Herbrand's Theorem, Skolemization and Proof Systems for First-Order Lukasiewicz Logic

Abstract

An approximate Herbrand theorem is established for first-order infinite-valued Łukasiewicz Logic and used to obtain a proof-theoretic proof of Skolemization. These results are then used to define proof systems in the framework of hypersequents. In particular, a calculus lacking cut elimination is defined for the first-order logic characterized by linearly ordered MV-algebras, a cut-free calculus with an infinitary rule for the full first-order Łukasiewicz Logic, and a cut-free calculus with finitary rules for its one-variable fragment.