[FOM] An Error in Tarski?

[Marion, Mathieu]

Dear Riki,

I do not know if this is relevant - I am replying on the spot - but did you know about this paper ?

David DeVidi and Graham Solomon, "Tarski on "Essentially Richer" Metalanguages", Journal of Philosophical Logic, Vol. 28, No. 1 (Feb., 1999), pp. 1-28.

It might be relevant, given that the share qualms about the same point.

As ever,

Mathieu 


On 2018-09-14, 7:20 PM, "fom-bounces at cs.nyu.edu on behalf of Richard Kimberly Heck" <fom-bounces at cs.nyu. Vol. 28, No. 1 (Feb., 1999), pp. 1-28edu on behalf of richard_heck at brown.edu> wrote:

    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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