Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Abstract

Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study (connections between) various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B 4 , viz. the rhombic dodecahedron, the tetrakis hexahedron, the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design.

Between Square and Hexagon in Oresme's Livre du Ciel et du Monde

“…My aim in this paper is to describe one particular manifestation of this dialectic, and to trace its historical roots. As such, the results of this paper not only constitute an interesting chapter in the historiography of logic, but they also provide valuable input for the contemporary systematic study of Aristotelian diagrams in logical geometry (Demey 2018(Demey , 2019aDemey and Smessaert 2017, 2018a, 2018bSmessaert and Demey 2014, 2015, 2017a.…”

Geometric and cognitive differences between logical diagrams for the Boolean algebra B 4 $\mathbb {B}_{4}$

“…Sauriol's tetra(kis-)hexahedron [78] and Moretti and Pellissier's tetraicosahedron [60,68] -are briefly compared to RDH in [82,84]. Furthermore, [28] provides a detailed comparative analysis of the correspondence between logical and geometrical distance (cf. Subsection 6.2 of the present paper) in the three cube-based visualizations as well as the tetrahedron-based visualization.…”

“…The resulting shape in NTH, by contrast, is a proper quadrilateral, with four different lengths for the (sub)contrariety and the subalternations and no parallel edges whatsoever. 28 The logical difference between the balanced and the unbalanced squares is thus visualized either as the minor geometrical difference between rectangle and parallelogram in RDH -cf. Figures 10(a-b) -, or as the more substantial geometrical difference between isosceles trapezium and quadrilateral in NTH -cf.…”

Aristotelian and Duality Relations Beyond the Square of Opposition

“…Finally, because of the ubiquity of the logical relations that it represents, the square is nowadays also frequently used outside the boundaries of philosophy and logic, in disciplines such as psychology, linguistics and computer science. A comprehensive overview of this wide diversity of applications (including many bibliographic references) can be found in [15] and [16]. The square of opposition visually represents the Aristotelian relations: contradiction, contrariety, subcontrariety, and subalternation.…”

“…The research program of logical geometry is concerned with the systematic study of logical diagrams in general, and Aristotelian diagrams and duality diagrams in particular. We investigate these diagrams using cognitive and geometric notions, such as informational vs. computational equivalence [12,14], Euclidean distance [16,44], vertex-first projections [10] and subdiagrams [6,42]. On the logical side, we focus on issues such as diagram informativity [43], logic-sensitivity [8], diagram classification [45] and Boolean structure [15,46].…”