A study of Monoidal t-norm based Logic

Abstract

The logical system MTL (for Monoidal t-norm Logic) is a formalism of the logic of left-continuous t-norms, which are operations that arise in the study of fuzzy sets and fuzzy logic. The objective is to investigate the important results on MTL and collect them together in a coherent form. The main results considered will be the completeness results for the logic with respect to MTL-algebras, MTL-chains (linearly ordered MTL-algebras) and standard MTL-algebras (left-continuous t-norm algebras). Completeness of MTL with respect to standard MTL-algebras means that MTL is indeed the logic of left-continuous t-norms. The logical system BL (for Basic Logic) is an axiomatic extension of MTL; we will consider the same completeness results for BL; that is we will show that BL is complete with respect to BL-algebras, BL-chains and standard BL-algebras (continuous t-norm algebras). Completeness of BL with respect to standard BL-algebras means that BL is the logic of continuous t-norms.