Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Aristotelian diagrams visualize the logical relations among a finite set of objects. These 1 diagrams originated in philosophy, but recently they have also been used extensively in artificial 2 intelligence, in order to study (connections between) various knowledge representation formalisms. 3 In this paper we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical 4 entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra 5 B4, viz. the rhombic dodecahedron, the tetrakis hexahedron, the tetraicosahedron and the nested 6 tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of 7 these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) 8 and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is 9 that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree 10 of correlation between logical and geometrical distance, the tetraicosahedron performs worse, and 11 the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed 12 new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by 13 appealing to the congruence principle from cognitive research on diagram design. 14