SUO: Re: Expostulation
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Matthew,
Finding myself with a spare half hour,
I will try to see if I can address some
of the remaining points in your last note.
Jon Awbrey wrote:
>
> Matthew West wrote:
> >
> > Dear Jon,
> >
> > > Matthew West wrote:
> > > >
> > > > MW: One reason I am pursuing this discussion
> > > > is that if there really are things it
> > > > cannot handle, I would really like
> > > > to know.
> > >
> > > Dear Matthew,
> > >
> > > I always take people at their word,
> > > so I take you really to want to know:
> > >
> > > The POV that you have been exposing here
> > > cannot handle 3-adic relations in any way,
> > > if by "handle" you mean "not destroying" the
> > > information contained in their apt expressions.
> > >
> > > The "Pat Hayes Universal Change Of Variables Formula" (PHUCOVF)
> > > has now been exposed as the "change of subject" (COS) formula
> > > that it really is at heart, and the author of it has admitted
> > > that nobody ever meant to say what some people once pretended
> > > that it said, as have you also in regard to a co-indited point.
> >
> > MW: Well I noticed that you claimed to have overturned Pat's proof,
> > but I am not sufficiently versed in these things to be sure that
> > you are not employing the same kinds of tricks that you claim Pat
> > has employed.
Well, at least we agree that some form of distracting prestidigitation
is involved, and it remains but to see who is raveling up their sleeve
and who is trying to dispell the spell of this sleight transliteration.
> > So I'm afraid that is not an arguement that will
> > convince me on its own. If you wish to do that
> > you will have to present a specific example of
> > where the information in some triadic relation
> > cannot be transformed into some set of binary
> > relations. If you provide the triadic relation,
> > I will provide the binary relations from which the
> > triadic relation can be constructed (or not and admit
> > you are right).
Once again, I have provided you not only with specific definitions
of the two most common forms of reduction that are usually met with
in this context, but I have given many examples of 3-adic relations,
indicating how they fall out with respect to these two typical FOA's.
I have now contributed to this working group what approaches a small
monograph on this topic, and I have just condensed and edited these
notes into a single essay on the "Reductions Among Relations" thread.
So far, nobody else has even defined another type of decomposition --
I do not yet recognize the "transliteration" that Pat put forth as
being up to the mark on that score -- and nobody else has provided
any extensional examples of relations, which most people who look
into it long enough find are best when it comes to settling these
sorts of questions in any concrete and definitive way. If people
prefer to continue with intensional examples then it is incumbent
on them to provide explicit definitions of these relations, but
so far I have just not seen any being 'given' for the 'taking',
if you will.
To summarize what is on the RAR thread:
1. If the genre of analysis/synthesis is relational de/composition,
then no 3-adic relations are reducible to 2-adic relations.
So if you seek examples under this heading, all 3-adics
are irreducible to 2-adics under relational composition.
2. If the genre of analysis/synthesis is projective de/construction,
then some 3-adics are reducible to 2-adics and some are not.
a. Two examples of projectively reducible 3-adic relations
are the ones from the "Story of A and B".
b. Two examples of projectively irreducible 3-adic relations
are the relations L^0 and L^1, that are discussed in this
excerpt from the RAR thread:
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Projectively Irreducible Triadic Relations, or
Triadic Relations Irreducible Over Projections:
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 triples
| of the form <x, y> in XxY:
|
| <0, 0>
| <0, 1>
| <1, 0>
| <1, 1>
| (L^0)<XZ> has these four triples
| of the form <x, z> in XxZ:
|
| <0, 0>
| <0, 1>
| <1, 1>
| <1, 0>
| (L^0)<YZ> has these four triples
| 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 triples
| of the form <x, y> in XxY:
|
| <0, 0>
| <0, 1>
| <1, 0>
| <1, 1>
| (L^1)<XZ> has these four triples
| of the form <x, z> in XxZ:
|
| <0, 1>
| <0, 0>
| <1, 0>
| <1, 1>
| (L^1)<YZ> has these four triples
| 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:
(L^0)<XY> = (L^1)<XY> = 1<XY> = BxB = B^2,
(L^0)<XZ> = (L^1)<XZ> = 1<XZ> = BxB = B^2,
(L^0)<YZ> = (L^1)<YZ> = 1<YZ> = BxB = B^2.
Therefore, L^0 and L^1 constitute examples of
"projectively irreducible triadic relations",
"triadic relations irreducible on projections".
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But lest you or anybody misunderstand what is going on here,
the nature of projective reduction is such that it freely
avails itself of another kind of 3-adic relation, namely,
the truth-functional connective 'and', in order to bind
the various 2-adic projections together into a whole.
In short, even this does not provide an example of
dispensing with 3-adic relations in the analysis.
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> Dear Matthew,
>
> We are all learning as we go, some of us one set of things, others another,
> but I think that Pat and I have finally settled down to the purely logical --
> in contradistinction to the "puerly rhetorical" (sic) -- facet of the task,
> and I am secure in the knowledge that both of us are dogged enough not to
> let it go until we flesh out some general satisfaction on this particular
> bone of contention. 'On jugera', as Evariste Galois said.
>
> Let me tell you where I think that you and I stand at present.
> I believe that we have agreed on the ground rules that nothing of
> any rational sense will come of arguing about any puported form of
> "analysis", "decomposition", "discombobulation", "reconstruction",
> "reduction", "transliteration", "transmogrification", or whatever,
> unless we have clearly defined the particular species of relation
> that we are talking about in a case at issue. Now, for my own part,
> I regard the argument as being effectively over right at this point,
> since I never knew a "form of analysis" (FOA) that pretended to be
> anything other than a relationship among one thing, the analysand,
> and at least two other things, the analytic components. One plus
> at least two implies at least three, and so the very idea of a FOA
> has us neck deep already in a 3-adic relation. As an object example,
> taken for the moment in the converse direction of synthesis, joining
> an x and a y to form the set {x, y} is already an element of a 3-adic
> relation, just as every other "binary operation" is a 3-adic relation.
>
> Now, there are good reasons for proceeding further, that is,
> once the omnipresence of 3-adic relations in every analytic
> project is conceded, but perhaps it would be best for me to
> halt at this point and see if we have any form of agreement
> about this much, at least.
>
> > > The only residual dross of the matter appears to turn on the
> > > circumstance that some people cannot yet distinguish a relation
> > > from an element of one.
> >
> > MW: Well this makes me think that you have a different definition
> > of what a relation is to me. This would easily explain how we
> > have been missing each other.
I believe that I use pretty much the same notion of a relation that
just about all logical and mathematical people employ, not to mention
just about all computer science and database people I ever encountered.
Jon Awbrey
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