SUO: Re: Irreducible Try-It-And-See
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Matthew West wrote:
>
> 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.
We went through this in gory detail way back in the last Millennium,
and I provided examples'o'plenty of everything that I asserted, but
I quite understand, as the amateur logician, I am rather accustomed
to ZZZ that meets the second or third paragraph or so of everything
that I might have to say on the subject, and so I am in the process
of going back through that crusty old "Reducibility Among Relations"
thread and trying to refurbish it for this Millennium's consumption.
But it is possible to say quite quickly right now, before I venture
into that chancey second paragraph, that this much, at least, is no
big demonstration at all, but merely follows from the definition of
the term "relational composition", or "relative product", and so on.
We have quite recently seen two pictures of how this multiplication
operates, the array picture and the bigraph picture, in the example
of John Sowa that I have been elaborating at rather great length on
the "Anxiety of Influenza" or "Virally Important Topics" thread, cf:
http://suo.ieee.org/email/msg04124.html
> MW: I'm, sorry, but without an example I have no idea what you mean.
Again, my use of the word "composition", and all of its diverting derivatives,
is purely definitional in this context, but more examples will be forthcoming.
> 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.
You probably have something else in mind,
aside from the "relational composition",
strictly speaking, when you say this,
and that is perfectly acceptable --
it remains to see what it is.
There can be a whole lot of
unanalyzed structure that
lies in a little word
like "plus".
> 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 gave examples last Autumn of each of these notions.
I have even sent you links to papers I wrote wherein
these very ideas were applied to a number of applied
topics in education, intelligent systems engineering,
organization, semiotics, social philosophy, etcetera.
Just by way of a reminder, here are the online links:
http://www.shss.montclair.edu/inquiry/fall95/awbrey.html
http://www.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm
http://www.door.net/arisbe/menu/library/aboutcsp/awbrey/integrat.htm
> > 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.
I am still here, too.
> 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.
For me, neither.
> > 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.
I really think that I have done so, but I will continue.
> 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.
Aside from that YAIOPF ("yet another instance of progressive fallacy")
that heralds the motif of its opening fanfare -- a fare for the fans
of that, I guess -- I am still trying to understand why Pat argues
the way he does and what precisely he thinks to come of it, more
because it is such a surprising phenomenon to me, personally,
that two people who share so much common literature as we do,
aside from Peirce, of course, can arrive at such discrepant
points of view, but this is a different matter from having
uncertainties about the implications of these definitions.
Moreover, it is a general principle of reasoning that one
does not have to approach a universal proposition, like
"All k-adic relations are 'analyzable', 'decomposable',
'reducible', in any given sense, to 2-adic relations",
or even be required to unravel the precise location
of the problems in a reputed proof, if one can find
or make a single counterexample to that proposition.
And I have given plenty of counterexamples so far,
in several specific senses of "decomposition" or
of "reduction", to show that it just ain't so.
But I will dust them off and trot them out
yet again.
> > 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.
I am still working my way through Pat's paper,
but one of the first questions that started
to worry me was right about at this point:
"What does the existential quantifier have to do with it?"
I am still wondering about that ...
> > (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)
> > ).
Here is the next thing that worries me about this argument:
See all of those little ampersands "&"? -- Each and every
one of those symbols is a stand-in pers "and", and "and",
as we know, is a 2-adic operation, and that means that
it invokes, to my "way of thinking" (WOT), anyway,
a 3-adic relation. <QED | WOT>! I have mentioned
this almost insignificant fact time 'and' time again,
before, but I have not bothered to bring it up this time
around because Pat, as I know, or, at least, the best that
I can gather so far, anyway, does not share this WOT, this
way of "functional thinking", with me -- he has his way of,
well, "non-functional thinking", I guess -- and so I have
been looking for an analysis or an argument that might be
independent and unbiased regarding WOT's on this score.
> 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.
This is okay, except for the part about
"represented as dyadic relations",
which remains as false as ever.
> 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.
The bit about reducing, really, it's more like "reconstituting",
any >3-adic relation as a composition of 3-adic relations gets
a bit tricky only because it depends on "splitting nodes" in
the corresponding graph, which means introducing exactly the
right sort of new relational domain. Here is the picture:
· · · ·
\ / \ /
o >>>--->>> o-·-o
/ \ / \
· · · ·
Of course, the graph-theoretical picture proves nothing.
It is merely intended to remind those who already grasp
the intention of its conventions what they already know.
Here is the animation (bottom right corner of the page):
http://www.iupui.edu/~peirce/web/desc/desc.htm
> 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, ...
Again, I have to ask who's "we"? Peirce did not
speak "at this rather loose level of discussion",
nor am I -- at least, in this connection, anyway.
> > 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.)
This points to many of the problems that arise when using either
natural language examples or the more formalized logical examples,
at least, in the way that some people use them, where there is no
explicit use of sign relations or an equivalent discipline of care
with quotation to constantly discern when a thing is being used as
a sign and when a thing is being denoted as an object, leading to
a general confusion of signs and expressions with the objects of
which they ought to serve as the signs and expressions. A person
working from nothing else but the signs and expressions of either
form of language, artificial or natural, will be able to get just
about nothing at all about the structure of a relation from looking
at the signs and expressions that denote it, since these signs and
expressions can have fairly arbitrary properties in terms of their
purely syntactic complexity and their puerly superficial numbers
of clauses, phrases, slots, variables, whatever. It is typically
only when a person already grasps something about the structure of
a relation that they imagine themselves to observe it exhibited in
some canonically pre-selected exemplar of that very form or property.
> 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.
John Sowa knows better than to concede this.
But there are other reasons for renunciation
of a quest, as Goethe so well taught us all.
> 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.
The graph-theoretical part of this was proved in the time of Euler.
Yes, most folks who see the sensibility of the appropriate mapping
of relations to graphs do not bother to go any further, but there
are supposed to be proofs in the logic lit for those who want to
go look -- I used to hear the name Herzberger (?) a lot in this
connection -- but having never doubted it, I never bothered.
Just according to my own personal taste in the matter,
I do not see that hauling in "yet another calculus" (YAC)
like lambda calculus, is really going to clear things up,
not so long as some people continue to confuse the things
accountable with their counters in the numerous calculi.
> 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.
on jugera ...
Jon Awbrey
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