From Blanché’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 Blanché’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.

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Correspondence to Henri Prade.

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Dubois, D., Prade, H. From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory. Log. Univers. 6, 149–169 (2012).

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Mathematics Subject Classification (2010)

  • Primary 68T30
  • Secondary 03A05
  • 03B05
  • 68T37


  • Square of opposition
  • Blanché hexagon
  • Piaget group
  • propositional connectives
  • formal concept analysis
  • possibility theory