On t-Norms as Basic Connectives for Fuzzy Logic

Abstract

In fuzzy logic in wider sense, i.e. in the field of fuzzy sets applications like fuzzy control and approximate reasoning, t-norms have reached a core position in recent times. And from a theoretical point of view fuzzy logic in the narrow sense, i.e. many-valued logic with a graded notion of entailment, is the main background theory e.g. for fuzzy reasoning. In pure many-valued logic, the Lukasiewicz systems, the Gödel sytems, and also the quite recent product logic all are t-norm based systems in the sense that the basic connectives of these systems can be defined starting with a suitable t-norm.

The present paper discusses the problem of the adequate axiomatizability for such t-norm based logical systems in general, surveying results of the last years. The main emphasis in the present paper is on propositional logic.