SUO: Relations Are Sets, People! (RASP!)
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
I was having this nightmare about Charlton Heston running
through the streets, screaming at every passerby in sight:
| "Soylent Relations Are Soylent Sets, People!"
But the sybilant rasp of that acronym (SRASSP) is something
that I am planning to save for later, when I really need it.
For the moment I will thread this thread for the sake
of focusing my volume, at least, on a tenet that each
of us tokenly assents to -- on extensional feast days --
but that we -- well some of us -- do not yet practice
the preaching of. Now, it is precisely for this fact
that our humain frailties put us in dire need of much
better "automated inference engines" (AIE's) -- since
they, at least, can be trained to pay attention to us,
and to the verbatim letter of what we say, when we go
to open our mouths, close our minds, and speak.
Just as the refulgent light of every good Cosmos must
rise from the dim excess dimensions of a priori Chaos,
I must begin with the dully instigatory bit of blight
on our e-mediate intelligensias, to wit, and juno who:
Pat Hayes wrote:
>
> [John Sowa wrote:]
> >
> > Pat,
> >
> > We went round and round on this issue before,
> > so I don't want to repeat the experience.
>
> I didnt go round and round on this issue.
> I wrote a review of Burch's thesis which
> I believe settled the matter conclusively.
> (J. Man-Machine Studies, 1995)
>
> But I just wanted to add a couple of new remarks:
>
> > > Peirce was misled by an analogy between logic and chemistry,
> > > which wasnt a bad idea in 1885, but seems kind of daft in
> > > hindsight. He seems to have thought that the associations
> > > between relations and their instances, which he was encoding
> > > as arcs in his graphs, were like valency in chemistry ...
>
> First of all, Peirce was a buddy of Sylvester's at
> Johns Hopkins University, and Cayley was a visitor
> there. Graph theory in all its glory was a big topic
> there. So he wasn't exactly naive about what can and
> cannot be done with graphs.
>
> I didnt say he was. His result is perfectly fine
> as a theorem in graph theory.
>
> > Second, Whitehead also remarked that dyadic relations
> > weren't sufficient for knowledge rep. Last week,
> > I scanned a few pages from Chapter 9 of his book
> > "Concept of Nature," but I don't want to bother
> > scanning the whole thing. However, if you have
> > it (or can find it), I recommend Ch. 8,
> > in which he remarks:
>
> | Other schools of philosophy admit relations but obstinately refuse
> | to contemplate relations that have more than two relata ... (p.150)
>
> > W. then goes on to discuss "percipient events", which require more
> > than 2 relata -- just the kind of examples that P. was dealing with.
>
> 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)) ).
>
> Since a binary relation has a name and relates two other things,
> it is conventionally called a 'triple' in data structure terminology
> (eg in RDF), and this ghost of trinaricity is where Peirce's simple
> graph-theoretic result can be glimpsed. His error was to conclude
> that this implied that a trinary *relation* was necessary; and
> that in turn was because he made the error of thinking that the
> assertion of identity between n things involved an irreducibly
> n-ary identity relation.
>
> Pat Hayes
>
> http://suo.ieee.org/email/msg04007.html
So much to say, so little volume in this or so many universes of discourse!
Aside from the gross ignorance of intellectual history, as to who knew what,
and when did they know it, the likes of which can only come from never having
bothered to read anything but the accounts of those who thought that our human
brain-pans did not retain the full-fledged volume of reason until the late great
Twentieth Century Limited steamed onto the scene of all negligible prior evolution,
well, and the spectacle of a Journal whose peers are so busy peering into the lint
of their navels that they do not even notice the greater half of the human race,
much less read what they deem fit to print -- but I must learn to speak more
plainly someday -- yet I am determined to stick to my logical guns today.
So let me limit myself to this excerpt from yesterday's voluminosity,
which I vaunt to believe may serve to settle the matter conclusively:
Jon Awbrey wrote:
>
> These examples, which were intended merely as explorations
> of certain dimensions of variation in the neighborhood of
> a trivial "change of variables" formula, analogous to the
> lambda calculus "alpha rule", I think, never had much to
> do with the analysis of any relation, since they do not
> bear on the structure of any sets, per se, but only on
> the structures of single tuples.
>
> Once again, the initial example said only this:
>
> | L(x) iff L(y) for some y = x.
>
> Equivalently, here are two other ways to say it:
>
> | L(x) iff there exists y such that L(y) and y = x.
> |
> | x in L iff there exists y in L such that y = x.
>
> Since x and y are k-tuples, we bring in
> the definition of equality for k-tuples:
>
> | x = y
> |
> | iff
> |
> | Proj<j>(x) = Proj<j>(y) for j = 1 to k.
>
> Just to be tricky, hopefully without a self-deception --
> but that all depends on what it is we wish to believe --
> we exploit a standardized abuse of notation and write:
>
> | L(x<1>, ..., x<k>)
> |
> | iff
> |
> | There exists y such that L(y) and Proj<j>(y) = x<j> for j = 1 to k.
>
> If you really want to try and pull a fast one,
> you can change the second L to a L', and thus:
>
> | L(x<1>, ..., x<k>)
> |
> | iff
> |
> | There exists y such that L'(y) and Proj<j>(y) = x<j> for j = 1 to k.
>
> Voila! Instant profundamento!
>
> | 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))
> |
> | http://suo.ieee.org/email/msg04007.html
>
> Now, one more time, we are not even up to the levels of "analyzing" --
> in other words "saying anything significant about the structure of" --
> a relation, because a relation is a set of tuples, and we are just
> talking about isolated tuples here. The fact that KIF seems to be
> a partner in this prestidigitation, to the extent that it does not
> help to expose the sleight-of-hand for what it is, is a thing that
> is probably unfair to blame on a language, in lieu of its speakers,
> but I guess that would take me hearing of another sort of speakers.
>
> I hope I have put this plain enough.
> If not, I promise to keep on trying.
>
> http://suo.ieee.org/email/msg04900.html
Okay, enough -- I hope -- and I hope, too, that we can all now begin
to crawl forward just a bit, on the off chance that getting anything
at all clear about the nature of relations might just turn out to be
of utility in the modules that we use to model that grander universe.
Looking Forward To It,
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤