Some Variants of Proof-theoretic Semantics and Their Relations with Intuitionistic Logic

We discuss the relationship between different variants of proof-theoretic semantics, particularly those stemming from Dag Prawitz’s initial ideas and some subsequent developments due to Peter Schroeder-Heister. We clarify that the known proofs of incompleteness of intuitionistic logic with respect to (monotonic or non-monotonic) proof-theoretic semantics of the latter kind do not apply to Prawitz’s original semantics. Nevertheless, we settle negatively the question of the completeness of intuitionistic logic with respect to Prawitz’s approach, thus refuting a conjecture he has made. Finally, we point to some features of the discussed variants of proof-theoretic semantics that we consider to be philosophically unsatisfactory.