# Is Tarski's theory of truth compatible with intuitionism?

Intuitionistically, truth is identified with provability: A is true means that it is possible to prove A. In his essay "Intuitionistic logic a philosophical challenge, Logic and Philoshophy" (1980) Prawitz affirms that Tarski's theory of truth is compatible with the intuitionistic position (p.3):

It may be thought that this theory should support classical logic. By combining the truth conditions for disjunctions and negations, we get that a sentence "A or not-A" is true if and only if the truth condition of A obtains or does not obtain. Since this truth condition just expresses the meaning of the sentence, its logical validity follows if we furthermore assume the principle that a truth condition either obtains or does not obtain, independently of our means of knowing which case is the actual one. But this principle, which we may call with Dummett the platonistic principle of truth, must of course not be taken for granted in a discussion of the validity of the law of excluded middle.

That is, Tarski's material condition is respected by intuitionism, since e.g.

• 'A or not A' is true if and only if the truth conditions of A obtains or not obtains.

is itself pretty neutral as long as haven't affirmed the validity of the excluded middle.

I have three questions:

1. Is this claim correct?
2. Also, is this argument enough to support it?
3. What does this has to say about the BHK interpretation?

Thanks!

According to Michael Dummett in his :

there are some issues with the adoption of Tarski's schema (T) for intuitionsitic logic:

[page 232] S is true iff A,

where an instance of the schema is to be formed by replacing "A" by some number-theoretic statement and "S" by a canonical name of that sentence, as, e.g., in:

"There are infinitely many twin primes" is true iff there are infinitely many twin primes.

[page 239] The obvious way to do this [to frame the condition for the intuitionistic truth of a mathematical statement] is to say that a mathematical statement is intuitionistically true if there exists an (intuitionistic) proof of it, where the existence of a proof does not consist in its platonic existence in a realm outside space and time, but in our actual possession of it. Such a notion of truth, obvious as it is, already departs at once from that supplied by the analogue of the Tarski-type truth-definition, since the predicate "is true", thus explained, is significantly tensed: a statement not now true may later become true [emphasis added]. For this reason, when "true" is so construed, the schema (T) is incorrect: for the negation of the right-hand side of any instance will be a mathematical statement, while the negation of the left-hand side will be a non-mathematical statement, to the effect that we do not as yet possess a proof of a certain mathematical statement, and hence the two sides cannot be equivalent.

• I think Prawitz disagrees with Dummett on how to understand the existence of proofs. The former understands it more platonistically, and Dummett denies that view in this quote. Commented Feb 23, 2016 at 0:19

intuitionistically truth is identified with proof

I'm not sure intuitionistic truth is identified with provability; but interpreted as such.

One might go back to the account by Plato of truth - justified true belief; and account the proof of a proposition, a justification; and because it is proved, true: so we have justified truth, and are justified in believing it.

• Knowledge was assumed to be justified true belief but agreeing with this definition would mean you agree that "true" and "justified belief" are different things (otherwise we woudn't say both). Commented Feb 22, 2016 at 17:34
• Thanks, but I don't think this is really an answer to the question above (maybe a comment would be more appropriate?) Also, "I'm not sure intuitionistic truth is identified with provability; but interpreted as such." If truth wasn't identified with provability, one notion should go beyond the other somehow. But this is very unclear. As far as I know, A is true simply means that there exists a proof of A. So truth is nothing more than provability and vice-versa. Cf. Martin-Löf, 1985 Truth of a proposition, evidence of a judgement, validity of a proof, p.413 Commented Feb 23, 2016 at 5:50
• Identity is complicated; see Liebniz snd indiscernables; also category theory and isomorphism, and equivalence. Commented Feb 23, 2016 at 20:57
• For sure - that a truth is to be justified, say by a proof, is now just part of our furniture of the mind; I was pointing out a connection when these things were first being thought about in a systematically way; I appreciate though, this might not be what you were looking for. Commented Feb 23, 2016 at 20:59