🚀 go-pugleaf

In article <1hntpkINNnp8@cat.cis.Brown.EDU> PL436000@brownvm.brown.edu (Jamie) writes:
>Randall,
>
>>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?

>
>But, I don't understand how using unreified predicates instead of
>objects helps avoid set theoretic paradoxes. Grelling's paradox
>uses only a predicate, and (unless I'm very confused) does not
>require reification.

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.

>
>You said earlier that the REAL trick is denying Separation.
>That seemed promising. Can I follow up a bit?

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.

>
>I can talk about everything all at once. But everything all at
>once is not a completed totality. Check.

In ZFC, this is the case.  In NFU, "everything all at once" _is_ a
completed totality (at least, it is reified; "completed" smacks of the
iterative hierarchy to me).

>
>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.

>
>(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.

>
>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.

>
>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.

>
>Where do you get off? (Is the problem inherent in speaking of a "range"?
>I will be surprised if it is.)

I think my whole point is that speaking of ranges is optional.

>
>Jamie



-- 
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