Combining Hilbert style and semantic reasoning in a resolution framework

Abstract

Many non-classical logics can be axiomatized by means of Hilbert Systems. Reasoning in Hilbert Systems, however, is extremely inefficient. Most inference methods therefore use the semantics of a logic in one kind or another to get more efficiency. In this paper a combination of Hilbert style and semantic reasoning is proposed. It is particularly tailored for cases where either the semantics of some operators is not known, or it is second-order, or it is just too complicated to handle, or flexibility in experimenting with different versions of a logic is required. First-order predicate logic is used as a meta-logic for combining the Hilbert part with the semantics part. Reasoning is done in a (theory) resolution framework. The basic method is applicable to many different (monotonic prepositional) non-classical logics. It can, however, be improved by treating particular formulae in a special way, as rewrite rules, as theory unification or theory resolution rules, even as recursive calls to a theorem prover. Examples for all these cases are presented in the paper.