SUO: Re: Thirdness
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Pierre,
The questions of reducibility among k-adic relations
divides into the following subquestions:
Q1. Reducibility of 3-adic relations to 2-adics.
Q1.1. Reducibility of 3-adic relations under relational composition?
A1.1. No 3-adic relations are relational composites of 2-adic relations.
Q1.2. Reducibility of 3-adic relations under projective construction?
This amounts to the composition of a 3-adic relation from the
3-adic relation & : B x B -> B plus a couple of 2-adics.
A1.2. Some 3-adic relations are projectively reducible to 2-adics,
and some 3-adic relations are projectively irreducible.
Examples of both types have been constructed.
Q2. Reducibility of higher k-adic relations to 3-adics?
A2. Peirce's arguments were always sufficient on this score.
More recently there are papers by Herzberger and Burch.
Comments.
The irreduciblity of all 3-adic relations under composition
was always just a matter of understanding the definition
of relational composition, and requires no elaborate proof.
The irreducibility of some 3-adic relations under projection
has now been "reduced", in the pedagogical sense of the word,
and in recent demonstrations in this forum, to the level of
ordinary set theory, like that taught in undergraduate math
courses, or the discrete math that is required in comp sci.
On the matter of reducing higher k-adics to 3-adic relations, Peirce's
first arguments are sufficient to the elementary status of this result.
Since the facts are evident once a person comprehends the definitions,
I have not read further, but the papers of Herzberger and Burch are out
there for those who desire the exercise. My scanning of fragments from
them leads me to guess that they made a lot of extra work for themselves
by trying to prove these theorems in syntactic formalisms that obstruct
the obvious, much as if one desired to prove the prime number theorem
or the pythagorean theorem from peano axioms alone. I am told that
such exercises build character, but I have enough character already.
I think that it should be a more moderately healthful exercise to
explain the reducibility of higher k-adics to 3-adics on the same
grounds of elementary set theory to which I have recently reduced
the irreducibility of 3-adic relations. So maybe I'll do that.
Betcha can't wait ...
Jon Awbrey
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Pierre Grenon wrote:
>
> > Pierre,
> >
> > > Well, hello John, that's no theorem. Your having
> > > stated a claim in an email doesn't turn the claim
> > > into a theorem the next time you make the claim.
> >
> > Of course not. The theorem was proved over a century
> > ago, but if you don't want to waste your time reading
> > it, I don't want to waste my time typing it.
>
> I was interested in arguments and proofs relating to your claims, not in
> reiteration of these claims.
>
> > > The examples you give (I mean the valid one, not
> > > the obfuscating ones) are picked out from a domain
> > > incorporating agents and agency. We're speaking
> > > domain ontology.
> >
> > I think that agents such as people and computers
> > are of sufficient interest to be valid topics
> > for an SUO.
>
> I think too, the point is that this is only a part of the SUO.
>
> > But you can define the scope of
> > what you want to do in any way you please.
>
> I don't have the pretention of defining anything. I've just tried to approach
> the PAR in a rational and scientific manner. It just took me too a long time to
> accept the fact that I was too serious about this group when what we really
> have here is but an immense pityful joke.
>
> > I think it's time for us to stop wasting
> > both our time in trying to communicate.
>
> It's nice to finally reach this consensus. I stand relieved,
> you may have the final word.
>
> > John
> >
> >
> --
>
> Pierre Grenon, IFOMIS Uni Leipzig
> http://people.ifomis.uni-leipzig.de/pierre.grenon/
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