Structures of opposition in possibility theory-based logic and related logics

The square of opposition represents interactions between logical quantifiers and involutive negations. About ten years ago it was noticed that the four set functions of possibility theory applied to opposite propositions induce a graded cube of opposition. This paper highlights new all-or-nothing cubes of opposition in possibilistic and related logics. We thus obtain three cubes that can be put side by side in a row. This construction extends to possibility theory valued on linear chains as well as on distributive lattices.