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

Abstract

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 The aim of this paper is to discuss and extend some of Béziau's (published and unpublished) results on the logical geometry of the modal logic S5 and the subjective quantifiers many and few.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. Mathematics Subject Classification (2000). Primary 03B45, 03B65, 52B10; Secondary 03C80, 52B05, 52B15, 68T30.

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

“…For example, Moretti [68] sketches the outlines of a comparison between the Aristotelian tetraicosahedron and nested tetrahedron, but it goes into far less geometrical detail than the present paper, and does not deal with the Aristotelian rhombic dodecahedron and tetrakis hexahedron. Similarly, Smessaert et al [63,69] compare the Aristotelian rhombic dodecahedron, tetrakis hexahedron and tetraicosahedron, but these studies again go into far less geometrical detail than the present paper and do not address the Aristotelian nested tetrahedron. Finally, Smessaert et al [55,57] offer an in-depth geometrical comparison of the Aristotelian rhombic dodecahedron and nested tetrahedron for B 4 , but these papers do not deal with the Aristotelian tetrakis hexahedron and tetraicosahedron.…”

“…Note that the number of vertices (V = 14) corresponds exactly to the number of contingent bitstrings of B 4 : this Boolean algebra has 2 4 = 16 bitstrings in total, so after discarding 1111 and 0000, we are left with 14 contingent bitstrings. The rhombic dodecahedron is a Catalan solid [80], and has as its symmetry group the octahedral group O h (of order 48), which it shares with its dual polyhedron, the cuboctahedron [63,81,82]. The latter is itself an Archimedean solid and has also been used as a polyhedral Aristotelian diagram [60].…”

“…The latter is itself an Archimedean solid and has also been used as a polyhedral Aristotelian diagram [60]. Furthermore, the rhombic dodecahedron is closely related to both the cube and the octahedron (which are themselves dual to each other), as shown in Figure 4 [63,69,83]. Just like the Hasse rhombic dodecahedron for B 4 discussed in the previous section, the Aristotelian rhombic dodecahedron for B 4 is also the vertex-first parallel projection of a tesseract, but now along the projection axis defined by (the vertices corresponding to) the bitstrings 1111 and 0000.…”

“…a stellar rhombic dodecahedron. The precise logical and geometrical relationship between Béziau's polyhedral Aristotelian diagram and the ones discussed in this paper is discussed in more detail in [63].) This case study is especially relevant in the context of artificial intelligence, since each of these four visualizations has recently been used in AI-related research.…”

Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

“…In contemporary research, Aristotelian diagrams have been used in various subbranches of logic, such as modal logic [4], intuitionistic logic [29], epistemic logic [24], dynamic logic [9] and deontic logic [28], and also even in metalogical investigations [12]. Furthermore, because of the ubiquity of the logical relations that they visualize, these diagrams are also often used in fields outside of pure logic, such as cognitive science [2,30,34], linguistics [1,17,41,43], philosophy [27,44], law [20,31,45] and computer science [10,13,15]. In sum, then, it seems fair to conclude that Aristotelian diagrams have come to serve "as a kind of lingua franca" [19, p. 81] for a highly interdisciplinary community of researchers who are all concerned, in some way or another, with logical reasoning.…”