From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: No Reification Here
Organization: Boise State University
Date: Fri, 1 Jan 1993 18:24:08 GMT

In article <1992Dec31.144204.18918@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>In article <1992Dec31.171138.18363@guinness.idbsu.edu>
>holmes@opal.idbsu.edu (Randall Holmes) writes:
>
>>In article <1hteqjINN85h@tamsun.tamu.edu> 
>>cmenzel@kbssun1.tamu.edu (Chris Menzel) writes:
>
>>>holmes@opal.idbsu.edu (Randall Holmes) writes:
>
>RH:
>>>>First-order logic does not commit one to abstract objects at all.
>
>CM:
>>>Seems hasty, Randall.  Don't you think the semantics for first-order
>>>logic (or better perhaps, a first-order language) commits you to
>>>(among other things) the semantic entities that interpret the
>>>constants and predicates of the language?  If so, then you're at least
>>>committed to sets or classes or, for friends of the forms, properties
>>>and relations as the meanings of predicates.
>
>RH:
>>First-order logic doesn't commit one to the semantics for first-order
>>logic, either.  The semantics for first-order logic certainly does
>>involve some set theory.
>
>Good.  And what does the semantics for set theory involve?

The intended interpretation of set theory doesn't have any semantics
(in the purely technical sense that it is not a _model_; its domain
and its primitive relations are not sets; thus technical semantics of
the kind Zeleny is discussing does not apply; I am not saying that the
terms and sentences of the intended interpretation do not have
referents and truth-values).

>
>RH:
>>Personally, I am committed to set theory, and so to semantics for
>>_some_ interpretations of first-order logic (all consistent theories
>>have countable models, after all).
>
>Nonsense.  ZFC^2 is consistent, and has no countable models.

All consistent first-order theories have countable models.  ZFC^2 can
be re-cast as a first-order theory with an additional sort of sets of
ZFC sets, and this first-order theory (the same as ZFC^2 from a
Quinean standpoint) has countable models (if it is consistent).  If we
accept higher-order logic as logic, this is an illegitimate maneuver,
of course.

>
>RH:
>>				    But the intended interpretation of
>>Cantorian set theory _cannot_ have semantics in the sense indicated.
>>Its domain of discourse is _not_ a set, and most of its predicates are
>>_not_ sets or relations (all are "too large").  This is not specific
>>to Cantorian set theory; the same holds for the set theories derived
>>from NF.  If one tries to get around this by using proper classes,
>>super-classes, and so forth, one still has the same problem at a
>>higher level (the theory of the classes, super-classes, and so forth).
>
>That's precisely my point: the true semantics of ZFC presupposes a
>proper class theory, which likewise presupposes a proper super-class
>theory, and so forth.  May I assume that you have conceded my point?

No, you continue to misunderstand me.  When I said that ZFC has
semantics, I made it very clear what I meant, exactly so as to avoid
allowing semantics for (the intended interpretation of) ZFC in this
technical sense, which I deny.  I reiterate that your position is
quite incompatible with the philosophy of Cantor.

>
>>>--Chris Menzel
>>>  Philosophy Department
>>>  Texas A&M University
>
>>-- 
>>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."
>


-- 
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