10 - Triangular norm-based mathematical fuzzy logics

Abstract

This chapter discusses the particular classes of infinite-valued propositional logics, which are strongly related to triangular norms (t-norms) as conjunction connectives and to the real unit interval as set of their truth degrees, and which have their implication connectives determined via an adjointness condition. Such systems have, in the last ten years, been of considerable interest, and the topic of important results. They generalize well-known systems of infinite-valued logic, and form a link to as different areas as linear logic and fuzzy set theory. Fuzzy sets, that is, sets with a graded notion of membership, have been used in the last decades quite successfully in engineering applications. It is a kind of standard choice in fuzzy set applications, to consider the real unit interval as the class of membership degrees. Furthermore, in discussions about the mathematical foundations for the set algebra of fuzzy sets, a kind of agreement has been reached to consider, for example, intersection and cartesian product operations for fuzzy sets which are based upon t-norm combinations of the membership degrees. The adjointness condition is in the present setting the suitable algebraic equivalent of the analytical notion of left-continuity, if one specifies the residuated lattice to be t-norm algebra.