SUO: Re: Irreducible Try-It-And-See
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Pat Hayes wrote:
>
> [Jon Awbrey wrote:]
> >
> > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> > >
> > > 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.
> >
> > That is true, but only because 'relational composition'
> > is very narrowly defined.
>
> That is precisely why it is useful.
> Just like addition, multiplication,
> all those other "narrowly defined"
> types of operations ...
>
> So Peircians claim. However, I have never seen any example
> of the utility of this calculus, or ever seen it applied
> in any useful sphere of human endeavour, other of course
> than Peircian scholarship.
This is really an absurd statement.
I cannot imagine what you mean to say,
or what you imagine I have been saying.
At its minimum, functional composition is
a special case of relational composition,
so all of category theory falls under it,
which mathematicians and many brands of
computer scientists use all of the time,
not to mention its utterly everyday use
in relational databases. And still more
mundanely, we use something informatively
equivalent to this "calculus", as you call
it, of "relative terms", as Peirce called it,
every time we work it out in our heads that
a father of a father is a grandfather, or
that a brother of a mother is an uncle.
The best that I can guess is that you
are attaching some extra meaning that
I do not know about to the notion of
a "calculus", as if to attrubute to
me some kind of "relational algebra"
approach that is really not my style
at all. I am only doing logic here,
as was Peirce, who did more to free
logic from the more misleading sorts
of algebraic analogies than anybody
I know about, before or since, still,
without fearing to exploit algebraic
analogies when and if they work and
without bearing any kind of grudge
against algebra-like laws applying
to special structures when they do.
Maybe this is an effect of projected
anachronism of modern interpretations.
Since I have taken the trouble to read
Peirce's original papers, I do not need
to read them through the any epigone's
eyes but my own.
> > For example, if you include all the inverses of the
> > composition operations, this result fails immediately,
> > since one of those inverses is a binary merge which
> > forms I3 from two copies of I2.
>
> I am not sure if we are using "inverse" and "merge"
> in the same way or not, so I hesitate to speculate.
> Could you provide me a simple, extensional example,
> just by way of locating the same page of notation?
>
> I don't have the page reference handy.
> The operator in question is what Peirce
> called the 'comma operation', I believe;
> it 'splits' a relational arc into two,
> making In into In+1. The inverse (in
> a conventional sense) of this is the
> operator I called 'binding' in the
> Burch review, which can be decribed
> in lambda-terms as
>
> Bi j ((lambda x1, ..., xn) A[x1, ..., xn]) =
> (lambda x1,...,xn-1) A[x1,...,xj-1, xi , xj+1 ... xn-1]
>
> This, notice, is an operation which reduces the total
> number of arcs in the graph, so it isnt in any obvious
> sense 'cheating', by inserting new links or anything
> like that. In graphical terms, it takes two free
> arc ends and merges them into one:
>
> | / .... | ____________
> | / ... | / ... /
> ---O ---- ==> ---O ---- /
> /|\ /|\______/
> / | \ ... / |
>
> Then I3 = B12(I2&I2):
>
> ---I2
> \___
> /
> ---I2
>
Now I know that you know the meaning of extensional ...
I will not complain about your introduction of
a logically problematic "calculus" -- I know
that it has its uses -- but did you really
chose the right caliber gun for the game?
> It even makes sense as chemistry: this kind of thing happens in a benzine ring.
You do know about molecular orbital theory?
Interactions got real "triply" with that.
> > > 2. Some 3-adic relations are reducible to, or reconstructible
> > > from 2-adic relations, in the sense of projective reduction.
> >
> > Er ... ALL n-adic relations, I think you mean there.
>
> No, think about a hollow sphere S1 and a solid sphere S2 in 3-space --
> these make up two 3-adic relations, S1, S2 c R^3, that have the same
> projections on the xy-, xz-, yz-planes, so no unique reconstruction.
>
> Ah, I obviously misunderstood what you meant by 'projective reduction'.
> Whatever that is, it isn't what we have all been talking about on this
> thread so far.
Since way last year, I have consistently been talking
about at least two different notions of reducibility,
"compositional" and "projective", among others.
> I have no idea what you mean in calling a 3-sphere a 3-adic relation.
> By a 3-adic relation I mean a relation with 3 arguments.
Relations do not have arguments.
Relational expressions or
relational formulae
have arguments.
> The extension of such a relation is a set of 3-tuples.
> This has nothing to do with spatial dimension
> or geometric projection.
I know that you know what an algebraic variety is ...
Any subset of a cartesian product space X<1> x ... x X<k>
can be represented as a body in a k-dimensional space,
and all of the various projections visualized in
a corresponding manner. It's just a picture.
> > > 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.
> > >
> > > 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. 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.
> >
> > On the other hand, Jon, it is the case that anything that can be said,
> > can be said using only 2-adic relations. This is a plain fact, known
> > to almost everyone who has had an elementary education in modern logic.
>
> My extremely rudimentary education in post*modern logic tells me that
> conjunction is a binary operation, and so a ternary relation, and so ...
>
> Conjunction is a connective, not a relation.
Can you give me an explanation of how you view "connectives"?
Yes, I did have this in school, and I do know how I view them --
I just need to know how exactly you understand them, theoretically,
if I am going to understand why I cannot presently understand you.
> (If you insist that it is a function on truthvalues
> and therefore a ternary relation, then I will concede
> that such things are needed. However, I will then assert
> even more strongly the utter banality and unimportance of
> this result. As far as I know, nobody, not even Scheffer,
> has ever asserted that one can do without binary connectives.)
Substituting identicals into what I hope is your transparent context:
Therefore nobody is saying that one can do without triadic relations.
The End.
Yes, Peirce considered this to be utterly banal.
That is kind of what an axiomatic truism is to
all of us "utterly common people in the street".
The air that we breathe is utterly banal,
and can be taken for granted as "given",
but only while there is lots and lots.
> > I know this goes against the canons of the Church of Peirce,
> > but eppur, si muevo, as I believe another old curmudgeon once said.
> >
> > The main point, however, is that this entire matter is NOT IMPORTANT.
> > The reduction to the semantic network case is one of a large collection
> > of such reducibility results of logic to particular subcases, all of which
> > have their utility and which add up to a collection of useful logical tricks,
> > a kind of sign-system hacker's guide. I don't mean to disparage such results,
> > but they are all elementary and have no deep metaphysical meaning or importance.
> > I still reel from encounters with Peircian groupies who seem to be so besotted
> > with the importance of the number three. Could this have anything to do with
> > any kind of Doctrine, I wonder??
>
> I do not really buy the semantic network analysis of relations,
> just going by your description of it, since representing them
> by "arcs" or "edges" seems to me that it pre-selects 2-adics.
>
> What is a 2-adic? Only a Peircian knows for sure.
> If that means relations with two arguments, then ...
Once again, relations do not have arguments.
People have arguments with their relations.
> 'pre'selection is irrelevant.
> The analysis shows that only binary relations are required.
> Claiming to not 'buy' a theorem isnt a very constructive attitude.
The "analysis" is inadequate. It fails to uncover hidden assumptions.
And a representation that leads one to believe false things is one
that I do not buy.
> > I suppose I should be grateful that CSP didnt choose seven.
>
> Actually, my anunciation hic et nunc is fixed on five,
> bearing on the Doctrine of the Pentaculate Conception,
> otherwise known as the Pragmatic Maxim.
>
> As usual, I have no idea what you are talking about.
D'oh!
I thought you were making a joke.
I was only attempting to join in.
The most frequent enunciation of
the Pragmatic Maxim bears five
cases of words deriving from
the Latin 'concipere'.
Jon Awbrey
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