Combinatorial Proof Semantics

The paper Proofs Without Syntax [Annals of Mathematics, to appear] introduced the notion of a combinatorial proof for classical propositional logic. The present paper uses combinatorial proofs to define a semantics for classical propositional sequent calculus, an inductive translation from sequent proofs to combinatorial proofs. The semantics is abstract and efficient: abstract in the sense that it identifies many sequent proofs, and efficient in the sense that combinatorial proofs are polynomial-time checkable and the inductive translation is polynomial.