Graded cubes of opposition and possibility theory with fuzzy events

Didier Dubois (1, 2) , Henri Prade (1, 2) , Agnés Rico (3)

1 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
2 CNRS - Centre National de la Recherche Scientifique
3 ERIC - Equipe de Recherche en Ingénierie des Connaissances

The paper discusses graded extensions of the cube of opposition, a structure that naturally emerges from the square of opposition in philosophical logic. These extensions of the cube of opposition agree with possibility theory and its four set functions. This extended cube then provides a synthetic and unified view of possibility theory. This is an opportunity to revisit basic notions of possibility theory, in particular regarding the handling of fuzzy events. It turns out that in possibility theory, two extensions of the four basic set functions to fuzzy events exist, which are needed for serving different purposes. The expressions of these extensions involve many-valued conjunction and implication operators that are related either via semi-duality or via residuation.

Mots clés: Possibility theory; Fuzzy event; Cube of opposition

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

Didier Dubois, Henri Prade, Agnés Rico. Graded cubes of opposition and possibility theory with fuzzy events. International Journal of Approximate Reasoning, 2017, 84, pp.168-185. ⟨10.1016/j.ijar.2017.02.006⟩. ⟨hal-01919020⟩