"i", "i">

A comparison of the corresponding projections for L(A) and L(B)
reveals that the distinction between these two 3-adic relations
is preserved under the operation that takes the full collection
of 2-adic projections into consideration, and this circumstance
allows one to say that this much information, that is, enough to
tell L(A) and L(B) apart, can be derived from their 2-adic faces.

However, in order to say that a 3-adic relation L on OxSxI
is "reducible" or "reconstructible" in the 2-dim projective
sense, it is necessary to show that no distinct L' on OxSxI
exists such that L and L' have the same set of projections,
and this can take a rather more exhaustive or comprehensive
investigation of the space of possible relations on OxSxI.

As it happens, each of the relations L(A) and L(B) turns
out to be uniquely determined by its 2-dim projections.
This can be seen as follows. Consider any coordinate
position <s, i> in the plane SxI. If <s, i> is not
in L_SI then there can be no element <o, s, i> in L,
therefore we may restrict our attention to positions
<s, i> in L_SI, knowing that there exist at least
|L_SI| = Cardinality of L_SI = eight elements in L,
and seeking only to determine what objects o exist
such that <o, s, i> is an element in the objective
"fiber" of <s, i>. In other words, for what o in O
is <o, s, i> in ((Proj_SI)^(-1))(<s, i>)? Now, the
circumstance that L_OS has exactly one element <o, s>
for each coordinate s in S and that L_OI has exactly
one element <o, i> for each coordinate i in I, plus
the "coincidence" of it being the same o at any one
choice for <s, i>, tells us that L has just the one
element <o, s, i> over each point of SxI. Together,
this proves that both L(A) and L(B) are reducible in
an informative sense to 3-tuples of 2-adic relations,
that is, they are "projectively 2-adically reducible".

Next time I will give examples of 3-adic relations
that are not reducible to or reconstructible from
their 2-adic projections.

Jon Awbrey

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

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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 11:48:02 -0500
X-Message-Number: 21

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

RAR. Note 4

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

Projective Reduction of Triadic Relations (cont.)

There are a number of preliminary matters that
will need to be addressed before I can proceed.

Last time I gave two cases of 3-adic (or triadic) relations
with projective reductions to 2-adic (or dyadic) relations,
by name, "triadics reducible over projections" (TROP's) or
"triadics reconstructible out of projections" (TROOP's).

Still, one needs to be very careful and hedgey about saying,
even in the presence of such cases, that "all is dyadicity".
I will make some attempt to explain why in the next episode,
and then I will take up examples of 3-adics that happen to
be irreducible in this sense, in effect, that are not able
to be recovered uniquely from their 2-adic projection data.
Call them "triadics irreducible over projections" (TRIOP's)
or "projectively irreducible triadics" (PIT's).

In the story of A and B, it appears to be the case
that that the triadic relations L(A) and L(B) are
distinguished from each other, and what's more,
distinguished from all of the other relations
in the garden of OSI, for the same O, S, I.

At least, so says I and my purported proof.
I am so suspicious of this result myself that
I will probably not really believe it for a while,
until I have revisited the problem and the "proof"
a few times, to see if I can punch any holes in it.

But let it pass for proven for now,
and let my feeble faith go for now.

For the sake of a more balanced account,
it's time to see if we can dig up any cases
of "projectively irreducible triadics" (PIT's).
Any such PIT relation, should we ever fall into one,
is bound to occasion another, since it is a porismatic
part of the definition that a 3-adic relation L is a PIT
if and only if there exists a distinct 3-adic relation L'
such that the 2-adic faces of L and L' are indiscernible.
In this event, then both L and L' fall into the de-genre
of PIT's together.

Well, PIT's are not far to find, once you think to look for them --
indeed, the landscape of "formal or mathematical existence" (FOME)
is both figuratively and litterally rife with them!

What follows is the account of a couple,
that I will dub "L_0" and "L_1".

But first, even though the question of projective reduction
has to do with 3-adic relations as a general class, and is
thus independent of their potential use as sign relations,
it behooves us to consider the bearing of these reduction
properties on the topics of interest to us for the sake
of communication and representation via sign relations.

| Nota Bene. On the Variety and Reading of Subset Notations.
|
| Let any of the locutions, L c XxYxZ, L on XxYxZ, L sub XxYxZ,
| substitute for the peculiar style of "in-line" or "in-passing"
| reference to subsethood that has become idiomatic in mathematics,
| and that would otherwise use the symbol that has been customary
| since the time of Peano to denote "contained in" or "subset of".

Most likely, any triadic relation L on XxYxZ that is imposed on
the arbitrary domains X, Y, Z could find use as a sign relation,
provided that it embodies any constraint at all, in other words,
so long as it forms a proper subset L of the entire space XxYxZ.
But these sorts of uses of triadic relations are not guaranteed
to capture or constitute any natural examples of sign relations.

Jon Awbrey

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

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

Subject: Re: Late Gothic Architecture
From: "Joseph Ransdell" <joseph.ransdell[…]yahoo.com>
Date: Mon, 25 Nov 2002 10:50:33 -0600
X-Message-Number: 22

William Sherman says:

> Strange as this may sound, I have seen real ghosts and spoken to real
spirit
> people (in literal conversation), but as I have been previously forbidden
by the
> list to speak on the subject I can say no more.


I don't recall forbidding you, qua manager of peirce-l, to speak on the
subject here unless it was in a context in which the force of this was to
forbid your doing so with no basis for it being regarded as Peirce-related,
which is the basic criterion of relevance in determining what is permissable
or not permissable to post here. That conditional constraint still obtains,
of course.

Joe Ransdell -- speaking as manager of peirce-l



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

Subject: Re: Identity & Teridentity
From: "Peter Brawley" <peter.brawley[…]artfulsoftware.com>
Date: Mon, 25 Nov 2002 09:13:09 -0800
X-Message-Number: 23

<snip>
<snip>

> And in the following paragraph note the relation between "a genuine three"
> and "a triad cannot be analyzed into dyads"
>
> 3.363. But it will be asked, why stop at three? Why not go on to find a
new
> conception in four, five, and so on indefinitely? The reason is that while
> it is impossible to form a genuine three by any modification of the pair,
> without introducing something of a different nature from the unit and the
> pair, four, five, and every higher number can be formed by mere
> complications of threes. To make this clear, I will first show it in an
> example. The fact that A presents B with a gift C, is a triple relation,
and
> as such cannot possibly be resolved into any combination of dual
relations.
> Indeed, the very idea of a combination involves that of thirdness, for a
> combination is something which is what it is owing to the parts which it
> brings into mutual relationship. But we may waive that consideration, and
> still we cannot build up the fact that A presents C to B by any aggregate
of
> dual relations between A and B, B and C, and C and A. A may enrich B, B
may
> receive C, and A may part with C, and yet A need not necessarily give C to
> B. For that, it would be necessary that these three dual relations should
> not only coexist, but be welded into one fact. Thus we see that a triad
> cannot be analyzed into dyads.

For some n-ary relations, n has a minimum of 3, eg 'A gave B to C'. That is
one proposition.

That some n-ary relations, where n=3, cannot be decomposed is a second
proposition, and obviously does not follow from n=3. Perhaps it follows from
something else, or perhaps Peirce thought it was intuitively obvious.

But database programmers think they decompose triadic relations like 'A gave
B to C' into binary relations every day, eg (i) A gave to C, (ii) the gift
was B. They think this because their relational language admits only dyadic
relations, and it works beautifully at decomposing n-ary relations where the
practical limit of n is 16 or even 32.

PB



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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 12:24:23 -0500
X-Message-Number: 24

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

RAR. Note 5

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

Projective Reduction of Triadic Relations (cont.)

Projectively Irreducible Triadic Relations, or
Triadic Relations Irreducible Over Projections:

In order to show what a projectively irreducible 3-adic relation
looks like, I now present a pair of 3-adic relations that have the
same 2-adic projections, and thus cannot be distinguished from each
other on the basis of this data alone. As it happens, these examples
of triadic relations can be discussed independently of sign relational
concerns, but structures of their basic ilk are frequently found arising
in signal-theoretic applications, and they are no doubt keenly associated
with questions of redundant coding and therefore of reliable communication.

Consider the triadic relations L_0 and L_1
that are specified in the following set-up:

| B = {0, 1}, with the "+" signifying addition mod 2,
| analogous to the "exclusive-or" operation in logic.
|
| B^k = {<x_1, ..., x_k> : x_j in B for j = 1 to k}.

In what follows, the space XxYxZ is isomorphic to BxBxB = B^3.
For lack of a good isomorphism symbol, I will often resort to
writing things like "XxYxZ iso BxBxB" or even "XxYxZ ~=~ B^3".

| Relation L_0
|
| L_0 = {<x, y, z> in B^3 : x + y + z = 0}.
|
| L_0 has the following four triples
| of the form <x, y, z> in B^3:
|
| <0, 0, 0>
| <0, 1, 1>
| <1, 0, 1>
| <1, 1, 0>

| Relation L_1
|
| L_1 = {<x, y, z> in B^3 : x + y + z = 1}.
|
| L_1 has the following four triples
| of the form <x, y, z> in B^3:
|
| <0, 0, 1>
| <0, 1, 0>
| <1, 0, 0>
| <1, 1, 1>

Those are the relations,
here are the projections:

Taking the dyadic projections of L_0
we obtain the following set of data:

| (L_0)_XY has these four pairs
| of the form <x, y> in XxY:
|
| <0, 0>
| <0, 1>
| <1, 0>
| <1, 1>

| (L_0)_XZ has these four pairs
| of the form <x, z> in XxZ:
|
| <0, 0>
| <0, 1>
| <1, 1>
| <1, 0>

| (L_0)_YZ has these four pairs
| of the form <y, z> in YxZ:
|
| <0, 0>
| <1, 1>
| <0, 1>
| <1, 0>

Taking the dyadic projections of L_1
we obtain the following set of data:

| (L_1)_XY has these four pairs
| of the form <x, y> in XxY:
|
| <0, 0>
| <0, 1>
| <1, 0>
| <1, 1>

| (L_1)_XZ has these four pairs
| of the form <x, z> in XxZ:
|
| <0, 1>
| <0, 0>
| <1, 0>
| <1, 1>

| (L_1)_YZ has these four pairs
| of the form <y, z> in YxZ:
|
| <0, 1>
| <1, 0>
| <0, 0>
| <1, 1>

Now, for ease of verifying the data, I have written
these sets of pairs in the order that they fell out
on being projected from the given triadic relations.
But, of course, as sets, their order is irrelevant,
and it is simply a matter of a tedious check to
see that both L_0 and L_1 have exactly the same
projections on each of the corresponding planes.

To summarize:

The relations L_0, L_1 sub B^3 are defined by the following equations,
with algebraic operations taking place as in the "Galois Field" GF(2),
that is, with 1 + 1 = 0.

1. The triple <x, y, z> in B^3 belongs to L_0 iff x + y + z = 0.
L_0 is the set of even-parity bit-vectors, with x + y = z.

2. The triple <x, y, z> in B^3 belongs to L_1 iff x + y + z = 1.
L_1 is the set of odd-parity bit-vectors, with x + y = z + 1.

The corresponding projections of L_0 and L_1 are identical.
In fact, all six projections, taken at the level of logical
abstraction, constitute precisely the same dyadic relation,
isomorphic to the whole of BxB and expressible by means of
the universal constant proposition 1 : BxB -> B. In sum:

1. (L_0)_XY = (L_1)_XY = 1_XY = BxB = B^2,

2. (L_0)_XZ = (L_1)_XZ = 1_XZ = BxB = B^2,

3. (L_0)_YZ = (L_1)_YZ = 1_YZ = BxB = B^2.

Therefore, L_0 and L_1 form an indiscernible couplet
of "triadic relations irreducible over projections"
or "projectively irreducible triadic relations".

Jon Awbrey

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

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

Subject: Re: Late Gothic Architecture
From: HGCALLAWAY[…]aol.com
Date: Mon, 25 Nov 2002 12:28:55 EST
X-Message-Number: 25

Peirce-l,

A bit more on the topic of late gothic architecture.

Part of the point of ghost stories, of course, is to emphasize the difference
between those departed who are at rest and those who are not at rest. The
ghost stories focus on the latter. Those who are resting in peace do not
bother us, no matter how supersititious some may be. Instead among the
supersititious, the departed who are not resting in peace make all the
trouble --things that go bump in the night and all that. (I say this though I
am myself not the least supersititious.)

So it was in the ghost story I saw last night about a haunted gothic palace
--of obvious late 19th century vintage. The great owner and Lord of the
palace had held children as captives and none of these dead could rest until
the injustice had been addressed. To give a more realistic example, based on
more or less the same pschology, I recall a friend who told me of speaking to
his dead father for some years or so, until he had thought through all the
unsolved problems in his relation-ship to his father.

But why do we associate 19th century gothic architecture with ghost stories?
Why was this so pre-dominant as the style of the second half of the 19th
century --a time in America, which culminated in the "gilded age." Why of a
sudden, would the Americans, who had long prided themselves on simplicity,
suddenly turn to an architectural style of elaborate excess of decoration and
gloomy expression? What suggests it self to me, at least, is that there was
much to hide. Elaborate styles tend to be favored in display of power, and
the power of the gilded age was certainly capable of excesses. The more the
excesses of practice in the use of power, the greater the need for elaborate
expressions of power to keep the lid on things.

In some sense, these were exactly the excesses of power which culminated in
WWI. Think of the contrast between the styles and attitudes before WWI and
those which prevailed afterward. What was the objective of the sudden shift
back toward simplicity and obviousness of function? What was to be avoided by
such a shift?
Isn't this late gothic style in some sense the expression of the great Empire
builders?

Howard

H.G. Callaway
(hgcallaway[…]aol.com)

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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 12:34:01 -0500
X-Message-Number: 26

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

RAR. Note 6

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

Projective Reduction of Triadic Relations (concl.)

We have pursued the "projective analysis" of 3-adic relations,
tracing the pursuit via a ready quantity of concrete examples,
just far enough to arrive at this clearly demonstrable result:

| Some 3-adic relations are and
| some 3-adic relations are not
| uniquely reconstructible from,
| or informatively reducible to,
| their 2-adic projection data.

Onward!

Prospects for a Compositive Analysis of Triadic Relations

We now take up the "compositive analysis" of 3-adic relations,
coining a term to prevent confusion, like there's a chance in
the world of that, but still making use of nothing other than
that "standardly uptaken operation" of relational composition,
the one that constitutes the customary generalization of what
just about every formal, logical, mathematical community that
is known to the writer, anyway, dubs "functional composition".

Jon Awbrey

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

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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 12:48:04 -0500
X-Message-Number: 27

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

RAR. Note 7

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

Compositional Analysis of Relations in General

The first order of business under this heading is straightforward enough:
to define what is standardly described as the "composition of relations".
For the time being I limit the discussion to 2-adic and 3-adic relations.

| Remark on the Ancestry, the Application, and
| The Anticipated Broadening of these Concepts.
|
| This is basically the same operation that C.S. Peirce described as
| "relative multiplication", except for the technical distinction that
| he worked primarily with so-called "relative terms", like "lover of",
| "sign of the object ~~~ to", and "warrantor of the fact that ~~~ to",
| rather than with the kinds of extensional and intensional relations
| to which the majority of us are probably more accustomed to use.
|
| It is with regard to this special notion of "composition", and it alone,
| that I plan to discuss the inverse notion of "decomposition". I try to
| respect other people's "reserved words" as much as I can, even if I can
| only go so far as to take them at their words and their own definitions
| of them in forming my interpretation of what they are apparently saying.
| Therefore, if I want to speak about other ways of building up relations
| from other relations and different ways of breaking down relations into
| other relations, then I will try to think up other names for these ways,
| or revert to a generic usage of terms like "analysis" and "combination".
|
| When a generalized definition of "relational composition" has been given,
| and its specialization to 2-adic relations is duly noted, then one will
| be able to notice that it is simply an aspect of this definition that
| the composition of two 2-adic relations yields nothing more than yet
| another 2-adic relation. This will, I hope, in more than one sense
| of the word, bring "closure" to this issue, of what can be reduced
| to compositions of 2-adic relations, to wit, just 2-adic relations.

A notion of relational composition is to be defined that generalizes the
usual notion of functional composition. The "composition of functions"
is that by which -- composing functions "on the right", as they say --
f : X -> Y and g : Y -> Z yield the "composite function" fg : X -> Z.

Accordingly, the "composition" of dyadic relations is that by which --
composing them here by convention in the same left to right fashion --
P c X x Y and Q c Y x Z yield the "composite relation" PQ c X x Z.

There is a neat way of defining relational composition, one that
not only manifests its relationship to the projection operations
that go with any cartesian product space, but also suggests some
natural directions for generalizing relational composition beyond
the limits of the 2-adic case, and even beyond relations that have
any fixed arity, that is, to the general case of formal languages.
I often call this definition "Tarski's Trick", though it probably
goes back further than that. This is what I will take up next.

Jon Awbrey

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

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

Subject: Re: Identity & Teridentity
From: John Collier <ag659[…]ncf.ca>
Date: Mon, 25 Nov 2002 11:53:51 -0500
X-Message-Number: 28

At 02:06 AM 25/11/2002, Howard wrote:
>Joe,
>
>You wrote, in reply to John,
>
>----quote----------
>I thought the basic claim is that representation is irreducibly triadic,
>John. Peirce recogmizes any degree of adicity you like, if it is only a
>matter of verbal form. So I understand you to be saying that you don't
>regard representation as triadic, other than verbally so. I don't know what
>you would have in mind in saying that Peirce convinced you of its
>triadicity, though, since I don't recall him trying to make any claim for it
>that wasn't a claim to its irreducibility to a complex of dyads.
>----end quote-----------
>
>Here I would just like to interject the observation that we may need to
>consider "reducibility" and "open to analysis" as importantly distinct. For,
>suppose that representation is, as you say, "irreducibly triadic," if we give
>an analysis of representation, say in terms of other relations, it would not
>follow that we have reduced representation to something else.

This is the point of Rosen's distinction between analytic and synthetic
models. Analytic models do not necessarily produce a solution for
every possible instance, but synthetic models do (since they are
basically summations of instances, logically). So we could have
a full analytic model (that might have fully computable solutions
for some god-like computer, like Boolos' Zeus demon -- see
Boolos and Jeffreys -- but not for us poor finite beings, or for
any other finite being), but it is not reducible to models of its the
components of the system it models and their relations. Nonetheless,
it is common in science to call an analytic model a reduction. I prefer
to call it a reductive model, and leave "reduction" for the case in
which there are analytic models of the system in question that are
also synthetic models.

I would suggest that the distinction, applied to relations (and I am talking
here of the sort of real world cases that Peirce gives in the passages
Joe cites that give cases of degenerate and non-degenerate triads,
not the sort of abstract case that Jon considers -- which so far lacks
textual support in this discussion) is such that the degenerate cases are
the ones that have synthetic models, and thus can be decomposed
completely into _indepedendent_ components. This contemporary
version of Peirce's categories seems to me to line up pretty well
with the sort of cases he discusses, including the chirality case
and representations, but it also includes biological organisms.

John



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

Subject: Re: Identity & Teridentity
From: John Collier <ag659[…]ncf.ca>
Date: Mon, 25 Nov 2002 12:11:15 -0500
X-Message-Number: 29

At 02:25 AM 25/11/2002, you wrote:
>John & list,
>
>In amongst a long reply of much interest, you said something that struck my
>eye and which I am inclined to question.
>
>----quote----------
>So, returning to the issue of what Peirce meant, my best guess is that he
>means that interpretation is non-linear and does not have a synthetic model.
>He was mis-taken in thinking that there are logically irreducible triadic
>relations.
>---end quote------
>
>As I said to Joe, there is some danger in equating analysis of a relation
>with reduction of a relation. So, I would rather say that the question of
>whether a given relation is open to some interesting analysis is a question
>for inquiry, depending in degree on how the subject-matter may develop --it
>is not simply a matter of set theory. I think that Quine merely offers a
>mathematical model of reduction to a dyadic relation, which model will remain
>without broad non-mathematical interest in many cases. Whether we will justly
>regard some analysis offered as a "reduction" is a distinct question. In
>consequence, I see no sufficient grounds for thinking, as of yet, at least,
>that Peirce "was mistaken in thinking that there are logically irreducible
>triadic relations." This, as it seems to me, is a rather strong claim in the
>present context.

I agree with the first part. The second part depends on what we mean
by "logically reducible". Quine did this for relations (provided an
analytic model scheme for reducing relations to dyadic relations),
so I take it that this is possible. If I had been more cautious I would have
said "He was mistaken if he thought that there are logically irreducible
triadic relations." I am beginning to think, from looking over the passages
that Joe posted, that Peirce never thought this in the sense that Quine
refuted. For me, to give an analysis (in terms of necessary and sufficient
conditions)is to give a logical reduction. It could be taken, more moderately,
to merely give a precise identification. What is does, on either account,
is the same. That is the key issue, not the description. I am inclined
to agree that a mathematical reduction is not a reduction, and for this
reason I prefer to call analytic models "reductive" rather than reductions,
but the literature on reduction uses both, more often the latter. I think
this is partly a heritage of the view that reduction just is mathematical
reduction. This view was shown to be false by the structuralists (Balzer
in particular), after initial enthusiasm that the structuralist view of
theories
could solve problems about theory reduction. The possibility of reduction
depends heavily on pragmatic aspects of interpretation. Stegmuller originally
thought that a "super-super-Montague" could give an analytical pragmatics.
if this were so, then representation could be reduced (sensu synthetic model)
to the dual coupled dyad that I mentioned previously, and Peirce would be
wrong. Stegmuller thought that the limits were on our abilities at
formalization.
It can't be proven exactly (because if it could be it wouldn't be true), but
there are a lot of reasons now to think the interpretation cannot be so
reduced. Basically, the irreducibility of Thirds must be established by
abduction, as was noted by someone here recently, and as I think that
Peirce said, though I have this only second hand.

John


John


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

Subject: Re: Identity & Teridentity
From: John Collier <ag659[…]ncf.ca>
Date: Mon, 25 Nov 2002 12:43:50 -0500
X-Message-Number: 30

At 02:40 AM 25/11/2002, Howard wrote:
>Jon,
>
>You wrote:
>
>----quote-----------
>Again, Jon seems to be assuming the irreducibility of teridentity, which was,
>as I recall from the subject line, exactly the point that was at issue.
>---end quote------
>
>I don't want to address the issue of what Jon said, or didn't say. But I
>think you usefully focus on a point at issue in this thread. Let us represent
>teridentity for present purposes, in this fashion:
>
>Ixyz,
>
>which we regard as true of every ordered triple, such that x=y and y=z. It
>seems plausible to say that we can analyze "Ixyz" in standard notation by
>reference to "x=y & y=z." There seems to be some advantage, or possible
>advantage, to the analysis, since we have axioms for identity theory,
>including e.g., "(x) (x=x)" and the substitutivity principle --if x=y, then
>whatever is true or x is true of y and vice versa.
>
>Now, I am less certain that this is a reduction than I am that it is a pretty
>good analysis. To say that it is a reduction, it seems, is just to say that
>once we under-stand the analysis, we see no further need of teridentity. That
>result I also regard as plausible, but I think there is an extra step to it,
>which is not always so plausible, regarding other relations. For example,
>suppose that representation is triadic and that we are able to give some
>analysis of it which does not depend on something equally triadic. It would
>not follow, as I see the matter, that we have no need of a triadic conception
>of representation. Some additional argument would seem to be needed there.

This seems correct to me. I would stress that I don't think that the argument
can be exclusively logical or mathematical, for the reasons that I gave
previously. The required argument, as I think it was you who suggested,
would have to be to the hypothesis that no synthetic model of representation
is possible. This argument is open if the analysis is into components that
are not independent in the sense that no component contains information
about the others (or more exactly, there is no analysis that is
a synthetic model). Whether this corresponds to Peirce's view I leave
to the experts. As you pointed out previously, I overstepped what
I have evidence for when I said that Peirce did not accept the possibility
of an analytic model of representation (which I called a logical reduction).
I would suggest, though, that his desire to establish teridentity as required
shows at least some ambiguity on his part about this.

John



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

Subject: Re: Identity & Teridentity
From: John Collier <ag659[…]ncf.ca>
Date: Mon, 25 Nov 2002 12:48:44 -0500
X-Message-Number: 31

At 03:10 AM 25/11/2002, Howard wrote:
>( I have changed the salutation on this posting, please excuse the typo, John,
>---HGC)
>
>John,

No worries. I found myself accidentally signing myself as Jon on
a recent post. It doesn't reflect any confusion about my identity,
but an error in using a sign. There is a difference.

John


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

Subject: Re: Reductions Among Relations
From: John Collier <ag659[…]ncf.ca>
Date: Mon, 25 Nov 2002 13:10:51 -0500
X-Message-Number: 32

So, if you project a triadic relation onto a dyadic relation
you loose information. Why is that surprising?

John


At 12:24 PM 25/11/2002, Jon wrote:
>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
>RAR. Note 5
>
>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
[SEE MESSAGE 24 ABOVE]

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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 13:10:35 -0500
X-Message-Number: 33

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

RAR. Note 8

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

Compositional Analysis of Relations in General (cont.)

There are one or two confusions that demand to
be cleared up before I can proceed any further.

We had been enjoying our long-anticipated breakthrough on the
allegedly "easy case" of projective reduction, having detected
hidden within that story of our old friends and usual suspects
A and B two examples of 3-adic relations, L(A) and L(B), that
are indeed amenable, not only to being distinguished, one from
the other, between the two of them, but also to being uniquely
determined amongst all of their kin by the information that is
contained in their 2-dimensional projections. So far, so good.
Had I been thinking fast enough, I would have assigned these the
nomen "triadics reducible in projections over dyadics" (TRIPOD's).
Other good names: "triadics reducible over projections" (TROP's),
or perhaps "triadics reconstructible out of projections" (TROOP's).

Then we fell upon two examples of triadic relations, L_0 and L_1,
that I described as "projectively irreducible triadics" (PIT's),
because they collapse into an indistinct mass of non-descript
flatness on having their dyadic pictures taken. That acronym
does not always work for me, so I will give them the alias of
"triadics irreducible by projections over dyadics" (TIBPOD's),
or perhaps "triadics irreducible over projections" (TIOP's).

I'm not accustomed to putting much stock in my own proofs
until I can reflect on them for a suitable period of time,
or until some other people have been able to go over them,
but until that day comes I will just have to move forward
with these results as I presently see them.

In reply to my notes on these topics, Matthew West
has contributed the following pair of commentaries:

1. Regarding L(A) and L(B)

| Whilst I appreciate the academic support for showing
| that any triadic relation can be represented by some
| number of dyadic relations, the real point is to use
| this fact to seek for an improved analysis based on
| more fundamental concepts. It is not the objective
| to do something mechanical.

2. Regarding L_0 and L_1

| I don't think you have shown very much except that reducing
| triadic relations to dyadic relations using the mechanical
| process you have defined can loose information. I am not
| surprised by this. My experience of doing this with real,
| rather than abstract examples, is that there are often
| extra things to do.

So I need to clarify that what I think that I showed was
that "some" triadic relations are "reducible" in a given
informational sense to the information that is contained
in their dyadic projections, e.g., as L(A) and L(B) were,
but that others are not reducible in this particular way,
e.g., as L_0 and L_1 were not.

Now, aside from this, I think that Matthew is raising
a very important issue here, which I personally grasp
in terms of two different ways of losing information,
namely:

1. The information that we lose in forming a trial model,
in effect, in going from the unformalized "real world"
over to the formal context or the syntactic medium in
which models are constrained to live out their lives.

2. The information that we lose in "turning the crank"
on the model, that is, in drawing inferences from
the admittedly reductive and "off'n'wrong" model
in a way that loses even the initial information
that it captured about the real-world situation.

To do it justice, though, I will need to return
to this issue in a less frazzled frame of mind.

This will complete the revision of this RARified thread from last Autumn.
I will wind it up, as far as this part of it goes, by recapitulating the
development of the "Rise" relation, from a couple of days ago, this time
working through its analysis and its synthesis as fully as I know how at
the present state of my knowledge. The good of this exercise, of course,
the reason for doing all of this necessary work, is not because the Rise
relation is so terribly interesting in itself, but rather to demonstrate
the utility of the functional framework and its sundry attached tools in
their application to a nigh unto minimal and thus least obstructive case.

Jon Awbrey

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Subject: Re: Reductions Among Relations
From: "Peter Brawley" <peter.brawley[…]artfulsoftware.com>
Date: Mon, 25 Nov 2002 10:31:59 -0800
X-Message-Number: 34

> So, if you project a triadic relation onto a dyadic relation
> you loose information. Why is that surprising?

Right, the relevant question seems to be, whether there is a proof that
there exists a triadic relation that cannot be losslessly reconstituted from
its dyadic constituents.

PB





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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 13:32:35 -0500
X-Message-Number: 35

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RAR. Note 9

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Compositional Analysis of Relations in General (cont.)

The good of this whole discussion, the use of it all,
the thing about it that makes it worth going through,
at least, for me, is not just to settle doubts about
the "banal", "common", or figuratively and literally
"trivial" (Latin for locus where "three roads" meet)
type of issue that may have appeared to be its point,
but, as I said in my recent reprise of justification,
to examine and explore "the extent to which it is possible to
construct relations between complex relations and simpler
relations. The aim here, once we get past questions of
what is reducible in what way and what not in no way,
is to develop concrete and fairly general methods
for analyzing the structures of those relations
that are indeed amenable to a useful analysis --
and here I probably ought to emphasize that
I am talking about the structure of each
relation in itself, at least, to the
extent that it presents itself in
extensional form, and not just
the syntax of this or that
relational expression".

So let me run through this development once more,
this time interlacing its crescendoes with a few
supplemental notes of showcasing or sidelighting,
aimed to render more clearly the aim of the work.

In order to accumulate a stock of ready-mixed concrete instances,
at the same time to supply ourselves with relatively fundamental
materials for building ever more complex and prospectively still
more desirable and elegant structures, maybe, even if it must be
worked out just a little bit gradually, hopefully, incrementally,
and even at times jury-rigged here and there, increasingly still
more useful architectronic forms for our joint contemplation and
and our meet edification, let us then set out once more from the
grounds of what we currently have in our command, and advance in
the directions of greater generalities and expanded scopes, with
the understanding that many such journeys are possible, and that
each is bound to open up on open-ended views at its unlidded top.

By way of a lightly diverting overture, let's begin
with an exemplar of a "degenerate triadic relation" --
do you guess that our opera is in an Italian manor? --
a particular version of the "between relation", but
let us make it as simple as we possibly can and not
attempt to analyze even that much of a case in full
or final detail, but leave something for the finale.

Let B = {0, 1}.

Let the relation named "Rise(2)"
such that Rise(2) c B^2 = B x B,
be exactly this set of 2-tuples:

| Rise(2)
| =
| {
| <0, 0>,
| <0, 1>,
| <1, 1>
| }

Let the relation named "Rise(3)"
such that Rise(3) c B^3 = BxBxB,
be exactly this set of 3-tuples:

| Rise(3)
| =
| {
| <0, 0, 0>,
| <0, 0, 1>,
| <0, 1, 1>,
| <1, 1, 1>
| }

Then Rise(3) is a "degenerate 3-adic relation"
because it can be expressed as the conjunction
of a couple of 2-adic relations, specifically:

Rise(3)<x, y, z> <=> [Rise(2)<x, y> and Rise(2)<y, z>].

But wait just a minute! You read me clearly to say already --
and I know that you believed me! -- that no 3-adic relation
can be decomposed into any 2-adic relations, so what in the
heck is going on!? Well, "decomposed" implies the converse
of "composition", which has to mean "relational composition"
in the present context, and this composition is a different
operation entirely from the "conjunction" that was employed
above, to express Rise(3) as a conjunction of two Rise(2)'s.

That much we have seen and done before, but in the spirit of
that old saw that "what goes up must come down" we recognize
that there must be a supplementary relation in the scheme of
things that is equally worthy of our note, and so let us dub
this diminuendo the "Fall" relation, and set to define it so:

Let the relation named "Fall(2)"
such that Fall(2) c B^2 = B x B,
be exactly this set of 2-tuples:

| Fall(2)
| =
| {
| <0, 0>,
| <1, 0>,
| <1, 1>
| )

Let the relation named "Fall<3>"
such that Fall<3> c B^3 = BxBxB,
be exactly this set of 3-tuples:

| Fall(3)
| =
| {
| <0, 0, 0>,
| <1, 0, 0>,
| <1, 1, 0>,
| <1, 1, 1>
| }

And on these notes ...
I must rest a bit ...
For an interval ...
To be continuo ...
Let it B ...

Jon Awbrey

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Subject: Re: Identity & Teridentity
From: "Peter Brawley" <peter.brawley[…]artfulsoftware.com>
Date: Mon, 25 Nov 2002 10:43:44 -0800
X-Message-Number: 36

<snip>

> For example a solid ball and a hollow sphere
> with the same center and radius have indiscernible projections
> on the XY, XZ, YZ planes.

Eh? A hollow sphere projects as an empty circle in all planes that go
through it, a full sphere projects as a filled circle in all planes that go
through it.

> So we get the follwing breakdown:
>
> a. All 3-adic relations are irreducible to compositions of 2-adic
relations.

You did not prove this (yet).

PB





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

Subject: Re: Identity & Teridentity
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 13:52:43 -0500
X-Message-Number: 37

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I&T. Note 25

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JA: For example a solid ball and a hollow sphere with the
same center and radius have indiscernible projections
on the XY, XZ, YZ planes.

PB: Eh? A hollow sphere projects as an empty circle in all planes
that go through it, a full sphere projects as a filled circle
in all planes that go through it.

You are confusing projection with intersection.

JA: So we get the follwing breakdown:

a. All 3-adic relations are irreducible to
compositions of 2-adic relations.

PB: You did not prove this (yet).

I repeat: It is not a matter of proof.
It is from the pertinent definition of
the operation of relative composition.

Exercise for the reader:

Using ordinary matrix multiplication,
find two square matrices A and B,
which multiplied together yield
an AB that is a cubic array.

Jon Awbrey

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

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Mon, 25 Nov 2002 14:10:05 -0500
X-Message-Number: 38

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RAR. Note 10

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Compositional Analysis of Relations in General (cont.)

Starting from the simplest notions of Rise and Fall,
I may easily have chosen to leave it as an exercise
for the reader to discover suitable generalizations,
say from Rise(k) and Fall(k) for k of order 2 and 3,
to the slightly more general case in which k is any
natural number, that is, finite, integral, positive.
But that is far too easy a calisthenic, and no kind
of a work-out to offer our band of fearless readers,
and so the writer picks up the gage that he himself
throws down, and for his health runs the easy track!

Let B = {0, 1}.

Let the relation Rise(k) c B^k
be defined in the following way:

| Rise(k)<x_1, ..., x_k>
|
| iff
|
| Rise(2)<x_1, x_2> and Rise(k-1)<x_2, ..., x_k>.

Let the relation Fall(k) c B^k
be defined in the following way:

| Fall(k)<x_1, ..., x_k>
|
| iff
|
| Fall(2)<x_1, x_2> and Fall(k-1)<x_2, ..., x_k>.

But let me now leave off, for the time being,
from the temptation to go any further in the
direction of increasing k than I ever really
intended to, on beyond 2 or 3 or thereabouts,
for that is not the aim of the present study.

Jon Awbrey

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