Commutative basic algebras and non-associative fuzzy logics

Abstract

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. Łukasiewicz, Godel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers and includes the class of MV-algebras. We show that the logic L CBA is very close to the Łukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization.