An Evaluation Semantics for Classical Proofs

Abstract

We show how to interpret classical proofs as programs in a way that agrees with the well-known treatment of constructive proofs as programs and moreover extends it to give a computational meaning to proofs claiming the existence of a value satisfying a recursive predicate. Our method turns out to be equivalent to H. Friedman's proof by "A-translation" of the conservative extention of classical over constructive arithmetic for $\Pi^{0}_{2}$ sentences. We show that Friedman's result is a proof-theoretic version of a semantics-preserving CPS-translation from a nonfunctional programming language (with the "control" (C, a relative of call/cc) operator) back to a functional programming language. We present a sound evaluation semantics for proofs in classical number theory (PA) of such sentences, as a modification the standard semantics for proofs in constructive number theory (HA). Our results soundly extend the proofs-as-programs paradigm to classical logics and to programs with C.