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

Abstract

The aim of this paper is to discuss and extend some of Béziau’s (published and unpublished) results on the logical geometry of the modal logic S5 and the subjective quantifiers many and few.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.