On the Relation Between Heyting's and Gentzen's Approaches to Meaning

Proof-theoretic semantics explains meaning in terms of proofs. Two different concepts of proof are in question here. One has its roots in Heyting’s explanation of a mathematical proposition as the expression of the intention of a construction, and the other in Gentzen’s ideas about how the rules of Natural Deduction are justified in terms of the meaning of sentences. These two approaches to meaning give rise to two different concepts of proof, which have been developed much further, but the relation between them, the topic of this paper, has not been much studied so far. The recursive definition of proof given by the so-called BHK-interpretation is here used as an explication of Heyting’s idea. Gentzen’s approach has been developed as ideas about what it is that makes a piece of reasoning valid. It has resulted in a notion of valid argument, of which there are different variants. The differences turn out to be crucial when comparing valid arguments and BHK-proofs. It will be seen that for one variant, the existence of a valid argument can be proved to be extensionally equivalent to the existence of a BHK-proof, while for other variants, attempts at similar proofs break down at different points.