Connecting systems of mathematical fuzzy logic with fuzzy concept lattices

Abstract

In this paper our aim is to explore a new look at formal systems of fuzzy logics using the framework of (fuzzy) formal concept analysis (FCA). Let L be an extension of MTL complete with respect to a given L-chain. We investigate two possible approaches. The first one is to consider fuzzy formal contexts arising from L where attributes are identified with L-formulas and objects with L-evaluations: every L-evaluation (object) satisfies a formula (attribute) to a given degree, and vice-versa. The corresponding fuzzy concept lattices are shown to be isomorphic to quotients of the Lindenbaum algebra of L. The second one, following an idea in a previous paper by two of the authors for the particular case of Gödel fuzzy logic, is to use a result by Ganter and Wille in order to interpret the (lattice reduct of the) Lindenbaum algebra of L-formulas as a (classical) concept lattice of a given context.