Home Philosophy Advances in Proof-Theoretic Semantics

## Truth NotionsThe second step towards a solution consists in looking for a plausible intuitive sense of (K) and (RAR), according to which not only (K), but also (RAR), becomes acceptable. Of course, there is a sense in which (RAR) is First, let me explain why, exactly, the question is crucial. If we read a formula of the language of classical propositional logic (CPL), it is natural and correct to read an occurrence in it of a propositional letter, say the language of IPL, since the key notion of the BHK-explanation is not the (bivalent) notion of truth. As a consequence, the simple occurrence of A plausible answer to this question is offered by Tarski's Convention T, in the case truth is expressed by a predicate. Tarski has proposed to consider a definition of truth as materially adequate if it entails every sentence of the form (5) N is true if and only if t, where 14If I understand it correctly, [15], p. 148, makes essentially the same point. easily extract an analogous condition for an operator: an operator O can be seen as a truth-operator if it is defined in such a way that it entails every sentence of the form (6) O where (7) O which is the usual version of what I shall call “The (T) Schema”. So, my proposal is that an operator is to be considered as a truth operator if its meaning is defined in such a way as to satisfy the (T) Schema. Before going on, let me examine an objection to this proposal raised by Dummett. In It is sometimes alleged that what makes a given notion a notion of truth is that it satisfies all instances of the (T) schema. This is wrong […]. If a constructivist proposes that the only intelligible notion of truth we can have for mathematical statements is that under which they are true just in case we presently possess a proof of them, he is offering a characterisation of truth for which the (T) schema fails, since truth, so understood, does not commute with negation.15 Let me try to make the argument explicit. Dummett is envisaging the case of a constructivist who equates the truth of a (mathematical) statement (8) T ¬ remarks that (8) is invalid when truth is equated to the actual possession of a proof (since from the fact that one does not possess a proof of T (9) (i) (ii) ¬ (iii) T ¬ It seems to me that Dummett's remark that (8) is invalid is not correct: ¬ T |

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