Structures of opposition induced by relations:the Boolean and the gradual cases

Davide Ciucci (1) , Didier Dubois (2, 3) , Henri Prade (2, 3)

1 DISCo - Dipartimento di Informatica Sistemistica e Comunicazione
2 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
3 CNRS - Centre National de la Recherche Scientifique

The paper shows that a cube of opposition, a structure that generalizes the square of opposition invented in ancient logic, can be generated from the composition of a binary relation with a subset, by the effect of set complementation on the subset, on the relation, or on the result of the composition. Since the composition of relations is encountered in many areas, the structure of opposition exhibited by the cube of opposition has a universal flavor. In particular, it applies to information processing-oriented settings such as rough set theory, possibility theory, or formal concept analysis. We then discuss how this structure extends to a fuzzy relation and a fuzzy subset, and the graded cube of opposition thus obtained provides an organized view of the different existing compositions of fuzzy relations. The paper concludes by pointing out areas of research where the cube of opposition, or its graded version are of interest.

Mots clés: Square of opposition; Fuzzy relation; Composition of relations; Multiple-valued logic

Domaines: Intelligence artificielle [cs.AI]; Apprentissage [cs.LG]; Logique en informatique [cs.LO]; Informatique et langage [cs.CL]

Davide Ciucci, Didier Dubois, Henri Prade. Structures of opposition induced by relations:the Boolean and the gradual cases. Annals of Mathematics and Artificial Intelligence, 2015, 76 (3), pp.351-373. ⟨10.1007/s10472-015-9480-8⟩. ⟨hal-01485022⟩