Home Philosophy Advances in ProofTheoretic Semantics


NeoVerificationist ApproachesDoes the paradox of knowability threaten the neoverificationist, who normally equates truth with knowability rather than with knowledge? Let us observe, first of all, that, even within the intuitionistic conceptual framework, it would be possible to suggest a definition of truth different from the one given above: Definition 6 TR α =def. ∃σ (proves(σ, α)). Of course, if this definition is proposed within the intuitionistic conceptual framework, the metalinguistic existential quantifier is to be understood intuitionistically: a proof of ∃σ (proves(σ, α)) is a procedure p whose execution yields, after a finite time, a pair (σ, π ), where σ is a construction30 and π is a proof of “σ proves α”. It is easy to see that Definition 6 is materially adequate. Define the following function f : if σ is a proof of α, f (σ ) is the following procedure p: (i) take σ ; (ii) effect the observation π that σ proves α; (iii) construct the pair (σ, π ). Since proofs are epistemically transparent, the observation π terminates after a finite time, and the pair (σ, π ) is therefore a proof of ∃σ (proves(σ, α)), i.e. of TR α. Conversely, define the following function g: if p is a procedure whose execution yields, after a finite time, a pair (σ, π ), where σ is a construction and π is a proof of “σ proves α”, then g( p) is σ ; since proof observation is factive, σ is a proof of α. Definitions 4 and 6 are not extensionally equivalent. Consider the sentence “Prime(n)”, where n is some very large natural number, and suppose that the primality test has never been applied to n. Then one of the two statements “Prime(n)” and “¬ Prime(n)” is true, according to Definition 6, since (i) we know the primality test, which is a procedure with the required properties, and (ii) we know that the primality test, if it were applied, would answer either that n is prime or that n is not prime. When truth is defined according to Definition 6, the truth of α is not a cognitive state, but empirical accessibility to a cognitive state which is a proof of α. On the other hand, neither “Prime(n) nor “¬ Prime(n)” is true, according to Definition 4, since we have a proof of neither statement, owing to the fact that the primality test has not been applied to n. Hence, according to Definition 6 there are statements that are true although they are not known now, and possibly not even in the future; while according to Definition 4 there are no statements of this kind: there are only unknown statements waiting to be made true (i.e. known) by our activity of proving mathematical statements or coming to know empirical statements. The essential point to notice in this connection is that the following formula is intuitionistically valid: (20) ∃σ (proves(σ, α)) ↔ K α, since both subformulas are equivalent to the same formula α. How is this possible? The validity of (20) puts dramatically into evidence a peculiarity of intuitionistic logic which deserves being stressed. Suppose that the procedure p described above, were it applied, would give as a result that n is prime. According to Definition 6, what determines the truth of “Prime(n)” before the execution of p is a mere fact (if it is a fact): the fact that the execution of p will give as a result a proof of “Prime(n)”. Now, the essential characteristic of intuitionistic logic, as Heyting conceives it, is its being a logique du savoir, opposed to classical logic as a logique de l'être31; this entails that the intuitionistic meaning of the logical constants, implication in particular, must be explained in terms of cognitive states instead of facts and relations between facts.32 Hence, the mere fact that “Prime(n)” is true before the execution of p plays no role in determining the assertibility or the nonassertibility of any formula of intuitionistic logic; in particular, it does not conflict with the validity of (20). As a consequence, if one wanted to define a notion of intuitionistic truth by means of Definition 6 instead of 4, one would face a dilemma: either to accept (20), whose validity follows from the fact that the biconditional is read intuitionistically, giving up the possibility of expressing the fact that there are true but unknown statements; or to insist that there are intuitionistically true but unknown statements, giving up the intuitionistic reading of the logical constants occurring in the semantic metalanguage. The moral drawn from this dilemma by the realist is clear: there are statements that are intuitionistically true but unknown; hence, as shown by the paradox, there are also statements that are intuitionistically true but unknowable; therefore (K) must be rejected. Equally clear is the moral drawn by the intuitionist: both linguistic and metalinguistic logical constants must be read intuitionistically, hence (20) is valid, and the notion of truth defined by Definition 6 either is to be rejected, or has intuitive consequences that cannot be expressed in an intuitionistic language. There is, however, a third answer that can be, and has been, proposed—an answer I should call “hybrid”: it consists in defining truth by Definition 6, in insisting that there are intuitionistically true but unknown statements, in giving up the intuitionistic reading of the metalinguistic logical constants, adopting for them a classical reading, and in accepting (K). This position is instantiated by whoever accepts (K) rejecting at the same time (RAR); for the reason why (RAR) is judged unacceptable can be only that it is understood as expressing the thought that every α is either false or known, i.e. is understood on the basis of the classical reading of the implication occurring in its formalization. Among the supporters of the hybrid position there are many neoverificationists, in my opinion. Be it as it may, it seems to me that this position incurs a paradox strictly analogous to the paradox of knowability. Assume (21) q ∧¬ K q; then, by Definition 6, there is a proof of q ∧¬ K q; let's call such proof σ ; then (22) proves(σ, (q ∧¬ K q)); by the definition of proof of a conjunction,33 (23) σ = (σ1, σ2), where (proves(σ1, q) & proves(σ2, ¬ K q)). Since σ1 proves q, and proofs are epistemically transparent, it is possible to perform the observation σ3 that σ1 proves q, and this observation is a proof of K p; then (24) proves(σ3, K q); if we now construct the pair σ t = (σ3, σ2), we have that (25) proves(σ t, (K q ∧¬ K q)), hence (26) 2:σ (proves(σ, ⊥)); on the other hand, the meaning of ⊥ is characterized by saying that there is no proof of ⊥, hence the formula (27) −2:σ (proves(σ, ⊥)) is assertible: a contradiction. Therefore, (28) −(q ∧¬ K q), from which (29) q ⊃ K q. A possible way out consists in giving up the intuitionistic idea that proofs are epistemically transparent; in this way the step from (23) to (24) is blocked. But the price to pay is very high: as proof, or more generally verification, is the keynotion of a neoverificationist theory of meaning, the nontransparency of proofs/verifications would create the same difficulties the neoverificationists impute to the realist theory of meaning because of the nontransparency of truthconditions (essentially, the nonsatisfiability of the manifestability requirement imposed onto knowledge of meaning). Another way out has been proposed by Dummett. As a matter of fact, Dummett has tackled the paradox in two papers,34 suggesting two different answers; since the former has been explicitly withdrawn by him,35 I will consider only the latter. Dummett's solution consists in accepting (30) α → ¬¬ K α, rejecting at the same time (2). This is legitimated, firstly, by the fact that only (30), not (2), follows intuitionistically from (1); secondly, by the fact that, if one reads negation intuitionistically, '¬¬ K α' means 'There is an obstacle in principle to our being able to deny that α will ever be known', in other words 'The possibility that α will come to be known always remains open'36 —which is precisely what the verificationist believes to hold good for every true α. Dummett does not explain why (2) should be rejected; he only remarks that what (2) says is "contrary to our strong intuition" (p. 51). As I remarked at the beginning, what (2) says is not contrary to our intuition if (2) is read intuitionistically; on the contrary, it certainly is contrary to our intuition if what it says is that either the fact that α does not obtain, or the fact that α is (or will ever be) known obtains; but this is precisely the classical reading of the implication occurring in (2). Hence, Dummett is reading classically the implication in (30), intuitionistically the double negation. Such a hybrid reading is not justified; as a consequence, Dummett's solution seems quite ad hoc. 
CONTENTS 

Related topics 