[FOM] Deflationism
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Wed Mar 22 21:32:46 EST 2006
On Wed, 22 Mar 2006 A.S.Virdi at lse.ac.uk wrote:
> It seems reasonable to suggest then that certain logico-mathematical
> principles like the soundness statement for PA are exactly the kind of
> statements that the deflationist wants in her arsenal. "All theorems of
> PA are true" finitarily expresses an infinite conjunction. To be unable
> to prove this from her theory of truth (plus, of course, her theory of
> natural number) would be an embarrassment for her.
Here's the rub. There is no such thing as "the" soundness statement for
PA; for there are various ways to express the thought in question. One way
to do so, eschewing the use of a truth-predicate altogether, is to state a
a reflection principle involving the provability-predicate for PA. Why
should that not be a good enough addition to the deflationist's "arsenal"?
She can avoid the alleged "embarrassment" without even invoking her theory
to truth, such as it is.
There is not much weight to be laid on the fact that a reflection
principle is schematic, while "All theorems of PA are true" is a single
sentence. After all, the deflationist's own T-schema is just that: a
schema.
I wonder also whether deflationists (or those who, like yourself, know
exactly where they stand, or at least where they are coming from) have
given any thought to the question whether the potentially infinite sets X
of sentences that one might wish to "conjoin" by saying "Every member of X
is true" could be required to be decidable? For, suppose one were to
impose that requirement. Then one could deal deflationarily with arbitrary
X as follows:
If X is decidable, then state "Every member of X is true"; but
if X is not decidable, then adopt the Schema
"if p can be shown to be in X, then p".
This approach would still enable the deflationist to avoid having to
say "Every theorem of PA is true", since the set of theorems of PA is
(if consistent) not decidable.
Neil Tennant
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