Logica Universalis

, Volume 6, Issue 1–2, pp 1–43 | Cite as

The Power of the Hexagon

Article

Abstract

The hexagon of opposition is an improvement of the square of opposition due to Robert Blanché. After a short presentation of the square and its various interpretations, we discuss two important problems related with the square: the problem of the I-corner and the problem of the O-corner. The meaning of the notion described by the I-corner does not correspond to the name used for it. In the case of the O-corner, the problem is not a wrong-name problem but a no-name problem and it is not clear what is the intuitive notion corresponding to it. We explain then that the triangle of contrariety proposed by different people such as Vasiliev and Jespersen solves these problems, but that we don’t need to reject the square. It can be reconstructed from this triangle of contrariety, by considering a dual triangle of subcontrariety. This is the main idea of Blanché’s hexagon. We then give different examples of hexagons to show how this framework can be useful to conceptual analysis in many different fields such as economy, music, semiotics, identity theory, philosophy, metalogic and the metatheory of the hexagon itself. We finish by discussing the abstract structure of the hexagon and by showing how we can swing from sense to non-sense thinking with the hexagon.

Mathematics Subject Classification

Primary 03B22 Secondary 03A05 03B44 03B45 00A30 

Keywords

Hexagon of opposition square of opposition contradiction negation quantification possibility modal logic deontic logic conceptual analysis structuralism truth a priori 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Badir, S.: How the semiotic square came. In [9], pp. 427–442 (2012)Google Scholar
  2. 2.
    Beziau J.-Y.: New light on the square of oppositions and its nameless corner. Log. Investig. 10, 218–232 (2003)MathSciNetGoogle Scholar
  3. 3.
    Beziau J.-Y.: The new rising of the square. In: Beziau, J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition., Birkhäuser, Basel (2012)CrossRefGoogle Scholar
  4. 4.
    Beziau J.-Y.: History of the concept of truth-value. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic, vol. 11, pp. 233–305. Elsevier, Amsterdam (2012)Google Scholar
  5. 5.
    Beziau J.-Y. et al.: Let be antilogic: anticlassical logic as a logic. In: Moktefi, A. (eds.) Soyons Logique, College Publication, London (2012)Google Scholar
  6. 6.
    Beziau J.-Y., (ed.): La Pointure du Symbole. Editions Pétra, Paris (2012)Google Scholar
  7. 7.
    Beziau, J.-Y., Jacquette, D. (eds.): Around and Beyond the Square of Opposition. Birkhäuser, Basel (2012)MATHGoogle Scholar
  8. 8.
    Beziau, J.-Y., Payette, G. (eds.): Special issue on the square of opposition. Logica Universalis 2(1) (2008)Google Scholar
  9. 9.
    Beziau, J.-Y., Payette, G. (eds.): The Square of Opposition—A General Framework for Cognition. Peter Lang, Bern (2012)Google Scholar
  10. 10.
    Blanché R.: Quantity, modality, and other kindred systems of categories. Mind 61, 369–375 (1952)CrossRefGoogle Scholar
  11. 11.
    Blanché R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953)CrossRefGoogle Scholar
  12. 12.
    Blanché R.: Opposition et négation. Revue Philosophique 167, 187–216 (1957)Google Scholar
  13. 13.
    Blanché R.: Sur la structuration du tableau des connectifs interpropositionnels binaires. J. Symb. Log. 22, 17–18 (1957)MATHCrossRefGoogle Scholar
  14. 14.
    Blanché R.: Structures intellectuelles. Vrin, Paris (1966)Google Scholar
  15. 15.
    Blanché R.: Sur le système des connecteurs interpropositionnels. Cahiers pour l’Analyse 10, 131–149 (1969)Google Scholar
  16. 16.
    Boethius, Institutio Musica. ca 505. English translation: C.M. Bower, Fundamentals of Music. Yale University Press, New Haven (1989)Google Scholar
  17. 17.
    Bréal M.: Essai de sémantique: science des significations. Hachette, Paris (1897)Google Scholar
  18. 18.
    Cartier, P.: How to take advantage of the blur between the finite and the infinite. Logica Universalis 6 (2012). doi:10.1007/s11787-012-0043-z
  19. 19.
    Dubois, D., Prade, H.: From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Logica Universalis 6 (2012). doi:10.1007/s11787-011-0039-0
  20. 20.
    Dufatanye, A.-A.: From the logical square to Blanché’s hexagon: formalization, applicability and the idea of the normative structure of thought. Logica Universalis 6 (2012). doi:10.1007/s11787-012-0040-2
  21. 21.
    Fellbaum C.: Wordnet, an Electronic Lexical Database for English. MIT Press, Cambridge (1998)Google Scholar
  22. 22.
    Granger G.G.: Pensée formelle et sciences de l’homme. Aubier, Paris (1960)Google Scholar
  23. 23.
    Grattan-Guinness I.: Omnipresence, multipresence and ubiquity: kinds of generality in and around mathematics and logics. Logica Universalis 5, 21–73 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Grigg R.: Lacan and Badiou: logic of the “pas-tout”. Filozofski vestnik 26, 53–65 (2005)Google Scholar
  25. 25.
    Grize J.-B.: Des carrés qui ne tournent pas rond et de quelques autres. Travaux du centre de recherches sémiologiques 56, 139–152 (1988)Google Scholar
  26. 26.
    Guitart, R.: A hexagonal framework of the field F4 and the associated Borromean Logic. Logica Universalis 6 (2012). doi:10.1007/s11787-011-0033-6
  27. 27.
    van Heijenoort J.: Review of [13]. J. Symb. Log. 24, 228 (1959)CrossRefGoogle Scholar
  28. 28.
    Hoeksema, J.: Blocking Effects and Polarity Sensitivity. In: Essays dedicated to Johan van Benthem on the Occasion of his 50th Birthday. Amsterdam University Press, Amsterdam (1999)Google Scholar
  29. 29.
    Horn L.R.: A Natural History of Negation. University Chicago Press, Chicago (1989)Google Scholar
  30. 30.
    Jaspers, D.: Logic and colour. Logica Universalis 6 (2012). doi:10.1007/s11787-012-0044-y
  31. 31.
    Jespersen O.: Negation in English and Other Languages. A.F. Host and Son, Copenhagen (1917)Google Scholar
  32. 32.
    Joerden, J.C.: Deontological square, hexagon, and decagon: a deontic framework for supererogation. Logica Universalis 6 (2012). doi:10.1007/s11787-012-0041-1
  33. 33.
    Kant, I.: Logik (Jäsche), 1800. English translation in Lectures on Logic, translated and edited by J. M. Young, Cambridge University Press, Cambridge (1992)Google Scholar
  34. 34.
    Lacan, J.: Le séminaire XX—Encore, Seuil, Paris, 1972–73. English translation: Jacques-Alain Miller (ed.) The Seminar XX, Encore: On Feminine Sexuality, the Limits of Love and Knowledge; transl. by Bruce Fink. W.W. Norton and Co., New York (1998)Google Scholar
  35. 35.
    Lacan J.: L’Étourdit. Scilicet Paris 6, 5–52 (1973)Google Scholar
  36. 36.
    Lévi-Strauss, C.: Les structures élémentaires de la parenté. PUF, Paris (1949)Google Scholar
  37. 37.
    Łukasiewicz J.: Logice trójwartościowej. Ruch Filozoficny 5, 170–171 (1920)Google Scholar
  38. 38.
    Łukasiewicz J.: Aristotle’s Syllogistic from the Standpoint of Modern Logic. Clarendon, Oxford (1951)MATHGoogle Scholar
  39. 39.
    Łukasiewicz J.: A system of modal logic. J. Comput. Syst. 1, 111–149 (1953)Google Scholar
  40. 40.
    Luzeaux D., Sallantin J., Dartnell C.: Logical extensions of Aristotle’s square. Logica Universalis 2, 167–187 (2008)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Moretti, A.: Geometry of modalities? Yes: through n-opposition theory. In: Beziau, J.-Y., Costa Leite, A., Facchini, A. (eds.) Aspects of Universal Logic, Travaux de logique, vol. 17, pp. 102–145, Neuchâtel (2004)Google Scholar
  42. 42.
    Moretti, A.: The geometry of logical opposition. PhD thesis, University of Neuchâtel (2009)Google Scholar
  43. 43.
    Moretti, A.: Why the logical hexagon? Logica Universalis 6 (2012). doi:10.1007/s11787-012-0045-x
  44. 44.
    Parsons, T.: The traditional square of opposition. Stanford Encyclopedia of Philosophy (2006, online)Google Scholar
  45. 45.
    Parsons T.: Things that are right with the traditional square of opposition. Logica Universalis 2, 3–11 (2008)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Pellissier R.: “Setting” n-opposition. Logica Universalis 2, 235–263 (2008)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Sesmat A.: Logique II. Les raisonnements, la logistique. Hermann, Paris (1951)Google Scholar
  48. 48.
    Seuren P.M.: The Logic of Language—Language from within Volume II. Oxford University Press, Oxford (2010)Google Scholar
  49. 49.
    Sigman M., Cecchi G.A.: Global organization of the Wordnet lexicon. Proc. Natl. Acad. Sci. 99, 1742–1747 (2002)CrossRefGoogle Scholar
  50. 50.
    Simons, P.: Approaching the alethic modal hexagon of opposition. Logica Universalis 6 (2012). doi:10.1007/s11787-012-0042-0
  51. 51.
    Smessaert, H.: The classical Aristotelian hexagon versus the modern duality hexagon. Logica Universalis 6 (2012). doi:10.1007/s11787-011-0031-8
  52. 52.
    Smith N.B.: The idea of the French hexagon. French Hist. Stud. 6, 139–155 (1969)CrossRefGoogle Scholar
  53. 53.
    Sullivan M.W.: Apuleian Logic. The Nature, Sources, and Influence of Apuleius’s Peri Hermeneias. North-Holland, Amsterdam (1967)MATHGoogle Scholar
  54. 54.
    Tarski A.: Drei Briefe an Otto Neurath. 1936 published in Grazer Philosophische Studien 43, 1–32 (1992)Google Scholar
  55. 55.
    Tiles M., Jinmei Y.: Could the Aristotelian square of opposition be translated into Chinese?. Dao J. Comp. Philos. 4, 137–149 (2004)Google Scholar
  56. 56.
    Vasiliev, N.: On partial judgements, the triangle of opposites and the law of excluded fourth. Scientific Papers of Kazan University (1910) (in Russia)Google Scholar
  57. 57.
    Wajsberg M.: Ein erweiterter Klassenkalkül. Monatshefte fur Mathematik und Physik 4, 113–126 (1933)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Weil, A.: Sur l’étude algébrique de certains types de lois du mariage. Appendix in [36], pp. 257–272 (1949)Google Scholar
  59. 59.
    Weil, A.: Souvenirs d’apprentissage. Birkhäuser, Basel (1991); English translation: The Apprenticeship of a Mathematician. Birkhäuser, Basel (1992)Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.CNPq-Brazilian Research CouncilUniversity of Brazil, Rio de Janeiro (UFRJ)Rio de JaneiroBrazil

Personalised recommendations