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On the Logical Geometry of Geometric Angles

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Abstract

In this paper we provide an analysis of the logical relations within the conceptual or lexical field of angles in 2D geometry. The basic tripartition into acute/right/obtuse angles is extended in two steps: first zero and straight angles are added, and secondly reflex and full angles are added, in both cases extending the logical space of angles. Within the framework of logical geometry, the resulting partitions of these logical spaces yield bitstring semantics of increasing complexity. These bitstring analyses allow a straightforward account of the Aristotelian relations between angular concepts. In addition, also relational concepts such as complementary and supplementary angles receive a natural bitstring analysis.

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Notes

  1. In this paper, we systematically use degrees to measure angles.

  2. See [20] for the introduction of this diagrammatic representation format for scalar structures.

  3. See [3, Section 3] for an analogous kite-based analysis of the mathematical terminology of compatibility and strong/weak contrariety.

  4. Aristotelian diagrams like these—as well as the JSB hexagon discussed above—are called \(\alpha \)-structures by Moretti [13]. See [4] for some further theoretical results on (the Boolean properties of) \(\alpha \)-structures, and [11] for another example of a decagonal \(\alpha \)-structure, which once again derives from the works of Schopenhauer.

  5. See [6, Section 4.3] for a more detailed analysis of the strong/weak Boolean subfamilies of the Aristotelian family of JSB hexagons.

  6. The case \(i = 3\) will be discussed in more detail later.

References

  1. Balbiani, P., Goranko, V., Kellerman, R., Vakarelov, D.: Logical theories for fragments of elementary geometry. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 343–428. Springer, New York (2007)

    Chapter  Google Scholar 

  2. Blanché, R.: Structures Intellectuelles. J. Vrin, Paris (1969)

  3. Demey, L.: Metalogic, metalanguage and logical geometry. Logique et Anal. (N.S.) 248, 453–478 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Demey, L.: From Euler diagrams in Schopenhauer to Aristotelian diagrams in logical geometry. In: Lemanski, J. (ed.) Language, Logic, and Mathematics in Schopenhauer, pp. 181–205. Springer, Cham (2020)

    Chapter  MATH  Google Scholar 

  5. Demey, L., Smessaert, H.: Duality in logic and language. In: Fieser, J., Dowden, B. (eds.) Internet Encyclopedia of Philosophy, pp. 1–37. University of Tennessee, Martin (2016)

    MATH  Google Scholar 

  6. Demey, L., Smessaert, H.: Combinatorial bitstring semantics for arbitrary logical fragments. J. Philos. Log. 47, 325–363 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Givant, S., Halmos, P.: Introduction to Boolean Algebras. Springer, New York (2009)

    MATH  Google Scholar 

  8. Horn, L.: A Natural History of Negation, 2nd edn. CSLI Publications, Stanford (2001)

    Google Scholar 

  9. Jacoby, P.: A triangle of opposites for types of propositions in Aristotelian logic. New Scholasticism 24, 32–56 (1950)

    Article  Google Scholar 

  10. Jaspers, D.: Logic and colour in cognition, logic and philosophy. In: Silva, M. (ed.) How Colours Matter to Philosophy, pp. 249–271. Springer, Cham (2017)

    Chapter  Google Scholar 

  11. Lemanski, J., Demey, L.: Schopenhauer’s partition diagrams and logical geometry. In: Basu, A., et al. (eds.) Diagrammatic Representation and Inference, pp. 149–165. Springer, Cham (2021)

    Chapter  MATH  Google Scholar 

  12. Moktefi, A.: Schopenhauer’s Eulerian diagrams. In: Lemanski, J. (ed.) Language, Logic, and Mathematics in Schopenhauer, pp. 111–127. Springer, Cham (2020)

    Chapter  Google Scholar 

  13. Moretti, A.: The Geometry of Logical Opposition. PhD thesis, Université de Neuchâtel, Neuchâtel (2009)

  14. Partee, B. (1987) Noun phrase interpretation and type-shifting principles. In: Groenendijk, J., de Jongh, D., Stokhof (eds.) Studies in Discourse Representation Theory and the Theory of Generalized Quantifiers, pp. 115–143. Dordrecht, Foris

  15. Pellissier, R.: Setting n-opposition. Log. Univers. 2(2), 235–263 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Renz, J., Nebel, B.: Qualitative spatial reasoning using constraint calculi. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 161–215. Springer, New York (2007)

    Chapter  Google Scholar 

  17. Schopenhauer, A.: The World as Will and Representation (translated and edited by J. Norman, A. Welchman and C. Janaway). Cambridge University Press, Cambridge (2010)

  18. Sesmat, A.: Logique II. Hermann, Paris (1951)

    Google Scholar 

  19. Seuren, P.A.M., Jaspers, D.: Logico-cognitive structure in the lexicon. Language 90(3), 607–643 (2014)

    Article  Google Scholar 

  20. Smessaert, H., Demey, L.: The unreasonable effectiveness of bitstrings in logical geometry. In: Béziau, J.Y., Basti, G. (eds.) The Square of Opposition: A Cornerstone of Thought, pp. 197–214. Springer, Berlin (2017)

    Chapter  Google Scholar 

  21. Weisstein, E.: Concise Encyclopedia of Mathematics. CRC Press, Boca Raton (1999)

    MATH  Google Scholar 

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Authors and Affiliations

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

    Hans Smessaert

  2. Center for Logic and Philosophy of Science, KU Leuven, Leuven, Belgium

    Lorenz Demey

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

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This research is supported via the ID-N research project Bitstring Semantics for Human and Artificial Reasoning (BITSHARE). The second author holds a research professorship (BOFZAP) at KU Leuven. Thanks to Jens Lemanski, Andrew Schumann and two anonymous reviewers for their feedback on an earlier version of this paper.

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Smessaert, H., Demey, L. On the Logical Geometry of Geometric Angles. Log. Univers. 16, 581–601 (2022). https://doi.org/10.1007/s11787-022-00315-7

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