Proof of Theorems in Fuzzy Logic Based on Structural Resolution

Abstract

An approach to proving theorems with fuzzy and not quite true argumentation is considered. Zadeh’s compositional rule of inference is used as the rule of provably correct reasoning, and its procedural implementation is enabled by a refutation mechanism. As such a mechanism, structural resolution (S-resolution) is proposed that is a generalization of the principle of resolutions to fuzzy statements. S-resolution is based on semantic indices of letters and their similarity. Semantic indices are essential in S-resolution. They contain data used as control information in the process of inference. And similarity implies finding letters to obtain an S-resolvent. Combining Zadeh’s compositional rule of inference and S-resolution allows, on the one hand, to withdraw the problem of correctness of resolvents in fuzzy logic and, on the other hand, to ensure the regularity of the process of a proof in two-valued and fuzzy logics.