This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite totally ordered scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a nonempty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

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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.

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