From Blanche's Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory

The paper first introduces a cube of opposition that associates the traditional square of opposition with the dual square obtained by Piaget's reciprocation. It is then pointed out that Blanche's extension of the square-of-opposition structure into an conceptual hexagonal structure always relies on an abstract tripartition. Considering quadripartitions leads to organize the 16 binary connectives into a regular tetrahedron. Lastly, the cube of opposition, once interpreted in modal terms, is shown to account for a recent generalization of formal concept analysis, where noticeable hexagons are also laid bare. This generalization of formal concept analysis is motivated by a parallel with bipolar possibility theory. The latter, albeit graded, is indeed based on four graded set functions that can be organized in a similar structure.