Re: Exception to the rule? (Tarski´s T-scheme)
From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 06/24/04
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Date: Thu, 24 Jun 2004 10:43:47 +0100
G. Frege wrote in message <924ld0hkvtmefdmd3lmpo1mf59hajt6t6j@4ax.com>...
>If I got Tarski right, the T-Schema is actually a definition (some sort
>of?) for the predicate (meta predicate) T().
In Der Wahrheitsbegriff and "The Semantic Conception of Truth", Tarski
referred to instances of the T-scheme as "partial definitions". But one can
only really say this if you have the Tarskian restriction of
non-self-application, which comes from the separation of object language and
meta-language. If you take *all* instances of the T-scheme in a sufficiently
rich language, then one instance will be Tr("L") <-> L where L is a liar
sentence such that L <-> ~Tr("L") is true. Of course, it was Tarski who
first pointed this out.
So, for any sufficiently rich theory D (roughly, if diagonalization is
possible), then the theory D + T-scheme is inconsistent. So, the
unrestricted T-scheme (all instances of Tr("phi") <-> phi) cannot be a
definition.
I always tell my students that Tarski's theory of truth is based on
*abandoning* the T-scheme.
However, instances of the T-scheme behave like a definition when the
Tarskian restriction is imposed. Indeed, for almost any theory D you
consider, D + restricted T-scheme is a conservative extension of D. Stewart
Shapiro and I both wrote papers about five years ago arguing that
deflationary theories of truth should be conservative. The issue for
deflationism of whether a theory of truth is conservative has now been
discussed quite widely. Tarski's theory of truth, however, is
non-conservative. If you add Tarski's theory of truth to PA, you can prove
"All theorems of PA are true", and thus you can prove Con(PA).
Furthermore, if you consider adding two T-schemes to such a theory D, say
Tr("phi") <-> phi
Tr*("phi") <-> phi
for all object language sentences phi, then an obvious consequence is that
you can prove, for each sentence phi, the biconditional
Tr("phi") <-> Tr*("phi").
So, this almost fixes the extension of Tr (this is what Quine meant when he
said that Tarski's Convention T fixes the extension of "true"). However, you
cannot prove the universal claim Ax(Tr(x) <-> Tr*(x)). This is an example of
omega-incompleteness, again something Tarski discussed in Der
Wahrheitsbegriff.
--- Jeff
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