Tarski's truth theorem asserts that a truth definition for a (reasonably strong) theory cannot be formalized within that theory.

I have seen that Tarski's theory of truth has received a lot of criticism. For examples, Hilary Putnam commented in “A comparison of something with something else”: As a philosophical account of truth, Tarski’s theory fails as badly as it is possible for an account to fail; Hartry Field in an influential discussion and diagnosis of what is lacking in Tarski’s account, pointed out that Tarski’s theory does not offer an account of reference and satisfaction at all. Rather, it only offers a number of disquotation clauses such as "‘Snow’ refers to snow".

So I wonder what is the status of Tarski's theory of truth in mathematical and philosophical logic. Is it widely accepted or not?

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    $\begingroup$ The second statement is false obviously. The first statement is dubious since "defined" is not defined. $\endgroup$
    – Mark Sapir
    Jun 26 at 3:15
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    $\begingroup$ What do you want to achieve? For an HSM SE user it is hard to beat Putnam. And even harder to beat Tarski. $\endgroup$
    – Mark Sapir
    Jun 26 at 4:01
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    $\begingroup$ This question is probably more appropriate for a philosophy forum. Tarski's theory of truth isn't mathematical, so it's neither "widely accepted or not" in mathematical logic. It makes its appearance in mathematical logic under the reflection principle, and it's a matter of interest how various theories react to the reflection principle. $\endgroup$
    – abo
    Jun 26 at 5:33
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    $\begingroup$ I think this question also belongs to HSM because Tarski's theory of truth was proposed in the 1930s and is part of mathematical history. $\endgroup$
    – hermes
    Jun 26 at 13:28
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    $\begingroup$ I am not sure what "widely accepted" means. Do generic 100 Putnams outwide one Tarski? $\endgroup$
    – Mark Sapir
    Jun 28 at 3:07

EDIT 07/07/21

I thought I would add more detail to my overly simplistic and perhaps somewhat misleading account of the philosophical criticism of Tarski’s theory of truth.

There is criticism of Tarski’s T-Scheme (and recursive definition) and there is criticism of Tarski’s philosophical view that our intuitive notion of truth is self-contradictory.

Historically, the earliest criticism of Tarski’s T-Scheme has its roots in the writing of the polymath F.P. Ramsey and goes under the name deflationism. Ramsey identified the equivalence principle which is present in Tarski’s T-Scheme as leading to the view that the idea of a truth predicate is redundant. Thus, when we say “snow is white” is equivalent to saying “it is true that snow is white”, the difference between them is merely stylistic and the truth predicate is redundant.

Other deflationists were quick to follow. For example, P.F. Strawson argued that while “snow is white” does express a proposition, “it is true that snow is white” does not express a proposition about a proposition, as Tarski would contend. Rather, it should be treated as a type of endorsement of the proposition “snow is white”.

Next to criticize Tarski was W.V.O Quine with a view called quotationalism. For Quine, when Tarski intends

”Snow is white” is true if and only if snow is white.

the quotation marks turn the sentence Snow is white into something noun-like. This leads to a type of circularity since adding the truth predicate gives us back a sentence whose truth we understand as well or poorly as we understood the original sentence itself. Quine gives his own analysis of truth predicates.

Another philosophical issue with Tarski’s T-Scheme goes under the name of indeterminacy. Problems here include, for example, problems of vagueness. Natural languages include many vague predicates - e.g., is bald, is tall, is a heap. This makes it hard, if not impossible to determine the truth of the axiom:

”Augustin-Louis Cauchy was bald” is true if and only if Augustin-Louis Cauchy was bald.

This is a taste of some of the early criticisms of Tarski’s philosophical theory. The Stanford Encyclopedia of Philosophy provides a detailed account.

(Original answer)

Regarding mathematical logic and the status of Tarski's semantic definition of truth and his theorem that there is no 1-ary relation that identifies the true formulae of formal arithmetic, these are fundamental to first-order logic. I'm certainly not aware of any controversy amongst mathematical logicians.

Regarding Tarski's philosophical ideas it is important to keep in mind that no philosophical theory is without controversy. Any metaphysical theory will have to confront the realist/anti-realist divide. In the case of Tarski's theory of truth, a "realist" will believe that a proposition is either true of false regardless of whether it is decidable, while an "anti-realist" will believe that an undecidable proposition has no truth value.

At first glance Tarski's semantic definition of truth may appear to suggest that Tarski was a realist, but this is not clear because, after all, this is philosophy so the idea of being clear is out of the question. Regardless of whether Tarski was a realist or anti-realist his theory would be subject to criticism from the other side.

TL/DR (and probably not relevant)

I'm not sure how well acquainted you are with philosophy, and I'm certainly no expert, but it may be helpful to keep in mind that Tarski's use of the word semantic is an archaic usage, prevalent amongst positivist philosophers in the 1930s. (Tarski's T-Schema make clear that Tarski had a positivist attitude.) The positivists used a trio of terms: syntactic, semantic, and pragmatic. Since this answer is so far unreferenced, I'll slip in:

A notion was called syntactic if it pertains to relations among words, semantic if it pertains to relations among words and things, and pragmatic if it pertains to relations among words, things, and people.

[Source: Truth, by Burgess & Burgess, Princeton Press, 2011].

So Tarski's use of the word semantic had nothing to do with its current use in linguistics as "meaning".

"Snow is white" is true iff snow is white

is semantic because it involves words and a thing (snow), but no people.

Since Tarski was a mathematician doing philosophy it is difficult to say whether he was a realist or anti-realist. As a mathematician he was most likely a realist (i.e., a Platonist) while as a philosopher he should be an anti-realist. So here we have another source of controversy: the mathematicians will tend to poo-poo the philosophers while the philosophers will tend to laugh at the mathematicians naivety.

  • $\begingroup$ @hermes Consider, for example, the continuum hypothesis of set theory. Both Gödel and Cohen, who together proved it to be undecidable, expressed the view that CH had a truth value. In fact I believe that Cohen considered CH to be "obviously true". It is not clear why you think this leads to a contradiction. Actually, I think he may have thought it to be obviously false. One or the other. $\endgroup$
    – Nick
    Jul 2 at 14:28
  • $\begingroup$ @hermes Undecidable Conjectures list CH. Also, '*... the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense. ..." from Undecidable Problem. Classical logic, the logic used by mathematics, includes the law of the excluded middle; namely that every proposition is either true or false. $\endgroup$
    – Nick
    Jul 2 at 15:18
  • $\begingroup$ @hermes When you say "compatibility" I assume this is a typo and that you mean "computability"? I am not a very sophisticated in my understanding of maths. I read music and computer science at university and have subsequently developed a casual interest in mathematics. Thanks for accepting my answer. $\endgroup$
    – Nick
    Jul 2 at 15:48

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