Logica Universalis

, Volume 3, Issue 2, pp 303–332 | Cite as

On the 3D Visualisation of Logical Relations

  • Hans SmessaertEmail author


The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (23) operators to a 4D algebra with sixteen (24) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations.

Mathematics Subject Classification (2000)

Primary 03B45 03B65 O3G05 Secondary 03E02 05C99 


logical square; logical hexagon aristotelian relations of opposition contradiction contrariety Boolean algebra modal logic 3D visualisation polyhedra 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of LinguisticsKatholieke Universiteit LeuvenLeuvenBelgium

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