The New Rising of the Square of Opposition

Abstract

In this paper I relate the story about the new rising of the square of opposition: how I got in touch with it and started to develop new ideas and to organize world congresses on the topic with subsequent publications. My first contact with the square was in connection with Slater’s criticisms of paraconsistent logic. Then by looking for an intuitive basis for paraconsistent negation, I was led to reconstruct S5 as a paraconsistent logic considering ¬□ as a paraconsistent negation. Making the connection between ¬□ and the O-corner of the square of opposition, I developed a paraconsistent star and hexagon of opposition and then a polyhedron of opposition, as a general framework to understand relations between modalities en negations. I also proposed the generalization of the theory of oppositions to polytomy. After having developed all this work I have begun to promote interdisciplinary world events on the square of opposition.

Keywords

Square of opposition Paraconsistent logic Negation Modal logic 

Mathematics Subject Classification

03B53 03B45 03B20 03B22 

References

  1. 1.
    Belnap, N.D.: A useful four-valued logic. In: Dunn, M. (ed.) Modern Uses of Multiple-Valued Logic, pp. 8–37. Reidel, Boston (1977) Google Scholar
  2. 2.
    Beziau, J.-Y.: L’holomouvement selon David Bohm. MD Thesis, Department of Philosophy, University Panthéon-Sorbonne, Paris (1986) Google Scholar
  3. 3.
    Beziau, J.-Y.: La logique paraconsistance C1 de Newton da Costa. MD Thesis, Department of Mathematics, University Denis Diderot, Paris (1990) Google Scholar
  4. 4.
    Beziau, J.-Y.: Review of [49]. Mathematical Rev., 96e03035 Google Scholar
  5. 5.
    Beziau, J.-Y.: What is paraconsistent logic? In: Batens, D., et al. (eds.) Frontiers of Paraconsistent Logic, pp. 95–111. Research Studies, Baldock (2000) Google Scholar
  6. 6.
    Beziau, J.-Y.: From paraconsistent to universal logic. Sorites 12, 5–32 (2001) MathSciNetGoogle Scholar
  7. 7.
    Beziau, J.-Y.: The logic of confusion. In: Arabnia, H.R. (ed.) Proceedings of the International Conference of Artificial Intelligence IC-AI’2002, pp. 821–826. CSREA, Las Vegas (2001) Google Scholar
  8. 8.
    Beziau, J.-Y.: Are paraconsistent negations negations? In: Carnielli, W., et al. (eds.) Paraconsistency: The Logical Way to the Inconsistent, pp. 465–486. Dekker, New York (2002) Google Scholar
  9. 9.
    Beziau, J.-Y.: S5 is a paraconsistent logic and so is first-order classical logic. Log. Investig. 9, 301–309 (2002) Google Scholar
  10. 10.
    Beziau, J.-Y.: New light on the square of oppositions and its nameless corner. Log. Investig. 10, 218–232 (2003) MathSciNetGoogle Scholar
  11. 11.
    Beziau, J.-Y.: Paraconsistent logic from a modal viewpoint. J. Appl. Log. 3, 7–14 (2005) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Beziau, J.-Y.: The paraconsistent logic Z—a possible solution to Jaskowski’s problem. Logic Log. Philos. 15, 99–111 (2006) MathSciNetMATHGoogle Scholar
  13. 13.
    Beziau, J.-Y.: Paraconsistent logic! (A reply to Salter). Sorites 17, 17–26 (2006) MathSciNetGoogle Scholar
  14. 14.
    Beziau, J.-Y.: Adventures in the paraconsistent jungle. In: Handbook of Paraconsistency, pp. 63–80. King’s College, London (2007) Google Scholar
  15. 15.
    Beziau, J.-Y., Payette, G. (eds.): Special Issue on the Square of Opposition. Logica Univers. 2(1) (2008) Google Scholar
  16. 16.
    Beziau, J.-Y.: A new four-valued approach to modal logic. Log. Anal. 54, 18–33 (2011) MathSciNetGoogle Scholar
  17. 17.
    Beziau, J.-Y.: The power of the hexagon. In: [18] Google Scholar
  18. 18.
    Beziau, J.-Y. (ed.): Special Double Issue on the Hexagon of Opposition. Logica Univers. 6(1–2) (2012) Google Scholar
  19. 19.
    Beziau, J.-Y., Payette, G. (eds.): The Square of Opposition—A General Framework for Cognition. Peter Lang, Bern (2012) Google Scholar
  20. 20.
    Blanché, R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953) CrossRefGoogle Scholar
  21. 21.
    Blanché, R.: Opposition et négation. Rev. Philos. 167, 187–216 (1957) Google Scholar
  22. 22.
    Blanché, R.: Sur la structuration du tableau des connectifs interpropositionnels binaires. J. Symb. Log. 22, 17–18 (1957) MATHCrossRefGoogle Scholar
  23. 23.
    Blanché, R.: Structures intellectuelles. Essai sur l’organisation systématique des concepts. Vrin, Paris (1966) Google Scholar
  24. 24.
    Bohm, D.: Wholeness and the Implicate Order. Routlege, London (1980) Google Scholar
  25. 25.
    Gardies, J.-L.: Essai sur la logique des modalités. PUF, Paris (1979) Google Scholar
  26. 26.
    Gödel, K.: Zum intuitionisticischen Aussagenkalkül. Akad. Wiss. Wien, Math.-Nat. Wiss. Kl. 64, 65–66 (1932) Google Scholar
  27. 27.
    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
  28. 28.
    Horn, L.R.: A Natural History of Negation. University Chicago Press, Chicago (1989) Google Scholar
  29. 29.
    Humberstone, L.: The logic of non-contingency. Notre Dame J. Form. Log. 36, 214–229 (1995) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Humberstone, L.: Modality. In: Jackson, F.C., Smith, M. (eds.) The Oxford Handbook of Contemporary Philosophy. Handbook of Analytical Philosophy, pp. 534–614. Oxford University Press, Oxford (2005) Google Scholar
  31. 31.
    Humberstone, L.: Contrariety and subcontrariety: the anatomy of negation (with special reference to an example of J.-Y. Béziau). Theoria 71, 241–262 (2005) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Jaśkowski, S.: Rachunek zdan dla systemów dedukcyjnych sprzecznych. Stud. Soc. Sci. Torun., Sect. A I(5), 55–77 (1948) Google Scholar
  33. 33.
    Kant, I.: Logik (1800) Google Scholar
  34. 34.
    The Kybalion: A Study of the Hermetic Philosophy of Ancient Egypt and Greece. Yogi Publication Society (1908) Google Scholar
  35. 35.
    Larvor, B.: Triangular logic. The Philosophers’ Magazine 44, 83–88 (2009) Google Scholar
  36. 36.
    Łukasiewicz, J.: A system of modal logic. J. Comput. Syst. 1, 111–149 (1953) Google Scholar
  37. 37.
    Luzeaux, D., Sallantin, J., Dartnell, C.: Logical extensions of Aristotle’s square. Logica Univers. 2, 167–187 (2008) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Omori, H., Wagarai, T.: On Beziau’s logic Z. Logic Log. Philos. 18, 305–320 (2008) Google Scholar
  39. 39.
    Marcos, J.: Nearly every normal modal logic is paranormal. Log. Anal. 48, 279–300 (2005) MathSciNetMATHGoogle Scholar
  40. 40.
    Marcos, J.: Modality and paraconsistency. In: Bilkova, M., Behounek, L. (eds.) The Logica Yearbook 2004, pp. 213–222. Filosofia, Prague (2005) Google 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. Université de Neuchâtel, Neuchâtel (2004) Google Scholar
  42. 42.
    Moretti, A.: The geometry of logical opposition. PhD Thesis, University of Neuchâtel (2009) Google Scholar
  43. 43.
    Mruczek-Nasieniewska, K., Nasieniewski, M.: Paraconsistent logics obtained by J.-Y. Beziau’s method by means of some non-normal modal logics. Bull. Sect. Log. 37, 185–196 (2008) MathSciNetGoogle Scholar
  44. 44.
    Mruczek-Nasieniewska, K., Nasieniewski, M.: Beziau’s logics obtained by means of quasi-regular logics. Bull. Sect. Log. 38, 189–203 (2009) MathSciNetGoogle Scholar
  45. 45.
    Osorio, M., Carballido, J.L., Zepeda, C.: Revisiting Z (to appear) Google Scholar
  46. 46.
    Pellissier, R.: “Setting” n-opposition. Logica Univers. 2, 235–263 (2008) MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Pellissier, R.: 2-Opposition and the topological hexagon. In: [18] Google Scholar
  48. 48.
    Montgomery, H., Routley, R.: Contingency and non-contingency bases for normal modal logics. Log. Anal. 9, 341–344 (1966) MathSciNetGoogle Scholar
  49. 49.
    Slater, B.H.: Paraconsistent logics. J. Philos. Log. 24, 451–454 (1995) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Smessaert, H.: On the 3D visualisation of logical relations. Logica Univers. 3, 303–332 (2009) MathSciNetCrossRefGoogle Scholar
  51. 51.
    Smessaert, H.: The classical Aristotelian hexagon versus the modern duality hexagon. In: [19] Google Scholar
  52. 52.
    Sokal, A.: Transgressing the boundaries: towards a transformative hermeneutics of quantum gravity. Social Text 46–47, 217–252 (1996) CrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.CNPq – Brazilian Research CouncilUFRJ – University of BrazilRio de JaneiroBrazil

Personalised recommendations