Constructive Truth and Circularity

We propose a constructive interpretation of truth which resolves the standard semantic paradoxes. 1. Heuristic concepts In intuitionism the law of excluded middle (LEM) is not accepted because the assignment of truth values to sentences is seen as a kind of open-ended process. Although the validity of any purported proof is supposed to be decidable, the truth value of a given sentence may not be decidable because one is not able to search through the infinite set of all potential proofs. Thus the failure of LEM is related to the intuitionistic rejection of a completed infinity. Of course the validity of any proof within a given formal system is decidable. But whether validity can really be considered a decidable property of proofs broadly understood, outside of any particular formal system, is debatable. The analogous claim is certainly not true of definitions. Indeed, suppose we could decide whether any given finite string of words constructively defines a natural number. Then in principle we would be able to unambiguously determine which numbers are constructively defined by a string of ten words by systematically examining all ten word long strings, and consequently “the smallest natural number not constructively definable in ten words” would be a valid constructive definition, which leads to a contradiction. We should conclude from this that definability is open-ended, but not in the way intuitionists suppose truth to be open-ended, i.e., not merely because one is unable to exhaustively search some infinite set. Rather, it is open-ended in the sense that given any well-defined class of accepted definitions we can always produce a new definition outside the family that we would also accept. I will say that “valid definition” is a heuristic concept [10]. This is different from Dummett’s notion of an indefinitely extensible concept since he takes concepts to be decidable ([2], p. 441). According to Troelstra it is “natural” to assume that the relation “c is a proof of A” is decidable, and besides “if we are in doubt whether a construction c proves A, then apparently c does not prove A for us” ([7], p. 7). But an identical argument could be made in support of the claim that validity of definitions is decidable (namely: if we are in doubt whether c constructively defines a number n, then c does not constructively define n for us). It is not a good argument because it assumes that we can decide whether there is any doubt about whether c proves A. To the contrary, incompleteness phenomena suggest that the general concept of a valid proof outside of any particular formal system is not decidable. For if we can accept, Date: April 13, 2010.