Herbrand schemes for first-order logic

Abstract This article provides a language-theoretic rendering of Herbrand’s theorem. To each first-order proof is associated a higher-order recursion scheme that abstracts the computation of Herbrand sets obtained through Gentzen-style multicut elimination. The representation extends previous results in this area by lifting the prenex restriction on cut formulas and relaxing the cut-elimination strategies. Features of the new approach are the interpretation of cut as simple composition and contraction as ‘call with current continuation’.