Theorem proving for prenex Gödel logic with Delta: checking validity and unsatisfiability

First-order Gödel logic with the projection operator △ (G∞) is an important many-valued as well as intermediate logic. In contrast to classical logic, the validity and the satisfiability problems of G∞ are not directly dual to each other. We nevertheless provide a uniform, computational treatment of both problems for prenex formulas by describing appropriate translations into sets of order clauses that can be subjected to chaining resolution. For validity a version of Herbrand’s Theorem allows us to show the soundness of standard Skolemization. For satisfiability the translation involves a novel, extended Skolemization method.