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Truth Notions

The 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 not acceptable; my question is whether there is a sense in which it is. A first component of such a sense has already been made explicit: it consists in giving the logical constants their (revised) BHKsense. The second component is of course the concept of truth, which (K) and (RAR) explicitly refer to. It is at this point that a problem mentioned above becomes relevant: at which conditions is a notion a notion of truth?

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 p, as “ p is true”; for example, the intuitive reading of an instance of the schema α + −α would be, “Either p is true or − p is true” (which, given the definition of “ p is false” as “− p is true”, is equivalent to “Either p is true or p is false”). This is correct because the key notion of the realistic explanation of the meaning of the logical constants is the realistic (i.e. bivalent) notion of truth; but it is no longer legitimate when we consider a formula of

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 p will not be sufficient to make reference to the truth of p: in order to make reference to the truth of p it will be necessary to use a truth-predicate, or a truth-operator. Notoriously, the choice between expressing truth with a predicate or an operator has an impact on many other things, in particular on the possibility of expressing semantic paradoxes; since the questions discussed in this paper are independent of such a possibility, I shall choose the simpler alternative of expressing truth with an operator. The question arises at this point: what makes an operator a truth operator?

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 N is the name of a sentence of the object language, and t is a translation of that sentence into the metalanguage. Since “materially adequate” means faithful to our intuitions about the notion of truth, we can take the validity of (5) as a criterion for a formally defined predicate to be a truth-predicate, i.e. a predicate defining a notion we are intuitively prepared to consider a notion of truth.14 From this we may

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 α if and only if t,

where α is a sentence of the object language and t is a translation of that sentence into the metalanguage. Finally, if we make the further simplifying assumption that the metalanguage is an extension of the object language, (6) is equivalent to

(7) O α if and only if α,

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 The Logical Basis of Metaphysics he writes:

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 α with the actual possession of a proof of α. The intuitionist may be seen as a case in point, and in a moment I myself shall explicitly endorse this view. At this point Dummett, assuming that a consequence of the (T) schema is that the following principle is valid:

(8) T ¬α if and only if ¬ T α,

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 α it does not follow that one possesses a proof of T ¬α), and concludes, by contraposing, that the (T) schema is not valid. Here is the derivation of (8) from the (T) schema:

(9)

(i)
T ¬α iff ¬α [from (7), replacing α with ¬ α]

(ii) ¬α iff ¬ T α [from (7), by contraposition]

(iii) T ¬α iff ¬ T α [from (i) and (ii), by transitivity].

It seems to me that Dummett's remark that (8) is invalid is not correct: ¬ T α does not mean that one does not possess a proof of T α, but that one possesses a method to transform every proof of T α into a contradiction. When one possesses such a method, one has a proof of ¬α; owing to the epistemic transparency of intuitionistic proofs, one can effect the (empirical) observation that what one has is a proof of ¬α, and this observation is a proof of K ¬α, i.e. of T ¬α under the present identification of truth with the actual possession of a proof. In conclusion, when the truth of a (mathematical) statement α is equated with the actual possession of a proof of α, truth does commute with intuitionistic negation.16

 
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