FOM: more on reflection, prosentences etc.
neilt at mercutio.cohums.ohio-state.edu
Mon Jan 11 20:57:07 EST 1999
Volker Halbach writes:
> I am not aware of an axiomatization of either the prosentences or the
> substitutional quantification, but I would expect that the result would be
> equivalent to the addition of a truth predicate with suitable axioms.
> Furthermore prosentences and substitutional variables are not contained in
> the language of T, because otherwise Tarski's Theorem would apply (This is
> again only a guess, because I don't know the appropriate axiomatizations of
> these devices).
Yes, that's right; prosentences are not in the language of T. The idea
would be to adopt a prosentential extension of T so that one could
mimic desirable results (in an extension of the original theory) that
could be had from the addition of a truth predicate. But I'd like to
see a proof that the theory-extension obtained by following this
prosentential strategy would actually be *equivalent* to the extension
obtained by adding a truth-predicate. It would depend on how one
defined "equivalence". Certainly, there are no prosentential analogues
of the T-sentences ("p" is T iff p) that would be needed in order to
ensure that the truth theory was materially adequate. So in one sense,
the extension of the original theory via a truth-predicate would be
stronger than a prosentential extension (or perhaps logically
incomparable to it).
In a recent paper entitled "Truth and Proof: Through Thick and Thin",
in the Journal of Philosophy, XCV, Oct. 1998, pp.493-521, my colleague
Stewart Shapiro pointed out how the truth-predicate-extension of PA
would allow one to prove Con(PA). This was part of an argument against
deflationism concerning truth. Shapiro's point is that the
truth-predicate-extension is non-conservative over the original
(unextended) language. This is in tension with the deflationist's
claim that there is nothing substantive to truth.
But it can be shown (or so I would argue) that a truth-avoiding
prosentential extension of PA allows one to formalize quite faithfully
the so-called "semantical argument" for the truth of the independent
G"odel sentence, and also the meta-argument for the consistency of
PA. In so far as prosententialism is very close in spirit to
deflationism, this might show deflationism to be less vulnerable to
refutation via the G"odel phenomena than Shapiro seems to suggest.
It would be interesting to know if anyone out there has any relevant
results to report, or comments to add.
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