Béziau’s Contributions to the Logical Geometry of Modalities and Quantifiers

  • Hans Smessaert
  • Lorenz Demey
Part of the Studies in Universal Logic book series (SUL)


The aim of this paper is to discuss and extend some of Béziau’s (published and unpublished) results on the logical geometry of the modal logic S5 and the subjective quantifiers many and few. After reviewing some of the basic notions of logical geometry, we discuss Béziau’s work on visualising the Aristotelian relations in S5 by means of two- and three-dimensional diagrams, such as hexagons and a stellar rhombic dodecahedron. We then argue that Béziau’s analysis is incomplete, and show that it can be completed by considering another three-dimensional Aristotelian diagram, viz. a rhombic dodecahedron. Next, we discuss Béziau’s proposal to transpose his results on the logical geometry of the modal logic S5 to that of the subjective quantifiers many and few. Finally, we propose an alternative analysis of many and few, and compare it with that of Béziau’s. While the two analyses seem to fare equally well from a strictly logical perspective, we argue that the new analysis is more in line with certain linguistic desiderata.


Logical geometry Modal logic S5 Subjective quantifiers Many/few Aristotelian diagram Stellated rhombic dodecahedron Cuboctahedron Rhombic dodecahedron 

Mathematics Subject Classification (2000)

03B45 03B65 52B10 03C80 52B05 52B15 68T30 



We thank Dany Jaspers, Alessio Moretti, Fabien Schang and Margaux Smets for their comments on earlier versions of this paper. The second author gratefully acknowledges financial support from the Research Foundation—Flanders (FWO).


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Research Group on Formal and Computational LinguisticsKU LeuvenLeuvenBelgium
  2. 2.Center for Logic and Analytic PhilosophyKU LeuvenLeuvenBelgium

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