Next Article in Journal
Chiral Separation in Preparative Scale: A Brief Overview of Membranes as Tools for Enantiomeric Separation
Previous Article in Journal
Polar Bear Optimization Algorithm: Meta-Heuristic with Fast Population Movement and Dynamic Birth and Death Mechanism
 
 
Article

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

1
Center for Logic and Analytic Philosophy, KU Leuven, 3000 Leuven, Belgium
2
Research Group on Formal and Computational Linguistics, KU Leuven, 3000 Leuven, Belgium
*
Author to whom correspondence should be addressed.
Academic Editor: Neil Y. Yen
Symmetry 2017, 9(10), 204; https://doi.org/10.3390/sym9100204
Received: 28 August 2017 / Revised: 14 September 2017 / Accepted: 27 September 2017 / Published: 29 September 2017
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. View Full-Text
Keywords: logical geometry; Boolean algebra; knowledge representation; bitstrings; rhombic dodecahedron; tetrakis hexahedron; tetraicosahedron; nested tetrahedron; Hamming distance; Euclidean distance; congruence principle logical geometry; Boolean algebra; knowledge representation; bitstrings; rhombic dodecahedron; tetrakis hexahedron; tetraicosahedron; nested tetrahedron; Hamming distance; Euclidean distance; congruence principle
Show Figures

Figure 1

MDPI and ACS Style

Demey, L.; Smessaert, H. Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation. Symmetry 2017, 9, 204. https://doi.org/10.3390/sym9100204

AMA Style

Demey L, Smessaert H. Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation. Symmetry. 2017; 9(10):204. https://doi.org/10.3390/sym9100204

Chicago/Turabian Style

Demey, Lorenz, and Hans Smessaert. 2017. "Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation" Symmetry 9, no. 10: 204. https://doi.org/10.3390/sym9100204

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop