[FOM] An Error in Tarski?
Richard Kimberly Heck
richard_heck at brown.edu
Thu Sep 13 23:14:30 EDT 2018
I've just been re-reading Tarski's paper "The Semantic Conception of
Truth and the Foundations of Semantics" formy course on theories of
truth. There's a remark he makes that has always somewhat puzzled me,
and now it seems to me that it must be wrong. The remark is one Tarski
makes during the discussion of 'essential richness':
If the condition of “essential richness” is not satisfied, it can
usually be shown that an interpretation of the meta-language in the
object-language is possible; that is to say, with any given term of the
meta-language a well-determined term of the object-language can be
correlated in such a way that the assertible sentences of the one
language turn out to be correlated with assertible sentences of the
other. As a result of this interpretation, the hypothesis that a
satisfactory definition of truth has been formulated in the
meta-language turns out to imply the possibility of reconstructing in
that language the antinomy of the liar; and this in turn forces us to
reject the hypothesis in question. (pp. 351-2)
I take there to be here an assertion of the following claim. Suppose
that a theory M is (relatively) interpretable in another theory O. Then
truth for the language of O cannot be defined in M (since then the liar
would be reproducible in M).
If Tarski were right, then there would be a vastly simpler argument for
one of the central results of my paper "Consistency and the Theory of
Truth" [2]: that, for (sufficiently strong, consistent) finitely
axiomatized theories T, T plus a Tarski-style compositional truth-theory
for T is never interpretable in T (Corollary 3.9). But so understood,
Tarski's claim is false.Ali Enayat and Albert Visser[1] showed that PA
plus a Tarski-style truth-theory for the language of arithmetic is
interpretable in PA. A simpler proof of the same result, for reflexive
theories generally, is given in my paper. Indeed, it is easy to see that
PA plus all instances of the T-scheme for the language of arithmetic is
interpretable in PA, and of course that theory defines truth for the
language of PA. (See my "The Logical Strength of Compositional
Principles" [3], theorem 2.3.)
Tarski's familiar unclarity about 'language' vs 'theory' makes it
unclear, however, exactly what he meant. But I take his claim to
concern /theories/ because he is largely responsible for the notion of
interpretation to which he is here alluding. (The paper in which Tarski
first introduces and studies this notion would not be published until
1953, however: nine years later, in Tarski, Mostowski, and Robinson.)
Moreover, I don't know of any coherent notion of interpretation for
/languages/, and his talk of "reconstructing in that language the
antinomy of the liar" certainly sounds like talk of provability. But
perhaps there is something else he had in mind.
Any ideas?
Riki
[1] http://dspace.library.uu.nl/bitstream/handle/1874/266885/preprint303.pdf
[2]
http://rkheck.frege.org/philosophy/online_papers.php#a8aac18228dfeb50731b573e84f8991f
[3]
http://rkheck.frege.org/philosophy/online_papers.php#4427415f70f3e68b532a07979436666f
--
----------------------------
Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University
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Website: http://rkheck.frege.org/
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