Home Philosophy Advances in Proof-Theoretic Semantics

## Neo-Verificationist ApproachesDoes the paradox of knowability threaten the neo-verificationist, 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:
Of course, if this definition is proposed within the intuitionistic conceptual framework, the metalinguistic existential quantifier is to be understood intuitionistically: a proof of ∃ It is easy to see that Definition 6 is materially adequate. Define the following function Definitions 4 and 6 are not extensionally equivalent. Consider the sentence “Prime The essential point to notice in this connection is that the following formula is intuitionistically valid: (20) ∃ since both subformulas are equivalent to the same formula 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 Among the supporters of the hybrid position there are many neo-verificationists, 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) then, by Definition 6, there is a proof of (22) proves by the definition of proof of a conjunction,33 (23) Since (24) proves if we now construct the pair (25) proves hence (26) on the other hand, the meaning of ⊥ is characterized by saying that there is no proof of ⊥, hence the formula (27) − is assertible: a contradiction. Therefore, (28) − from which (29) 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 key-notion of a neo-verificationist theory of meaning, the non-transparency of proofs/verifications would create the same difficulties the neo-verificationists impute to the realist theory of meaning because of the non-transparency of truth-conditions (essentially, the non-satisfiability 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) 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 —which is precisely what the verificationist believes to hold good for every true (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 |

< Prev | CONTENTS | Next > |
---|

Related topics |

Academic library - free online college e textbooks - info{at}ebrary.net - © 2014 - 2019