Concept lattices and order in fuzzy logic

The theory of concept lattices (i.e. hierarchical structures of concepts in the sense of Port-Royal school) is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of vague data—the propositions “object x has attribute y”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better.