From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: rec.arts.books,sci.philosophy.tech
Subject: Re: A note on Modal Logic that has nothing to do with Ikky Sex
Summary: still nothing to do with ikky sex
Date: 19 Dec 92 01:58:41 GMT
Organization: The Phallogocentric Cabal

Randall, evidently smarting from having had to concede a point to a
lowly perpetual undergraduate, valiantly jumps into the fray:

In article <1992Dec18.163009.28035@guinness.idbsu.edu>
holmes@opal.idbsu.edu (Randall Holmes) writes:

>In article <1992Dec18.015229.18660@husc3.harvard.edu>
>zeleny@husc10.harvard.edu (Michael Zeleny) writes:

>>In article <1gr2fmINNl8s@cat.cis.Brown.EDU> 
>>PL436000@brownvm.brown.edu (Jamie) writes:

>>>zeleny@husc10.harvard.edu (Michael Zeleny) said:

MZ:
>>>>Finally, logicians quantify over all sets all the time; that's what
>>>>the notion of a proper class is for.

RH:
>Not precisely.  One quantifies over all sets in ZFC proper, in which
>there is no essential reference to proper classes (all such references
>are eliminable).

If you want precision, then be precise.  References to proper classes
are *only* eliminable in the object language, because they are confined
to the metalanguage.  Consider, for instance, the discussion in the
beginning of Kunen's _Set Theory_, I.8.  I agree with Church that any
(genetic) set theory gives rise to the set, class, super-class, etc....
hierarchy in one way or another; I differ from him in not regarding it
as an absolute shortcoming, relative to type theory.  (Moreover, I
suspect that some corresponding stratification obtains within type-free
theories; but my ignorance prevents me from pressing this point further

>[...]

J:
>>>I suppose we can quantify over all the sets. But can we quantify over
>>>everything at once? Intuitively we certainly can. "When I say
>>>'everything,' I mean 'EVERYTHING.'" Didn't I just quantify over
>>>everything at once? EVERYTHING at once?

RH:
>That's what "for all x, P" means; and it is certainly true that for
>all x, x = x, and I just said it.

You know better than that, Randall.  As a good Quinean (or are you?),
you should be aware that reference is inscrutable between theories,
whilst intratheoretical reference depends on all of your cognitive
baggage.  What you say may be meaningful to you, but it is utterly
meaningless to all right-thinking ZF patriots.  Methinks you ought to
reconsider ontological relativity...

MZ:
>>I have another one, which you may contest: any instance of predication
>>*seems* to presuppose a universal applicability.  But we all know what
>>the semantical antinomies do to that one...

RH:
>Predication does not involve application, because the predicate is not
>necessarily understood to be an object.

As you discuss predicates in metatheory, your singular terms commit you
to recognizing them as objects, in the same way the formulation of
first-order logic commits you to a second-order metatheory.  Methinks
you ought to meditate on what there is...

>[...]

J:
>>>Does Zeleny think it is possible to quantify over everything?

MZ:
>>I think not.  "Everything" is an inconsistent multiplicity, which does
>>not admit of reference, much less quantification.  (The reason we can
>>understand this sort of discourse, is because it has the semantics of a
>>figurative circumlocution.)  There is no such thing as the universe, or
>>else if there is one, it cannot be designated.

RH:
>{x|x = x}.

Wrong again, my friend.  Syntactically, abstraction terms are
contextually eliminable in favor of existential statements, e.g. via
Church's schema.  What you just said is not a term, but a declarative
sentence, and one that is demonstrably false in the *true* set theory.
Inscrutability of reference, and all that...

MZ
>>						(Am I sounding like
>>Gorgias yet?)  Naturally, Randall Holmes will disagree, but I am
>>speaking as a _bona fide_ ZFC chauvinist.

RH:
>No, you aren't.  ZFC is irrevocably committed to quantification over
>absolutely everything; if quantifiers in ZFC have to be restricted to
>sets, one obtains a theory with the precise strength of the theory of
>types; for example, one cannot prove the existence of aleph-omega.

Vide supra.

RH:
>The proof of the consistency of NFU relative to ZFC shows that
>"everything" is not an inconsistent totality.  Get it through your
>head.  Nothing speculative about it; this is cold, hard, mathematical
>fact.  "Everything" inherits the inconsistent nature of the Russell
>class if one assumes the axiom of separation; it is not in itself
>paradoxical.

But the principle of separation is true, as a consequence of the Axiom
of Replacement!  No assumptions here.  Look, we have been through all
this before; why not agree to disagree, as I suggested above?

>[...]

MZ:
>>I believe that it is implicitly qualified to the contextually relevant
>>universe of discourse.  Which is not to say that one cannot quantify
>>over all sets, or all classes, or all cumquats, or all pubic hairs,
>>or...

RH:
>The "everything" which is an inconsistent totality in ZFC is the
>implicit universe of discourse itself, not some larger totality.

Huh?

>>>Jamie

MZ:
>>ObBook: Michael Hallett, _Cantorian set theory and limitation of size_.
>>Reasonably accessible to the dedicated amateur, willing to look things
>>up.

RH:
>It's a good book.  But there are other approaches.

Indeed there are.  Unfortunately, either they are false, or just not
sufficiently developed to offer a convincing philosophical alternative;
at this time, I doubt that anyone can tell whether the former or the
latter is, in fact, true.

>-- 
>The opinions expressed		|     --Sincerely,
>above are not the "official"	|     M. Randall Holmes
>opinions of any person		|     Math. Dept., Boise State Univ.
>or institution.			|     holmes@opal.idbsu.edu

ObBooks: Anything by Quine.  A class act, even if it is dead wrong most
of the time.

ObNonexistentBooks: Church's _Introduction to Mathematical Logic_,
Volume II.  Will trade a xerox copy of Chapter VII for manuscripts of
any other parts.
 
cordially,
mikhail zeleny@husc.harvard.edu
"Nothing can be said truly of what does not exist."