zeleny at math.ucla.edu
Tue Mar 23 16:46:29 EST 1999
From: Charles Silver <csilver at sophia.smith.edu>
>On Tue, 23 Mar 1999, Randall Holmes wrote:
>>Dear Dr. Kanovei,
>>I reply to your letter point by point.
>>a) 2nd order logic axiomatizes N caregorically
>>through appeal to subsets of N.
>>This is not the best way to see it. I prefer to say that one appeals
>>to properties rather than sets (because of the "topic-neutral" character
>>required of a logic).
> One thing that bothers me about talking of properties is that
>there don't seem to be desirable, *extensional* identity conditions for
>them. For example, two sets are equal if they have all their elements in
>common. I don't know of any such conditions for properties. Take the
>property of being under ten feet tall and the property of being under nine
>feet tall. Every human that has one property has the other. But, are the
>properties themselves the same? I don't think so. I would be more
>comfortable talking of properties if I knew some nonintensional way of
>identifying them. A long time ago, I fiddled with this a little, but
>nothing seemed satisfactory. Does anyone know of such conditions?
Church's Logic of Sense and Denotation identifies three categories of
extensional identity conditions for properties and relations, which
are construed as special cases of unary truth functions of order N+1,
encoding N-ary truth functions via the Schoenfinkel reduction. See
the publications in Nous of 1973-4 and 1994, and Charles Parsons' JSL
account of one of Church's alternatives.
>(I tend to think attaching possible-world considerations would only make
>matters worse, but maybe I'm wrong.)
The earlier Nous publication outlines a possible worlds model, whereas
the latter proves consistency via a class of set-theoretic models.
Cordially -- Mikhail Zeleny at math.ucla.edu * MZ at ptyx.com ** www.ptyx.com
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