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The Interaction Between Logic and Geometry in Aristotelian Diagrams

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Diagrammatic Representation and Inference (Diagrams 2016)

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Abstract

We develop a systematic approach for dealing with informationally equivalent Aristotelian diagrams, based on the interaction between the logical properties of the visualized information and the geometrical properties of the concrete polygon/polyhedron. To illustrate the account’s fruitfulness, we apply it to all Aristotelian families of 4-formula fragments that are closed under negation (comparing square and rectangle) and to all Aristotelian families of 6-formula fragments that are closed under negation (comparing hexagon and octahedron).

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Notes

  1. 1.

    See www.logicalgeometry.org.

  2. 2.

    See [15] for some historical background on this nomenclature.

  3. 3.

    One counterexample is Chow [7], who studies Aristotelian diagrams that satisfy the logical condition, but not the geometrical condition.

  4. 4.

    In [20, p. 77] this formula is applied to a fragment of 4 formulas (so \(n=2\)).

  5. 5.

    In this paper, we will mainly focus on regular polygons and polyhedra (the only exception being the brief discussion of rectangles in Sect. 4). However, this restriction is only made for reasons of space; in principle, the account presented here can be applied to regular and non-regular shapes alike.

  6. 6.

    The idea of working up to symmetry can already be found in [4, p. 315], where it is stated that Aristotelian squares that are symmetrical variants of each other should be “counted as being of the same type”. The assumed irrelevance of symmetry considerations for diagram design is also in line with work on other types of diagrams, such as Euler diagrams [27]: several of their visual characteristics have been investigated [1, 3], but it has been found that rotation has no significant influence on user comprehension of Euler diagrams [2].

  7. 7.

    Note that both fractions have the same numerator (since the two Aristotelian diagrams have the same logical properties, viz. they both visualize the fragment \(\mathcal {F}\)), but different denominators (since the two diagrams have different geometrical properties, viz. \(\mathcal {P}\) is less symmetric than \(\mathcal {P}'\)).

  8. 8.

    These considerations can be viewed as an application of the congruity/isomorphism principle in diagram design [18, 36]: the visual properties of the diagram should closely correspond to the logical properties of the visualized fragment.

  9. 9.

    We are using the term ‘square’ in a strictly historical sense here, regardless of its concrete geometrical properties.

  10. 10.

    In particular: (i) a contrariety between \(\varphi \) and \(\psi \) yields subalternations from \(\varphi \) to \(\lnot \psi \) and from \(\psi \) to \(\lnot \varphi \); (ii) a subcontrariety between \(\varphi \) and \(\psi \) yields subalternations from \(\lnot \varphi \) to \(\psi \) and from \(\lnot \psi \) to \(\varphi \); (iii) a subalternation from \(\varphi \) to \(\psi \) yields a contrariety between \(\varphi \) and \(\lnot \psi \) and a subcontrariety between \(\lnot \varphi \) and \(\psi \).

  11. 11.

    For reasons of space, our discussion of the visualizations of these 5 families will be fairly brief; however, much more can (and should) be said about each of them.

  12. 12.

    This result is in line with earlier work on visualizations for JSB fragments [13, 16].

  13. 13.

    Hexagons 1, 3 and 6 in [6, pp. 131–132] visualize an SC fragment using three distinct fundamental forms, viz. those shown in Figs. 2(b), 5(e) and (f), respectively.

  14. 14.

    The hexagons in Fig. 6(a) and (c) strike a balance between the (sub)contrarieties and subalternations, by distributing visual prominence equally among them.

  15. 15.

    The term ‘polytope’ is a generalization of the terms ‘polygon’ and ‘polyhedron’ to arbitrary dimensions [9].

  16. 16.

    One might object that higher-dimensional cross-polytopes can be visualized, by considering their lower-dimensional projections. This objection fails to take into account, however, that such projections will break some of the cross-polytopes’ symmetries, and thus not be useful for our current purposes. This phenomenon can already be observed in very low dimensions; for example, the hexagon in Fig. 2(a) can be seen as the 2D projection of the octahedron in Fig. 3(c), but this projection drastically reduces the number of symmetries (from the octahedron’s 48 to the hexagon’s 12).

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Acknowledgements

We would like to thank Dany Jaspers and Margaux Smets for their valuable feedback on an earlier version of this paper. The first author holds a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO).

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Authors and Affiliations

  1. Center for Logic and Analytic Philosophy, KU Leuven, Leuven, Belgium

    Lorenz Demey

  2. Department of Linguistics, KU Leuven, Leuven, Belgium

    Hans Smessaert

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  1. Lorenz Demey
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  2. Hans Smessaert
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Correspondence to Lorenz Demey .

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Editors and Affiliations

  1. The University of Cambridge , Cambridge, United Kingdom

    Mateja Jamnik

  2. The University of Tokyo , Tokyo, Japan

    Yuri Uesaka

  3. Millersville University , Millersville, Pennsylvania, USA

    Stephanie Elzer Schwartz

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Demey, L., Smessaert, H. (2016). The Interaction Between Logic and Geometry in Aristotelian Diagrams. In: Jamnik, M., Uesaka, Y., Elzer Schwartz, S. (eds) Diagrammatic Representation and Inference. Diagrams 2016. Lecture Notes in Computer Science(), vol 9781. Springer, Cham. https://doi.org/10.1007/978-3-319-42333-3_6

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  • DOI: https://doi.org/10.1007/978-3-319-42333-3_6

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