Proof-Theoretic Semantics: Some Basic Ideas

Abstract

This manuscript of 2003 describes and compares two basic approaches to what the author has called “proof-theoretic semantics”. The “standard” approach, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. The second one, which is based on ideas by Hallnäs and the author, understands proofs semantically by reading the application of certain proof rules directly as semantical steps. Whereas the first one is a global approach, dealing with proofs as a whole and imposing requirements on them, the second one is local as is interprets individual proof steps without demanding from the very beginning that a proof composed of such single steps has special features. -- The attached manuscript on proof-theoretic vs. constructive consequence argues that the proof-theoretic notion fully specifies reduction procedures and is nearer to term rewriting than the more general constructive notion.