Model Existence for Higher Order Logic

Abstract

In this paper we provide a semantical meta-theory that will support the development of higher-order calculi for automated theorem proving like the corresponding methodology has in first-order logic. To reach this goal, we establish classes of models that adequately characterize the existing theorem-proving calculi and we present a standard methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of machine-oriented calculi with respect to this model classes. We further parameterize the introduced semantical structures and the corresponding abstract consistency properties with an order k. This provides a finer proof methodology which can be used to show that the primitive substitution rule in resolution calculi as discussed in [And71] or [BK97] can be restricted with respect to the order of the input problems without losing completeness.