Logica Universalis

, Volume 6, Issue 1–2, pp 149–169 | Cite as

From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory

  • Didier Dubois
  • Henri Prade


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.

Mathematics Subject Classification (2010)

Primary 68T30 Secondary 03A05 03B05 68T37 


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


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institut de Recherche en Informatique de Toulouse (IRIT)Université Paul SabatierToulouse Cedex 9France

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