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## Internal and Intuitive TruthThe next question to consider is whether the validity of the (T) Schema picks out a unique notion of truth. Tarski seems to hold that it does. In [21] he expresses the conviction that the material adequacy condition imposed onto the definition of truth, is capable to select the classical Aristotelian notion of truth as correspondence. The conviction is not explicitly stated, but it can be inferred from the following facts: (i) In section I.3 Tarski expresses an intention: We should like our definition to do justice to the intuitions which adhere to the
(ii) In section I.4 the same intention is made precise by requiring that the definition satisfies the material adequacy condition. Hence, the material adequacy condition 'does justice' to the intuitive notion of truth as correspondence. The question whether the intuitive notion of truth as correspondence is bivalent is not explicitly addressed by Tarski; an affirmative answer from him is suggested by the fact that in [20] he derives the principle of bivalence from the (materially adequate) definition of truth.18 However, the derivation crucially uses Excluded Middle, as is made clear by the following steps: (10) (i) α [the law of Excluded Middle](ii) T (iii) T So, Tarski's conviction is correct only under the premiss that the metalanguage is associated to a metatheory whose semantics validates Excluded Middle. If this principle is not valid in the metatheory, it is possible to exhibit counterexamples to Tarski's conviction, namely it is possible to define a non-bivalent notion of truth satisfying the (T) Schema. I shall now show how.19 Let us adopt a metatheory in which the logical constants are read according to the (revised) BHK-explanation (which, of course, does not validate the Excluded Middle). We are looking for a materially adequate definition, i.e. such that all the equivalences (11) T are logical consequences of it (where T is the intended truth operator). The definition I suggest is the following20:
where the meaning of K is defined by Definition 2. Definition 4 is materially adequate: (2t) is valid for the reasons explained above; and the converse (12) K is valid as well: it expresses the expectation of a function Concluding, it is true both (i) that the validity of the (T) schema expresses our essential intuition about the notion of truth, and (ii) that our most common intuitive notion of truth is realistic; but the reason why (ii) holds is bivalence, not the (T) schema: the validity of the (T) schema is neutral among different intuitive notions of truth. It is therefore possible, and necessary, to introduce a clear distinction between the condition at which an operator is a truth operator and the condition at which an operator reflects our realistic intuitions about the notion of truth; the former consists in the validity of the (T) schema,22 the latter may be epitomized into the slogan of truth as correspondence and consequently into the validity of the law of bivalence. A notion satisfying the former condition is capable to play (at least some of) At this point it should be clear that an intuitive sense of the principle (RAR), according to which it becomes acceptable, does exist: for, if the logical constants are understood according to the BHK-explanation, and truth is understood according to Definition 4, then (RAR) is a tautology, saying that every known statement is known. The intuitionistic solution of the paradox consists therefore in accepting (RAR) as obvious when the logical constants are understood intuitionistically and truth as internal. Is the idea of equating truth to knowledge consistent? There is an argument— called by [16] “The Standard Argument”—that purports to show that it is not.23 It consists in the following derivation of ∃ (13)
) ∨ (¬ p ∧¬K ¬ p ) ∃α(α ∧¬K α) ∃α(α ∧¬K 3, 4α) ∃ where ∧¬K ¬ p ¬ K ∧¬K ¬ p ¬ K ¬
¬ p ∧¬K ¬ p
) ∨ (¬ p ∧¬K ¬ p ) ( p ∧¬K p ) ∨ (¬ p ∧¬K ¬ 1, 2p )
22The claim that an operator O is a truth operator iff it satisfies the schema (7) should not be confounded with the minimalist claim that (7) is a good definition of the meaning of O. The former claim is perfectly compatible with the idea, embraced above, that the validity of (7) is not the definition, but the material adequacy condition of the definition, of O. 23See [16], p. 275. Now, if we observe that there are statements My answer consists in observing that the argument is valid but unsound, since ¬ K [¬ K (14) ⊥ 2 ⊥ 3
¬ The formula ∃ The intuitionistic inconsistency of ¬ K itionist should accept, that there are undecided, hence unknown, statements. Here it is important, again, to pay attention to the intuitionistic meaning of the logical constants, in particular of negation. The assertibility of ¬ 24An example is “Prime inference of T |

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