The Relationship between Aristotelian and Hasse Diagrams

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

“…One of the crucial insights in this area is that Aristotelian diagrams can also be fruitfully seen as truly geometrical entities and studied by means of tools and techniques such as projection matrices, Euclidean distance, symmetry groups, etc. [54][55][56][57]. (This hybrid perspective on diagrams, treating them simultaneously as diagrammatic visualizations of an underlying abstract structure and as geometrical entities by themselves, can also be found in crystallography [58,59].)…”

Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

“…9 This can be explained by noting that the octagon for F ‡ contains exactly 6 squares, 10 and each square corresponds exactly to a statement in AX : the square is classical (i.e. not degenerated) iff the corresponding statement is an axiom in the logical system with respect to which the octagon is defined.…”

“…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.…”

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

“…Finally, it has recently been discovered that this visualisation issue was already discussed in full detail in the 1960s by P. Sauriol, who made use of a so-called tetrahexahedron [22]. 12 Of these three, only the rhombic dodecahedron is canonically discussed in the mathematical literature on polyhedra [9] and does equal justice to its cube and octahedron components [27]; see Figure 11. Furthermore, as was already noted above, the rhombic dodecahedron is geometrically related to both of the three-dimensional Aristotelian diagrams that were discussed in the previous section: Béziau's stellar rhombic dodecahedron is its first stellation (recall Figure 5b), while Moretti's cuboctahedron is its dual polyhedron (recall Figure 8c).…”

“…Furthermore, as was already noted above, the rhombic dodecahedron is geometrically related to both of the three-dimensional Aristotelian diagrams that were discussed in the previous section: Béziau's stellar rhombic dodecahedron is its first stellation (recall Figure 5b), while Moretti's cuboctahedron is its dual polyhedron (recall Figure 8c). Finally, the rhombic dodecahedron fits naturally in a unified perspective on Aristotelian diagrams and Hasse diagrams [12]. Because of these reasons, we will henceforth use the rhombic dodecahedron as the canonical representation of the logical geometry of S5.…”