Open Archive TOULOUSE Archive Ouverte (OATAO) The Cube of Opposition-A Structure underlying many Knowledge Representation Formalisms

Didier Dubois, Henri Prade, Agnès Rico
2015 24th International Joint Conference on Artificial Intelligence (IJCAI 2015)   unpublished
OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : Eprints ID : 15502 The contribution was presented at : To cite this version : Dubois, Didier and Prade, Henri and Rico, Agnés The Cube of Opposition: A Structure Underlying Many Knowledge Representation Formalisms. Abstract The square of opposition is a
more » ... f opposition is a structure involving two involutive negations and relating quantified statements, invented in Aristotle time. Rediscovered in the second half of the XX th century, and advocated as being of interest for understanding conceptual structures and solving problems in paraconsistent logics, the square of opposition has been recently completed into a cube, which corresponds to the introduction of a third negation. Such a cube can be encountered in very different knowledge representation formalisms, such as modal logic, possibility theory in its all-or-nothing version, formal concept analysis, rough set theory and abstract argumentation. After restating these results in a unified perspective, the paper proposes a graded extension of the cube and shows that several qualitative, as well as quantitative formalisms, such as Sugeno integrals used in multiple criteria aggre-gation and qualitative decision theory, or yet belief functions and Choquet integrals, are amenable to transformations that form graded cubes of opposition. This discovery leads to a new perspective on many knowledge representation formalisms, laying bare their underlying common features. The cube of opposition exhibits fruitful parallelisms between different formalisms, which leads to highlight some missing components present in one formalism and currently absent from another.