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

  • Lorenz Demey
  • Hans Smessaert
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Abstract

Aristotelian diagrams are used extensively in contemporary research in artificial intelligence. The present paper investigates the geometric and cognitive differences between two types of Aristotelian diagrams for the Boolean algebra \(\mathbb {B}_{4}\). Within the class of 3D visualizations, the main geometric distinction is that between the cube-based diagrams (such as the rhombic dodecahedron) and the tetrahedron-based diagrams. Geometric properties such as collinearity, central symmetry and distance are examined from a cognitive perspective, focusing on diagram design principles such as congruence/isomorphism and apprehension. The cognitive effectiveness of the different visualizations is compared for the representation of implication versus opposition relations, and for subdiagram embeddings.

Keywords

Logical geometry Knowledge representation Rhombic dodecahedron Tetrahedron Opposition and implication Congruence principle Central symmetry 

Mathematics Subject Classification (2010)

68T27 51M20 52B10 03B05 03B45 

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Notes

Acknowledgements

A preliminary version of this paper was presented at the Diagrams 2016 conference in Philadelphia, PA, USA. We would like to thank the audience of that presentation, as well as Koen Roelandt, Margaux Smets and two anonymous reviewers of this journal for their useful feedback. The first author is financially supported by a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO).

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Authors and Affiliations

  1. 1.Center for Logic and Philosophy of ScienceKU LeuvenLeuvenBelgium
  2. 2.Department of LinguisticsKU LeuvenLeuvenBelgium

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