SUO: Re: Ducibility Among Relations
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Pat Hayes wrote:
>
> [Jon Awbrey wrote:]
> >
> > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> >
> > I am going to string together a cleaned-up composite version of
> > an earlier thread on the topic of reducibility among relations --
> > viewed another way, on the extent to which it is possible to
> > construct relations between complex relations and simpler
> > relations. The aim here, once we get past questions of
> > what is reducible in what way and what not in no way,
> > is to develop concrete and fairly general methods
> > for analyzing the structures of those relations
> > that are indeed amenable to a useful analysis --
> > and here I probably ought to emphasize that
> > I am talking about the structure of each
> > relation in itself, at least, to the
> > extent that it presents itself in
> > extensional form, and not just
> > the syntax of this or that
> > relational expression.
> >
> > By way of a lightly diverting overture, let's begin
> > with an examplar of a "degenerate triadic relation",
> > a particular version of the "between relation", but
> > let us make it as simple as we possibly can and not
> > attempt to analyze even that much of a case in full
> > or final detail, but leave something for the finale.
> >
> > Let B = {0, 1}.
> >
> > Let the relation named "Rise<2>"
> > such that Rise<2> c B^2 = B x B,
> > be exactly this set of 2-tuples:
> >
> > | Rise<2> = {<0, 0>,
> > | <0, 1>,
> > | <1, 1>}
> >
> > Let the relation named "Rise<3>"
> > such that Rise<3> c B^3 = BxBxB,
> > be exactly this set of 3-tuples:
> >
> > | Rise<3> = {<0, 0, 0>,
> > | <0, 0, 1>,
> > | <0, 1, 1>,
> > | <1, 1, 1>}
> >
> > Then Rise<3> is a "degenerate 3-adic relation"
> > because it can be expressed as the conjunction
> > of a couple of 2-adic relations, specifically:
> >
> > Rise<3><x, y, z> iff [Rise<2><x, y> and Rise<2><y, z>].
> >
> > But wait just a minute! You read me clearly to say already --
> > and I know that you believed me! -- that no 3-adic relation
> > can be decomposed into any 2-adic relations, so what in the
> > heck is going on!? Well, "decomposed" implies the converse
> > of "composition", which has to mean "relational composition"
> > in the present context,
>
> Why does it *have to mean* that?
In some sense, I reckon, just because I "declared" it
to possess that definition in the "declaration section"
of this, my present "programme" -- words are nigh unto
indefinitely reusble, of course, at least, I am almost
cretain that I once heard somebody mentioning that most
fortunate fact about them. But my choice was not some
arbitrary whim: it conforms to the extremely standard
form of "functional composition" to describe the special
case of "relational composition" that it does indeed form.
And here I have even deferred to more contemporary usage
in preference to Peirce's use of "relative product, yes,
more due to the slight distinction that yet remains here,
but still, there are only so many words around, and we
always have the very generic terms of "analysis" and
of "synthesis" if we desire to speak very loosely.
> (There are other notions of decomposition.)
I did not know that.
> And what exactly *is* that?
I defined it informally many times.
I referred to my sources, where it
was defined formally 130 years ago.
No, it shan't be on the final exam,
so you do not ever have to read it,
but I did supply a sufficient clue.
> I suspect that what you are leading
> towards (and as usual without actually
> getting there) is some kind of relational
> algebra, where relations are things and there
> are operations which compose new relations from
> old ones. If my suspicions are correct, then PLEASE
> don't take us on any more of these roundabout tours
> through gardens of trivial examples rendered in ASCII
> art, but cut to the chase. Tell us the operations
> of the relational algebra you propose we should take
> seriously. It can probably be done in a page or so.
> There are many possibilities out there, and you may
> have some new ones; but we can't make any useful
> progress until we know which one you are talking
> about.
Pat, I do not tell you what to write.
I am just thinking like a programmer here.
Programs are recursive things, as you know.
Until one has the base hammered down, it is
just plain foolhardy to take that first step.
And I see no indication from the general tenor
of the torts that I have heard about hereabouts
that we have risen up even as far as to dare and
contemplate the evolution of the lower moulds yet.
I have given more than enough informal defs
and explicit refs for one who already knows
the subject to know what I am talking about,
but those who know the subject would never
had said this stuff about reducing 3 to 2,
without giving very careful definitions of
the sense of reduction that they intended.
> > Okay, there is a lot more to say, even about such a simple example,
> > but I have a feeling that this much is just about enough for today.
>
> No, it wasnt anywhere near enough. It was just enough
> to be irritating and just not quite enough to convey
> any actual content. If exasperation were music, Jon,
> you would have perfect pitch.
Then I am an icon of the cosmic eriphony,
And have dutifully performed my function,
So thank you for the gracious compliment.
Jon Awbrey
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