Proof Nets for Herbrand’s Theorem

Abstract

This article explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence” of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (Herbrand nets) as an analytic proof system with syntactic cut-elimination. Herbrand nets can also be seen as a natural generalization of Miller’s expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LK H ; each net corresponds to an equivalence class of LK H proofs under natural proof transformations. A surprising property of our cut-reduction algorithm is that it is non-confluent despite not supporting the usual examples of non-confluent reduction in classical logic.