From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory

Abstract

The paper first introduces a cube of opposition that associates the traditional square of opposition with the dual square obtained by Piaget’s reciprocation. It is then pointed out that Blanché’s extension of the square-of-opposition structure into an conceptual hexagonal structure always relies on an abstract tripartition. Considering quadripartitions leads to organize the 16 binary connectives into a regular tetrahedron. Lastly, the cube of opposition, once interpreted in modal terms, is shown to account for a recent generalization of formal concept analysis, where noticeable hexagons are also laid bare. This generalization of formal concept analysis is motivated by a parallel with bipolar possibility theory. The latter, albeit graded, is indeed based on four graded set functions that can be organized in a similar structure.

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References

  1. 1.

    Anonymous. Syllogisme. In: Encyclopédie ou Dictionnaire Raisonné des Sciences, des Arts et des Métiers, de D. Diderot and J. (le Rond) d’Alembert. 1751–1772. http://portail.atilf.fr/encyclopedie/Formulaire-de-recherche.htm

  2. 2.

    Belnap N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic., D. Reidel, Dordrecht (1977)

    Google Scholar 

  3. 3.

    Benferhat S., Dubois D., Kaci S., Prade H.: Modeling positive and negative information in possibility theory. Int. J. Intell. Syst. 23, 1094–1118 (2008)

    MATH  Article  Google Scholar 

  4. 4.

    Béziau J.-Y.: New light on the square of oppositions and its nameless corner. Logical Invest. 10, 218–233 (2003)

    Google Scholar 

  5. 5.

    Blanché R.: Quantity, modality, and other kindred systems of categories. Mind. LXI(243), 369–375 (1952)

    Article  Google Scholar 

  6. 6.

    Blanché R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953)

    Article  Google Scholar 

  7. 7.

    Blanché R.: Opposition et négation. Revue Philosophique de la France et de l’Étranger. CXLVII, 187–216 (1957)

    Google Scholar 

  8. 8.

    Blanché R.: Structures Intellectuelles. Essai sur l’Organisation Systématique des Concepts. Vrin, Paris (1966)

    Google Scholar 

  9. 9.

    Boole G.: An investigation of The Laws of Thought on which are founded The Mathematical Theories of Logic and Probabilities. Chap. XV: The Aristotelian Logic and its Modern Extensions, Examined by the Method of its Treatise. Macmillan, 1854. Reprinted by Dover, NewYork (1958)

    Google Scholar 

  10. 10.

    Castañeda, H.-N.: Review of “A negação. Revista Brasileira de Filosofia, 7, 448–457, 1957, by L. Hegenberg”. J. Symb. Logic 25(3), 265 (1960)

    Google Scholar 

  11. 11.

    Cohn, A.G., Gotts, N.M.: The ‘egg-yolk’ representation of regions with indeterminate boundaries. Proc. GISDATA Specialist Meeting on Geographical Objects with Undetermined Boundaries, vol. 2, pp. 71–187. Francis Taylor (1996)

  12. 12.

    De Morgan, A.: On the structure of the syllogism. Trans. Camb. Phil. Soc. VIII, 379–408 (1846). Reprinted in: On the Syllogism and Other Logical Writings (Heath, P., ed.). Routledge and Kegan Paul, London (1966)

  13. 13.

    Djouadi, Y., Dubois, D., Prade, H.: Possibility theory and formal concept analysis: context decomposition and uncertainty handling. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) Computational Intelligence for Knowledge-Based Systems Design (IPMU 2010). Springer, LNCS 6178, 260–269 (2010)

  14. 14.

    Dubois D.: On ignorance and contradiction considered as truth-values. Logic J. IGPL 16(2), 195–216 (2008)

    MATH  Article  Google Scholar 

  15. 15.

    Dubois D.: Degrees of truth, ill-known sets and contradiction. In: Bouchon-Meunier, B., Magdalena, L., Ojeda-Aciego, M., Verdegay, J.-L. (eds.) Foundations of Reasoning under Uncertainty. Studies in Fuzziness and Soft Computing, vol. 249., pp. 65–83. Springer, Berlin (2010)

    Google Scholar 

  16. 16.

    Dubois D., Dupin de Saint-Cyr F., Prade H.: A possibility-theoretic view of formal concept analysis. Fundam. Inf. 75, 195–213 (2007)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Dubois D., Esteva F., Godo L., Prade H.: An information-based discussion of vagueness: six scenarios leading to vagueness. In: Cohen, H., Lefebvre, C. (eds.) Handbook of Categorization in Cognitive Science, chap. 40, pp. 891–909. Elsevier, Amsterdam (2005)

    Google Scholar 

  18. 18.

    Dubois D., Hajek P., Prade H.: Knowledge-driven versus data-driven logics. J. Logic Lang. Inf. 9, 65–89 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Dubois D., Prade H.: Possibility theory: qualitative and quantitative aspects. In: Gabbay, D., Smets, P. (eds.) Quantified Representation of Uncertainty and Imprecision. Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 1., pp. 169–226. Kluwer, Dordrecht (1998)

    Google Scholar 

  20. 20.

    Dubois, D., Prade, H.: Squares of opposition in generalized formal concept analysis and possibility theory. In: Béziau, J.-Y., Gan-Krzywoszyńska, K. (eds.) Hand Book of the 2nd World Congress on the Square of Opposition, Corte, Corsica, June 17–20, 2010, p. 42. Università di Corsica, Pasquale Paoli (2010)

  21. 21.

    Dubois, D., Prade, H.: De l’organisation hexagonale des concepts de Blanché à l’analyse formelle de concepts et à la théorie des possibilités. Actes 5iemes Journées d’Intelligence Artificielle Fondamentale, Lyon, 8–10 Juin, 113–129 (2011)

  22. 22.

    Dubois, D., Prade, H.: Possibility theory and formal concept analysis: characterizing independent sub-contexts. Fuzzy Sets Syst. (2012, to appear)

  23. 23.

    Dupleix, S.: De l’opposition des énonciations. Book IV, chap 10. In: La Logique ou Art de Discourir et de Raisonner. 1607. Reprinted by Fayard, Paris (1984)

  24. 24.

    Fodor J., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht (1994)

    Google Scholar 

  25. 25.

    Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer (1999)

  26. 26.

    Gentilhomme Y.: Les ensembles flous en linguistique. Cahiers de Linguistique Théorique et Appliquée (Bucarest) 5, 47–63 (1968)

    Google Scholar 

  27. 27.

    Gottschalk W.H.: The theory of quaternality. J. Symb. Logic 18, 193–196 (1953)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Horn L.R.: Hamburgers and truth: why Gricean explanation is Gricean. In: Hall, K., Koenig, J.-P., Meacham, M., Reinman, S., Sutton, L.A. (eds.) Proc. of 16th Annual Meeting of the Berkeley Linguistics Society., pp. 454–471. Berkeley Linguistics Society, Berkeley (1990)

    Google Scholar 

  29. 29.

    Jacoby P.: A triangle of opposites for types of propositions in Aristotelian logic. New Scholasticism XXIV(1), 32–56 (1950)

    Google Scholar 

  30. 30.

    Hamblin C.L.: The modal “Probably”. Mind 60(270), 234–240 (1959)

    Article  Google Scholar 

  31. 31.

    Klement E.P., Mesiar R., Pap E.: Triangular Norms. Kluwer, Dordrecht (2000)

    Google Scholar 

  32. 32.

    Luzeaux D., Sallantin J., Dartnell C.: Logical extensions of Aristotle’s square. Log. Univers. 2(1), 167–187 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Moretti A.: Geometry for modalities? Yes: through n-opposition theory. In: Béziau, J.-Y., Costa Leite, A., Facchini, A. (eds.) Aspects of Universal Logic. Special issue of Travaux de Logique, n. 17., pp. 102–145. Université de Neuchâtel, Neuchâtel (2004)

    Google Scholar 

  34. 34.

    Moretti, A.: The geometry of logical opposition. Ph.D. dissertation, University of Neuchâtel, Switzerland, March 2009. http://alessiomoretti.perso.sfr.fr/

  35. 35.

    Moretti A.: The geometry of standard deontic logic. Log. Univers. 3(1), 19–57 (2009)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Moretti, A.: From the “logical square” to the “logical poly-simplexes”. A quick survey of what happened in between. In: Béziau, J.-Y., Payette, G. (eds.) New Perspectives on the Square of Opposition. Peter Lang, Bern (to appear)

  37. 37.

    Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Fall 2008 Edition). http://plato.stanford.edu/archives/fall2008/entries/square/

  38. 38.

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

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Pellissier, R.: 2-opposition and the topological hexagon. In: New Perspectives on the Square of Opposition, (J.-Y. Béziau, G. Payette, eds.), Peter Lang, Bern (to appear)

  40. 40.

    Piaget J.: Traité de Logique. Essai de Logistique Opératoire. Armand Colin, Paris (1949)

    Google Scholar 

  41. 41.

    Piaget J.: Essai sur les Transformations des Opérations Logiques: Les 256 opérations ternaires de la logique bivalente des propositions. Presses Universitaires de France, Paris (1952)

    Google Scholar 

  42. 42.

    Roubens M., Vincke Ph.: Preference Modeling. LNEMS 250. Springer, Berlin (1985)

    Google Scholar 

  43. 43.

    Shackle G.L.S.: Decision, Order, and Time in Human Affairs. Cambridge University Press, UK (1961)

    Google Scholar 

  44. 44.

    Sesmat, A.: Logique II: Les Raisonnements, la Logistique. §115 to §135. Hermann, Paris (1951)

  45. 45.

    Smessaert H.: On the 3D visualisation of logical relations. Log. Univers. 3, 303–332 (2009)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Varzi A.C., Warglien M.: The geometry of negation. J. Appl. Non-Classical Logics 13, 9–19 (2003)

    MATH  Article  Google Scholar 

  47. 47.

    Zadeh L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Zadeh L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Henri Prade.

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Dubois, D., Prade, H. From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory. Log. Univers. 6, 149–169 (2012). https://doi.org/10.1007/s11787-011-0039-0

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Mathematics Subject Classification (2010)

  • Primary 68T30
  • Secondary 03A05
  • 03B05
  • 68T37

Keywords

  • Square of opposition
  • Blanché hexagon
  • Piaget group
  • propositional connectives
  • formal concept analysis
  • possibility theory