From: PL436000@brownvm.brown.edu (Jamie)
Newsgroups: sci.philosophy.tech,sci.logic
Subject: Quantifiers
Date: 30 Dec 92 20:09:11 GMT
Organization: Brown University - Providence, Rhode Island  USA
NNTP-Posting-Host: brownvm.brown.edu
News-Software: BNN via BNN_POST v1.0 beta

>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
Newsgroups: sci.philosophy.tech,sci.logic