The Cube of Opposition: A Structure Underlying Many Knowledge Representation Formalisms
Résumé
The square of opposition is a structure involving two involutive negations and relating quantified statements, invented in Aristotle time. Redis-covered in the second half of the XX th century,and advocated as being of interest for understanding conceptual structures and solving problems inparaconsistent logics, the square of opposition hasbeen 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 asmodal logic, possibility theory in its all-or-nothingversion, formal concept analysis, rough set theoryand abstract argumentation. After restating theseresults in a unified perspective, the paper proposes agraded extension of the cube and shows that severalqualitative, as well as quantitative formalisms, suchas Sugeno integrals used in multiple criteria aggregation and qualitative decision theory, or yet belieffunctions and Choquet integrals, are amenable totransformations that form graded cubes of opposition. This discovery leads to a new perspective onmany knowledge representation formalisms, layingtheir underlying common features. The cube of opposition exhibits fruitful parallelisms between different formalisms, which leads to highlight somemissing components present in one formalism andcurrently absent from another.