Newsgroups: rec.arts.books,sci.philosophy.tech
Subject: Re: A note on Modal Logic that has nothing to do with Ikky Sex
From: zeleny@husc10.harvard.edu (Michael Zeleny)
Date: 22 Dec 92 01:33:55 EST
Followup-To: sci.philosophy.tech
Organization: The Phallogocentric Cabal

I am directing the follow-up to a more dessicated venue.

In article <1992Dec21.180311.14665@guinness.idbsu.edu>
holmes@opal.idbsu.edu (Randall Holmes) writes:

>In article <1992Dec18.205843.18689@husc3.harvard.edu>
>zeleny@husc10.harvard.edu (Michael Zeleny) writes:

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

RH:
>	I conceded a very small fraction of a point, actually...

Given my readiness to discount the disagreements stemming from our
divergence on the subject of the One True Set Theory, I am most
satisfied with the extent of your concession.  Who knows, if we do
this sort of thing more often, you might become some sort of pluralist
after all...

>>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:

RH:
>[...]

What, Randall, an actual ellipsis?

MZ:
>>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

RN:
>It is possible to work in ZFC without referring to classes at all.
>There is no need ever to refer to super-classes, etc.

I have no idea why you think that this observation would vitiate any of
my points.  Sure you can work in ZFC without *referring* to classes at
all, -- but only if you are a formalist concerned solely with the
syntactical aspects of the theory, in particular ignoring all semantical
considerations.  More precisely, if you truly espouse the Quinean
extensionalism, you may eliminate all reference to metatheoretical
entities like super-classes; however you will purchase this liberty at
the cost of inscrutability of reference.

To continue the Quinian idiom, you are proud as Punch of the proxy
function which injects the universe of ZFC into that of NFU(+AxInf+AC).
However, given that your function is not a bijection, you have stopped
short of effecting an ontological reduction.  Therefore I contend that
your interpretation of ZFC in NFU is, in fact, a misinterpretation of a
theory built on incommensurable premisses.  In view of the above
considerations, you are in no position to refute my claim.

>[...]

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.

MZ:
>>You know better than that, Randall.  As a good Quinean (or are you?),

RH:
>Merely a fellow-traveller.

Careful where you draw the line!

MZ:
>>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...

RH:
>"For all x, x = x" is a well-formed and true statement of ZFC.  It
>says something about every object without exception.

Indeed.  Why do you think that this contradicts any of my claims?

On second thought, it does, in the following way.  Consider a set
theorist who, perhaps unlike yourself, believes that ZFC has semantics.
Thus his version of set theory properly consists of a unified language,
consisting of (say) the first-order L_{ZFC} and the first-order language
of the model theory of the same, L_{M(ZFC)}.  Now, the latter happens to
contain a recursive truth definition for the former, as well as a a name
for each of its terms and sentences.  (I emphasize that it is a certain
kind of union of both languages, that our hero regards as comprising his
Set Theory.  Were he an intensionalist, he would modify my list by
adding a recursive *sense* definition for each term, which he would then
use to express a truth definition.)  Clearly, the (Tarskian, to be sure)
lesson of this situation is that our hero is not at liberty to regard
the variables of L_{ZFC} as ranging over *everything*, including the
terms and sentences of L_{M(ZFC)}, at least as long as he remains
unwilling to render inconsistent his Grand Unified Set Theory.  At this
point, he has two reasonable solutions to his predicament, one of which
is a special case of the other.  For more details, I refer you to
Church's 1975 JSL article I have cited elsewhere.

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.

MZ:
>>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...

RH:
>On the contrary, predicates are best understood via incomplete
>sentences in meta-theory (not "as" incomplete sentences; the whole
>point is that predicates do not have to be reified, which is
>fortunate, since some cannot be reified).  If one quantifies over
>predicates, one is then forced to admit that one is referring to
>something.

Please elaborate.  "Nothing can be said truly of what does not exist."

>[...]

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.

MZ:
>>Vide supra.

RH:
>I haven't seen any relevant comment.  ZFC does indulge in the
>definition of sets via unrestricted quantification.  If one deprives
>oneself of this indulgence, one is stuck in "bounded Zermelo set
>theory", which is essentially the theory of types.

Randall, I do not recall having contested unrestricted quantification in
*your* sense; I am merely arguing that taking seriously the *semantics*
of ZFC, whose existence you are evidently prepared to deny, makes such
quantification depend on a Tarskian hierarchy of classes, super-classes,
etc....  Recall that any quantification over the objects of a Tarskian
theory, must necessarily exclude from its range the objects particular
to its metatheory.  This claim is so simple and so uncontroversial, that
I wonder why you would bother to challenge it at all.

Please explain the terrible logical flaws contained in my argument, or
concede my point.

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.

MZ:
>>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?

RH:
>The axiom of replacement is also an illicit assumption in this
>context.  The universe is not an inconsistent totality; it is
>incompatible with the axiom of separation or the axiom of replacement,
>to be sure.

One man's illicit assumption is another man's analytic truth.
Inscrutability of reference strikes again!

>[...]

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 domain that quantifiers range over in ZFC is the class {x|x = x},
>and, assuming the axioms of ZFC, this is an illegitimate totality.
>There is nothing to indicate that the contextually relevant domain in
>ZFC is _not_ the sum total of everything, and this is even less clear
>in ZFA (ZF with atoms) (the Great Pyramid of Gizeh then does not stand
>accused of being a well-founded set).  Quantifiers in ZFC are not
>limited in any way which is relevant inside the theory, and there is
>little reason to believe that there is anything outside the domain of
>ZFC (or at least ZFA).

Again, your sense of being "inside the theory" is quite alien to my
theory of language.  This is a basic point of disagreement between us,
and your failure to acknowledge it only supports my charges of a Quinian
_advocatus diaboli_.

>>>>>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.

MZ:
>>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.

RH:
>I can.

Show me.  Describe a model of NFU, which is *philosophically*
competitive with the iterative hierarchy, a.k.a. the intended
interpretation of ZFC.  Keep in mind that I have successfully explained
the latter to audiences whose average mathematical background did not
exceed the curriculum of a semi-decent American (!) high school.  Should
you succeed in doing so, I promise to renounce all claims of superiority
of the limitation of size doctrine.

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

RH:
>Hah!

ObBooks: Julius Moravcsik, _Thought and Language_.  Puts Quine in his
place, as if it were necessary after the wily pragmatist was forced to
eat his words on the subject of Church's intensional logic (see the
successive editions of "Reference and Modality").

>[...]

>-- 
>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

cordially,
mikhail zeleny@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."