From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Quantifiers
Organization: Boise State University
Date: Wed, 30 Dec 1992 23:04:19 GMT

In article <1hsvljINNb0h@cat.cis.Brown.EDU> PL436000@brownvm.brown.edu (Jamie) writes:
>>From: holmes@opal.idbsu.edu (Randall Holmes)
>
>Oh, good, he's back.
>
>I'm only going to reply to this one, not to the later two.
>
>>>>I do not agree with Zeleny that quantification involves ontological
>>>>commitment to the domain over which one is quantifying as a completed
>>>>totality.  It merely involves commitment to each of the objects in the
>>>>domain.  I'll admit that I am committed to an ontology including all
>>>>of the quantification ranges, since there is only one, the universe,
>>>>which is in fact an object!  But this involves taking unfair advantage
>>>>of my NF advocacy; even from a ZFC standpoint, it can be observed that
>>>>quantification ranges are not necessarily objects; they can be
>>>>non-reified predicates, and they can be "referred to" using
>>>>syntactical means.
>>>
>>>I hope I don't have to ask for the longer reply!
>>
>>Well, yes, you do.  Do you want it?
>
>Yes (he said with some trepidation).
>
>>I'm not sure which paradox is Grellings's.  It is possible to state
>>semantic paradoxes using quoted predicates instead of reified
>>predicates; for instance "'Yields a falsehood when preceded by its
>>quotation' yields a falsehood when preceded by its quotation".  What
>>"gives" here is the existence of the truth (or falsehood) predicate
>>for sentences.  But these are not set-theoretical paradoxes; the set
>>theoretical paradoxes all involve reification.
>
>Ok. You won't get a set-theoretic paradox unless the work is done
>with sets. I should have understood what you were saying, but
>I didn't.
>
>You now know what Grelling's paradox is. I suppose it won't be the
>Truth predicate that "gives," then, but the "satisfaction"
>predicate.
>
>>Not exactly.  The trick I favor is to adopt the axiom of stratified
>>comprehension, which implies that separation must be false.  I have
>>been at pains to point out that my objections to Zeleny's arguments on
>>this thread have the same force from the usual ZFC standpoint.
>>Russell's paradox shows that Separation must fail in any set theory
>>with a universal set, but I see no virtue in abandoning Separation
>>_per se_; I abandon it in pursuit of something else.
>
>Ok.
>
>>>I can restrict my quantifiers, too. I suppose that denying Separation
>>>means that there are some ways I cannot restrict them. For example,
>>>I CAN say something like, "Everything blue is also square." May
>>>I also say, meaningfully, "Every quantifier that has another
>>>quantifier in its range commits its employer to abstract objects"?
>>
>>No.  First-order logic does not commit one to abstract objects at all.
>
>No, but I thought you had no compunctions about abstract objects.
>So I'm not following you here.
>Oh, wait, I get it.
>As the sequel shows, I wasn't asking whether I could say that TRULY.
>
>>>(I am not at all interested, right now, in whether I can say that
>>>TRULY, only MEANINGFULLY.)
>>
>>I conjecture that you can say it meaningfully, but I think that it
>>comes down to some kind of statement about syntax.  I'm not certain
>>about this.  This is related to Zeleny's question about whether one
>>can state the principle of ontological commitment.
>
>My question was whether I could (a) quantify over quantifiers, and
>(b) restrict my quantifiers by using any meaningful predicate I choose.
>
>If I can, then I essentially have Separation for quantifiers.
>
>I will explain shortly (but maybe not terribly clearly).
>
>>>Can I meaningfully say, "Some quantifiers have themselves in their
>>range."?
>>
>>Eh?  A quantifier is a syntactical device (a character string or a
>>Godel number); it falls in the range of lots of quantifiers.
>
>So, I suppose the answer is "Yes."
>
>>>I won't insult you by leading you down such an obvious path. You can
>>>see better than I can where this is going.
>>
>>As I said, I'm not certain.
>
>Ok, here goes.
>
>Quantifier Separation:
>For any meaningful quantifier, Q, and any meaningful open formula, F,
>there is another meaningful quantifier, Q*, such that Q* ranges
>over all and only those things x such that (Q ranges over x AND
>x satisfies F).
>
>Quantifier Universality:
>There is an unrestricted quantifier, one which ranges over everything.
>
>These two seem to be inconsistent.
>Let F be the formula "__does not range over itself." Let Q be the
>Unrestricted Quantifier. Then Q* (from Quantifier Separation)
>is a quantifier that ranges over all and only those things that
>do not range over themselves. But this is impossible, since
>Q* would then range over itself iff it did not.
>
>Let me guess.
>Stratification, right? For quantifiers?
>
>Jamie

Restricted quantifiers, in ZFC or NF, are nothing more than formulae
with one free variable; (for all x in range Q)(P) means nothing more
than (for _all_ x)(if "Q holds of x" then P).  There is no other way
of introducing restricted quantifiers in either theory.

Not stratification, since the universal quantifier exists in ZFC, too.
Your principle of Quantifier Separation is no problem.  Your
quantifiers might as well be "formulae with one free variable" (in the
language of ZFC + constants for every object, for the sake of
argument): your principle of quantifier separation is then seen to
assert that for each formula Q and formula F, there is a formula Q*
satisfied by exactly those x such that x satisfies Q and x satisfies
F.  This is true: the formula is "Q and F".  But your purported
formula "x does not range over x" is not a formula, because it
translates to "formula x does not satisfy formula x", and the notion
of satisfaction cannot be expressed by any formula.  Your argument is
just flat wrong.

Give it up:  universal quantification is _not_ paradoxical.




-- 
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
Newsgroups: sci.philosophy.tech