SUO: RE: Re: Irreducible Try-It-And-See
Dear Jon,
> Matthew West wrote:
> >
> > Dear Lee,
> >
> > I thought we had gotten beyond this.
> > Indeed I thought we had got to the
> > point where it was demonstrated
> > (by Pat, but by me previously)
> > that in representation terms
> > anything can be reduced to
> > dyadic relations.
> >
> > This caused a change in tack to saying that there are axioms that
> > require more than 2 arguments (not a particular surprise to me).
> > The question remaining is then whether there are any axioms that
> > cannot be reduced to some set of independent axioms that involve
> > only 3 elements.
> >
> > Regards
> >
> > Matthew
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> Matthew,
>
> No, that is not an accurate summation
> of the general state of understanding
> on this topic, and no such result has
> been demonstrated, not in those terms.
>
> What we have demonstrated so far is limited to this:
>
> 1. No 3-adic relations are reducible to composites or products
> of 2-adic relations, in the sense of relational composition.
MW: I'm, sorry, but without an example I have no idea what you mean.
What I will say is that if you give me a triadic relation, I would
expect to be able to provide in return some number of dyadic
relations plus perhaps some axioms that convey the same meaning.
This is what I understood Pat has done, and I have been doing quite
happily for many years being fully able to express all the semiotics
that I have seen here.
>
> 2. Some 3-adic relations are reducible to, or reconstructible
> from 2-adic relations, in the sense of projective reduction.
MW: More long words I don't understand without examples.
>
> I do not understand the reasons behind
> the persistence of this error, except
> out of some dogma of reductionism or
> just plain wishful thinking that the
> world be less complex than it is.
MW: I am simply pursuing an inquiry into something I don't understand.
I would have thought you above all would have understood that.
MW: I come from a distant and foreign land where our experiences and
mannerisms are somewhat different for what seems to me to be
arbitrary reasons. I am trying to discover if the differences are
indeed arbitrary or not. I don't mind what the outcome is, but
because Peirce says so isn't good enough.
>
> Furthermore, any attempt by folks "way out here"
> to canonize this account by methods other than
> the ordinary methods of reasoned inquiry, say,
> by enscouncing it in some liturgical doctrine
> to "record this as an 'official' outcome of
> the SUO group", no doubt soon to be joined
> by the complementary "Index of Books" that
> nobody but the duly-appointed Censors may
> read, for fear of being contaminated with
> alien doctrines that may weaken the Faith
> of those too naive to think for themselves,
> well, such a course would only bring ridicule
> on the SUO Effort, and by those who mince their
> words far less finely than I do.
MW: I am rather more of the "there are two ways
of talking about this - here's how they relate"
brigade personally. But you should know this by now.
> Please try to
> understand, a statement like "anything can be
> reduced to dyadic relations" is just bound to
> sound to whole communities of folks who work
> with this stuff every day like you just said
> that rectangular matrices are not closed with
> respect to matrix multiplication, and I am just
> trying to prevent you and the SUO Group as a whole
> from being subject to these embarrassments. People
> have to seek out their own authorities, if that is
> what it takes, but the "arguments" that have been
> cited so far on behalf of this putative reduction
> suffer from an "ignoratio elenchi" that is really
> quite astounding, for all of its cleverness and
> its diligence in racing down the wrong track.
MW: Then you need to explain what the errors are.
Pat's explaination was:
>Had Peirce or Whitehead lived a little longer maybe they would have
>become aware of the fact that any n-ary relation can be defined in
>terms of binary relations, with the aid of the existential
>quantifier. The translation, as I know you know, John, is this:
>
>R(t1,...,tn) ---> (exists e)(R(e) & first(e, t1) & second(e, t2)
>&...& nth(e, tn))
>
>where 'first', 'second', etc., are some fixed set of binary
>relations. (In case grammar these correspond to cases such as
>'agent', 'subject' and so on, and the 'e' is something like an event
>or a situation, of type R, corresponding to the verb of the simple
>sentence, as in:
>Gave(John, Book, Mary, yesterday)
> --->
>(exists e)(Giving(e) & agent(e, John) & subject(e, book) &
>recipient(e, Mary) & time(e, yesterday)) ).
MW: After some discussion John said:
>Example 3: "John initiated an act of giving. The giving
> had a book as object. The giving had Mary as recipient."
>
>This example does capture the three-way relationship, but
>only by creating another entity "giving", which itself has
>the open slots in its definition -- formally speaking,
>any representation of "giving" must have "frame-like" or
>"lambda-calculus-like" representation, which contains
>three inner variables or slots. Then each of the three
>sentences instantiates one of those slots.
>
>The only point that Peirce, Whitehead, and I have been trying
>to make is that there are concepts in English, such as Give,
>which cannot be defined without using a frame, a lambda
>expression, or some such formal device that contains three
>distinct slots, variables, boxes, or whatever.
MW: Well it seems to me there is not much to argue about here.
The conclusion seems to be that for things like "give" whilst
it can be represented as dyadic relations, you would need some
lambda calculus that involved 3 variables to tie it together.
MW: This is absolutely fine with me, I never thought you didn't
need that. I just wasn't tied to using triadic relations as the
representation form to achieve this.
MW: On the other hand I see no special reason why 3 is a magic
number. Why are there not relations of arbitrary/Lambda calculus
of arbitrary adicity? Of course this may be my ignorance, in
which case I seek enlightenment.
MW: Pat's view is:
>OK, if THAT is the point, then of course I will agree with it
>immediately: we will need to write axioms that mention more than two
>things, or use more than two relation symbols, or whatever. But now I
>am puzzled by the emphasis on threeness for a different reason, since
>it seems to me for example that if we are to really capture the full
>meaning of 'give' we will probably need to relate it to about 7 or
>more other concepts (whether they are expressed formally as relations
>or names or whatever). If we are at this rather loose level of
>discussion, I don't see how one can insist that "threeness" is
>particularly important. A giving involves two agents, a thing given,
>an intention (maybe several), a time, a place (usually), probably a
>reason, maybe an overarching larger event or circumstance (such as a
>birthday), etc. . Most things have many connections to other things,
>and (depending on how strictly one understands 'full meaning') their
>full meaning cannot be stated without making reference to more than
>three of them. So why do you stop at three?
>(Let me say what my suspicion is, so you can refute it if I am wrong.
>I think that when it comes to reducing n>3 to 3, y'all want to cite
>Peirce's result as showing conclusively and mathematically that the
>reduction can be done; but when it comes to going from 3 to 2, you
>want to get all kind of sketchy, and argue that that last step is
>just a dry formal result with no real meaning, and if you look behind
>the mere mathematics you will still see trinities everywhere. My
>problem is that if I stay mathematical and strict I can reduce it all
>to 2, and if I follow your intuitive perspective I can't reduce it to
>3.)
MW: After some further exchanges John said:
>I agree. I don't want to stop at three, but for the moment,
>I am happy that we all agree that there are more than 2.
MW: I don't see this as a concession, simply a holding position.
MW: So my final question is that there has been talk that anything
can be reduced to triadic form. Now it seems to me that the rules
have changed slightly (what is sauce for the goose is sauce for the
gander) so this now means not the kind of reduction that allows
a triadic relation to be reduced to a dyadic relation, but
something where not only are there only triadic relations, but there
is no lambda calculus required that involves more than 3 variables
as well.
MW:If this can be demonstrated within the constraints that Pat
outlines above, then something useful is established. If not, then we
simply have some different custom and practice about how situations
are modelled by different cultures, to which I would have thought
you should have been sympathetic.
>
> Many Regards,
>
> Jon Awbrey
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
Regards
Matthew
===============================================================
Matthew West http://www.matthew-west.org.uk/
Principal Consultant Shell Visiting Professor
Operations & Asset Management The Keyworth Institute
Shell Services International The University of Leeds
http://www.shellservices.com/ http://www.keyworth.leeds.ac.uk/
H3229, Shell Centre, London, SE1 7NA, UK.
Tel: +44 207 934 4490 Fax: 7929 Mobile: +44 7796 336538
===============================================================