SUO: Re: Focus and Volume
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Pat Hayes wrote:
>
> > Phil, Jon, Jim, et al.,
> >
> > I agree that the points Jon A. is trying to make are often obscure,
> > and I also agree that Jon would make his case much stronger if he
> > would state what to him may seem obvious. But I also believe that
> > the Donald Duck example is very pertinent to the discussion.
> >
> > Philip Jackson wrote:
> >
> > > Perhaps Jon can explain the relevance of his postings which seem
> > > quite frivolous, such as information about Donald Duck's nephews...
>
> Really, shouldnt the SUO list provide enough evidence that
> it is unwise to suggest that Jon explain *anything*?
Those of you who are determined to instruct me in the fine art of plain speaking,
please give some kind of a hint as to whether this sort of statement from Pat is:
1. blunt.
2. obnoxious.
3. typical.
4. inadvertently missing the requisite number of happy faces.
5. all of the above.
Since I am determined to model my participation in this discourse
on the best established standards, I earnestly await your advice.
Thanks In Prospect --
I now return to the program already in progress ...
> > Following is the relevant excerpt, stated in KIF:
> >
> > 1.
> >
> > | (forall (?x ?y ?z)
> > | (<=> (TitanicTrio ?x ?y ?z)
> > | (exists (?e)
> > | (and (TitanicTrio ?e)
> > | (Member1 ?e ?x)
> > | (Member2 ?e ?y)
> > | (Member3 ?e ?z)
> > | ) ) ) )
> >
> > 2.
> >
> > | (forall (?x ?y ?z)
> > | (<=> (TitanicTrio ?x ?y ?z)
> > | (exists (?e)
> > | (and (Donald ?e)
> > | (1st Nephew ?e ?x)
> > | (2nd Nephew ?e ?y)
> > | (3rd Nephew ?e ?z)
> > | ) ) ) )
I have almost forgotten what this was all about,
and Matthew and I have long since moved on to
more Nuts'n'Bolts types of issues -- you are,
of course, free to speculate on which of us
is which -- but I do remember the intention
to put these two examples in a better form:
> > 1.
> >
> > | (forall (?x ?y ?z)
> > | (<=> (TitanicTrio ?x ?y ?z)
> > | (exists (?e)
> > | (and (TitanicTrio ?e)
> > | (MallardATrois1 ?e ?x)
> > | (MallardATrois2 ?e ?y)
> > | (MallardATrois3 ?e ?z)
> > | ) ) ) )
> >
> > 2.
> >
> > | (forall (?x ?y ?z)
> > | (<=> (TitanicTrio ?x ?y ?z)
> > | (exists (?e)
> > | (and (DonaldDuck ?e)
> > | (Nepotismaniac1 ?e ?x)
> > | (Nepotismaniac2 ?e ?y)
> > | (Nepotismaniac3 ?e ?z)
> > | ) ) ) )
Okay, I feel much better now. Now I know that it was really a safe bet,
because charitable and sensible people tend to interpret expressions in
the light of many reasonable presumptions of an appropriate context for
them, that you all knew that I meant "DonaldDuck" when I wrote "Donald",
but, of course, those extra four bytes are enough to narrow the choices,
from a big mess down to one only, even if, as a kid, you did have a pet
duck that you naturally named "Donald", like, most likely, thousands of
other children, but that cannot possibly be admitted as a viable option,
and so you may take the bare suggestion of it as the most outlandish of
canards against the absolute information-theoretic certainties of being
able to identify unique, even if fictional, individuals with naught but
a finite number of bits. But that was only my sleightest worry, anyway.
The other problems that I felt I should fix were these:
1. Even though there was probably no chance
that "Element1", "Element2", "Element3"
would have been read in a set-theoretic
sense, instead of naming the components
of a 3-tuple, I think that it is best to
avoid any potential for confusion, since
it would be downright silly to think that
these sorts of expressions would serve as
free-standing axioms, that is, without some
medium of set-theoretic or kinship-theoretic
axioms to sustain them.
2. I needed to correct the absence of the properly run-on
concatenation style in the forms for the "j^th Nephew".
Okay, it is pretty trivial stuff, but you never know when a seemingly
infinitesimal bit of sloppiness might evolve into a major catastrophe.
The only new thought that I have here, in the light of Pat's remarks,
is that probably the symmetry of these parallel cases would be broke
in the magnetic field, as it were, of the suitable environing axioms
about sets, tuples, kinship relations, and so on, but that's just an
off-the-cuff guess, and may take quite a bit more thought on my part.
> > These two KIF statements have exactly the same structure, except that
> > the first one assumes the existence of an abstract entity ?e of type
> > TitanicTrio (i.e., something very much like a set, which most of us
> > have agreed to be abstract) while the second one assumes the existence
> > of an entity ?e that happens to be a duck named Donald.
> >
> > I believe that this example is a good answer to the claim
> > that triadic relations can be reduced to dyadic relations
> > by postulating the existence of some new entity ?e. Although
> > that point may be formally true, Jon's example shows that there
> > is no way of knowing whether that entity is something abstract,
> > such as a set or some hypothetical TitanicTrio, or something
> > concrete such as a Duck.
>
> I had not seen this before (all email from JA is filtered out of my
> SUO mailbox), but it deserves some comment. In fact, the claim that
> is risky here is that the first of these refers to an abstract entity
> and the second to a duck. That might be what the writer intended,
> but it isn't what the KIF says. Being isomorphic, these two pieces
> of KIF, taken in isolation, have precisely the same models: any
> interpretation that makes one true can be easily transformed into
> an isomomorphic interpretation, over the same universe, that makes
> the other true.
>
> I presume that what John means is that the *intended*
> intepreretations are different in the way he describes.
> But the intentions of the writers of axioms have no place
> in any kind of formal analysis of the meanings of those axioms
> (other than by exclusion, in the sense that one can formally show
> that the actual meaning does not correspond to the intended meaning,
> cf. the previous paragraph.) So this kind of example can hardly be
> claimed by Jon (or anyone else) to be any kind of *demonstration*
> of some kind of irreducibility. At most, it could show that some
> distinction that Jon (or someone) has in mind, cannot be represented
> in the formal notation. To get from that to an irreducibility result
> requires showing that the distinction *can* be represented in some
> other formalism. Good luck with the mathematical theory of ducks,
> whoever sets out to show this.
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 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 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, once more time, we are not even up to the level 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.
Jon Awbrey
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