Boolean Differences between Two Hexagonal Extensions of the Logical Square of Oppositions

  • Hans Smessaert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7352)


The classical Aristotelian Square characterizes four formulae in terms of four relations of Opposition: contradiction, contrariety, subcontrariety, and subalternation. This square has been extended into a hexagon by two different strategies of inserting intermediate formulae: (1) the horizontal SB-insertion of Sesmat-Blanché and (2) the vertical SC-insertion of Sherwood-Czeżowski. The resulting visual constellations of opposition relations are radically different, however. The central claim of this paper is that these differences are due to the fact that the SB hexagon is closed under the Boolean operations of meet, join and complement, whereas the SC hexagon is not. Therefore we define the Boolean closure of the SC hexagon by characterizing the remaining 8 (non-trivial) formulae, and demonstrate how the resulting 14 formulae generate 6 SB hexagons. These can be embedded into a much richer 3D Aristotelian structure, namely a rhombic dodecahedron, which also underlies the modal system S5 and the propositional connectives.


square of oppositions hexagon of oppositions logical geometry Boolean closure 3D visualisation 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hans Smessaert
    • 1
  1. 1.Department of LinguisticsUniversity of LeuvenBelgium

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