FOM: The Epistemic Status of Arbitrary Objects
Sean C Stidd
sean.stidd at juno.com
Thu Jan 31 09:35:03 EST 2002
I take it (on the basis of a note from Professor Silver) that part of the
point of the Fine/Silver defense of arbitrary objects is that we 'need'
them conceptually to make sense of universal quantiification - that
somehow the notion of an arbitrary object is relatively basic to our
human understanding of proofs. The following argument of Professsor
Friedman's might seem to bear that out:
>CONSERVATIVE EXTENSIONS.
>
>We can now ask
>
>**Is this extension of first order predicate calculus conservative
>over
>first order predicate calculus?**
>
>The obvious answer is yes. This is because given any structure
>interpreting
>everything except Arb, we can interpret Arb to hold of and only of
>any
>particular object in the domain.
Is it mere pettifogging to suppose that "given any structure interpreting
everything except Arb" has roughly the same meaning as "let S be an
arbitrary structure interpreting Arb"? If that is correct, then Professor
Friedman here is employing the notion of arbitrary object to show that
arbitrary objects are a conservative extension of the predicate calculus.
If we think that "let a be an arbitrary object" is really interpretable
in terms of "any", "all," and universal quantifiers, as I take it it
would be in a fully formalized proof, AND that we have a better grasp on
the latter notions than the former, then arbitrary objects can be gotten
rid of in the epistemic lexicon; quantifiers over given domains would
constitute an analysis of the "arbitrary object"-locution. But if you
think that we can only make sense of quantifiers in mathematics on the
basis of some antecedent grasp of the notion of arbitrary objects,
there's still a (philosophical) problem here.
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