Skip to main content
SpringerLink
Log in

Structures of Opposition and Comparisons: Boolean and Gradual Cases

  • Published:
Logica Universalis Aims and scope Submit manuscript

Abstract

This paper first investigates logical characterizations of different structures of opposition that extend the square of opposition in a way or in another. Blanché’s hexagon of opposition is based on three disjoint sets. There are at least two meaningful cubes of opposition, proposed respectively by two of the authors and by Moretti, and pioneered by philosophers such as J. N. Keynes, W. E. Johnson, for the former, and H. Reichenbach for the latter. These cubes exhibit four and six squares of opposition respectively. We clarify the differences between these two cubes, and discuss their gradual extensions, as well as the one of the hexagon when vertices are no longer two-valued. The second part of the paper is dedicated to the use of these structures of opposition (hexagon, cubes) for discussing the comparison of two items. Comparing two items (objects, images) usually involves a set of relevant attributes whose values are compared, and may be expressed in terms of different modalities such as identity, similarity, difference, opposition, analogy. Recently, J.-Y. Béziau has proposed an “analogical hexagon” that organizes the relations linking these modalities. Elementary comparisons may be a matter of degree, attributes may not have the same importance. The paper studies in which ways the structure of the hexagon may be preserved in such gradual extensions. As another illustration of the graded hexagon, we start with the hexagon of equality and inequality due to R. Blanché and extend it with fuzzy equality and fuzzy inequality. Besides, the cube induced by a tetra-partition can account for the comparison of two items in terms of preference, reversed preference, indifference and non-comparability even if these notions are a matter of degree. The other cube, which organizes the relations between the different weighted qualitative aggregation modes, is more relevant for the attribute-based comparison of items in terms of similarity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Notes

  1. In the crisp case it means \(\overline{A}\cap (A\cup B) = B\) if A and B are disjoint.

  2. In previous papers [16, 17, 32], we wrongly credited Reichenbach [49] for inventing this cube in the setting of syllogisms, because we omitted to consider additional constraints of non-equality of involved predicates introduced by him. This point is discussed later on in the next subsection.

  3. The actual octagon of Johnson and Keynes does not materialize the bold lines expressing mutual exclusiveness, while other lines relating A to a and E to e appear and are labeled “complementaries”; likewise, lines relating I to i and O to o appear and are labeled “sub-complementaries”. For instance, A= all P’s are Q and a = all Q’s are P complement each other in the sense that if they both hold, P and Q are identical. Interestingly, in the Reichenbach cube, both A and a, and E and e, are mutually exclusive (and called “opposite”), since P is supposed not to be equal to Q. Moreover yet other lines relating A to o, E to i, O to a and I to e are labeled “contra-complementaries”. Observe that these different forms of complementarity do not appear in the traditional square.

References

  1. Alsina, C.: On a family of connectives for fuzzy sets. Fuzzy Sets Syst. 16(3), 231–235 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amgoud, L., Prade, H.: A formal concept view of abstract argumentation. In: van der Gaag, L.C. (ed.) Proceedings of the European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’13), Utrecht, July 8–10. LNCS, vol. 7952, pp. 1–12. Springer, Berlin (2013)

  3. Barbot, N., Miclet, L., Prade, H.: A new perspective on analogical proportions. In: Kern-Isberner, G., Ognjanović, Z. (eds.) Proceedings of the 15th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’19), Belgrade, Sept 18–20. LNCS 11726, pp. 163–174. Springer, Berlin (2019)

  4. Barbot, N., Miclet, L., Prade, H.: The analogical cube of opposition. In: Béziau, J.Y., Buchsbaum, A., Vandoulakis, I. (eds.) Handbook of Abstracts of the 6th World Congress on the Square of Opposition, Crete, Nov 1–5, pp. 9–10 (2018)

  5. Béziau, J.Y., Buchsbaum, A., Correia, M. (eds.): Handbook of Abstracts of the 5th World Congress on the Square of Opposition, Easter Island, Nov 11–15 (2016)

  6. Béziau, J.Y., Gan-Krzywoszyńska, K. (eds.): Handbook of Abstracts of the 2nd World Congress on the Square of Opposition, Corte, Corsica, June 17–20 (2010)

  7. Béziau, J.Y., Gan-Krzywoszyńska, K. (eds.): Handbook of Abstracts of the 3rd World Congress on the Square of Opposition, Beirut, Lebanon, June 26–30 (2012)

  8. Béziau, J.Y., Gan-Krzywoszyńska, K. (eds.): Handbook of Abstracts of the 4th World Congress on the Square of Opposition, Roma, Vatican, May 5–9 (2014)

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

    MathSciNet  MATH  Google Scholar 

  10. Béziau, J.-Y.: The power of the hexagon. Log. Univ. 6(1–2), 1–43 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Béziau, J.-Y.: There is no cube of opposition. In: Beziau, J.-Y., Basti, G. (eds.) The Square of Oppositions: A Cornerstone of Thought, Studies in Universal Logic, pp. 179–193. Birkaüser Verlag, Basel (2017)

    Chapter  Google Scholar 

  12. Béziau, J.-Y.: An analogical hexagon. Int. J. Approx. Reason. 94, 1–17 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Béziau, J.Y., Payette, G. (eds.): The Square of Opposition. A General Framework for Cognition. Peter Lang, Bern (2012)

    Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  16. Ciucci, D., Dubois, D., Prade, H.: Structures of opposition in fuzzy rough sets. Fundam. Inform. 142, 1–19 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ciucci, D., Dubois, D., Prade, H.: Structures of opposition induced by relations. The Boolean and the gradual cases. Ann. Math. Artif. Intell. 76(3), 351–373 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Coletti, G., Petturiti, D., Vantaggi, B.: Fuzzy weighted attribute combinations based on similarity measures. In: Antonucci, A., Cholvy, L., Papini, O. (eds.) Proceedings of the 4th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’17), Lugano, July 10–14. LNCS 10369, pp. 364–374. Springer, Berlin (2017)

  19. de Bovelles, Ch.: L’Art des Opposés. Texte et traduction par P. Magnard précédé d’un essai, Soleil Noir, avec 37 reproductions. Librairie Philosophique J. Vrin, Paris (1984)

  20. Diderot, D., d’Alembert, J. (eds.): Encyclopédie ou Dictionnaire Raisonné des Sciences des Arts et Métiers (1751–1772)

  21. Dubois, D., Prade, H., Rico, A.: Fuzzy extensions of conceptual structures of comparison. In: Medina, J., Ojeda-Aciego, M., Verdegay Galdeano, J.L., Pelta, D.A., Cabrera, I.P., Bouchon-Meunier, B., Yager, R.R. (eds.) Proceedings of the 17th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’18), Cádiz, June 11–15, Part I, Communications in Computer and Information Science, vol. 853, pp. 710–722. Springer, Berlin (2018)

  22. Dubois, D., Prade, H., Rico, A.: The cube of opposition: a structure underlying many knowledge representation formalisms. In: Yang, Q., Wooldridge, M. (eds.) Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI’15), Buenos Aires, July 25–31, pp. 2933–2939. AAAI Press (2015)

  23. Dubois, D.: Generalized probabilistic independence and its implications for utility. Oper. Res. Lett. 5(5), 255–260 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dubois, D., Prade, H.: Weighted minimum and maximum operations in fuzzy set theory. Inf. Sci. 39(2), 205–210 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dubois, D., Prade, H.: Possibility Theory—An Approach to Computerized Processing of Uncertainty. Plenum Press, New York (1988)

    MATH  Google Scholar 

  26. Dubois, D., Prade, H.: From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Log. Univers. 6, 149–169 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dubois, D., Prade, H.: Gradual structures of oppositions. In: Esteva, F., Magdalena, L., Verdegay, J.L. (eds.) Enric Trillas: Passion for Fuzzy Sets, Studies in Fuzziness and Soft Computing, vol. 322, pp. 79–91. Springer, Berlin (2015)

    Chapter  Google Scholar 

  28. Dubois, D., Kerre, E., Mesiar, R., Prade, H.: Fuzzy interval analysis. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets Series, pp. 483–581. Kluwer, Boston, MA (2000)

    Chapter  MATH  Google Scholar 

  29. Dubois, D., HadjAli, A., Prade, H.: Fuzziness and uncertainty in temporal reasoning. J. Univers. Comput. Sci. 9, 1168–1194 (2003)

    MathSciNet  Google Scholar 

  30. Dubois, D., Prade, H., Rico, A.: Residuated variants of Sugeno integrals: towards new weighting schemes for qualitative aggregation methods. Inf. Sci. 329, 765–781 (2016)

    Article  MATH  Google Scholar 

  31. Dubois, D., Prade, H., Rico, A.: Organizing families of aggregation operators into a cube of opposition. In: Kacprzyk, J., Filev, D., Beliakov, G. (eds.) Granular, Soft and Fuzzy Approaches for Intelligent Systems, Dedicated to Professor Ronald R. Yager, Studies in Fuzziness and Soft Computing, vol. 344, pp. 27–45. Springer, Berlin (2017)

    Chapter  Google Scholar 

  32. Dubois, D., Prade, H., Rico, A.: Graded cubes of opposition and possibility theory with fuzzy events. Int. J. Approx. Reason. 84, 168–185 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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)

  34. Fodor, J.C.: Contrapositive symmetry of fuzzy implications. Fuzzy Sets Syst. 69, 141–156 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  36. Frank, M.J.: On the simultaneous associativity of $F(x, y)$ and $x+y - F(x, y)$. Aequ. Math. 19, 194–226 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  38. Johnson, W.E.: Logic. Part I, p. 142. Cambridge University Press, Cambridge (1921). (chap. IX)

    Google Scholar 

  39. Keynes, J.N.: Studies and Exercises in Formal Logic, Part II, 3rd edn, p. 144. MacMillan, New York (1894). (chap. IV)

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Moretti, A.: The geometry of opposition. Ph.D. thesis. University of Neuchâtel (2009)

  43. Moretti, A.: Geometry for modalities? Yes: Through N-opposition theory? In: Béziau, J.-Y., Costa Leite, A., Facchini, A. (eds.) Aspects of Universal Logic, pp. 102–145. University of Neuchâtel, Neuchâtel (2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University, Stanford (2008)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  47. Pellissier, R.: 2-opposition and the topological hexagon. In: Béziau, J.-Y., Payette, G. (eds.) New Perspectives on the Square of Opposition. Peter Lang, Bern (2012)

    Google Scholar 

  48. Prade, H., Richard, G.: From analogical proportion to logical proportions. Log. Univers. 7, 441–505 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Reichenbach, H.: The syllogism revised. Philos. Sci. 19(1), 1–16 (1952)

    Article  Google Scholar 

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

  51. Ruspini, E.H.: A new approach to clustering. Inf. Control 15, 22–32 (1969)

    Article  MATH  Google Scholar 

  52. Sauriol, P.: Remarques sur le théorie de l’hexagone logique de Blanché. Can. Philos. Rev. 7(3), 374–390 (1968)

    Google Scholar 

  53. Sesmat, A.: Logique II: Les Raisonnements, la Logistique. Hermann, Paris (1951)

    Google Scholar 

  54. Van de Walle, B., De Baets, B., Kerre, E.E.: Characterizable fuzzy preference structures. Ann. Oper. Res. 80, 105–136 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  55. W. of Sherwood, William of Sherwood’s Introduction to Logic. University of Minnesota Press, Minneapolis, with translation, introduction and notes by N. Kretzmann (1966)

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author thanks Lorenz Demey (Center for Logic and Philosophy of Science, Leuven, Belgium) for interesting discussions on cubes of opposition in Kolymbari (Crete) in early November 2018 at the 6th world congress on the Square of Opposition. The authors are indebted towards an anonymous reviewer for pointing out references to Johnson–Keynes octagon as the proper ancestor of their cube.

Author information

Authors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse (IRIT), Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Didier Dubois & Henri Prade

  2. Equipe de Recherche en Ingénierie des Connaissances - ERIC, Université Claude Bernard Lyon 1, Lyon, France

    Agnès Rico

Authors
  1. Didier Dubois
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henri Prade
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Agnès Rico
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henri Prade.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors acknowledge a partial support of ANR-11-LABX-0040-CIMI (Centre International de Mathématiques et d’Informatique) within the Program ANR-11-IDEX-0002-02, Project ISIPA.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dubois, D., Prade, H. & Rico, A. Structures of Opposition and Comparisons: Boolean and Gradual Cases. Log. Univers. 14, 115–149 (2020). https://doi.org/10.1007/s11787-020-00241-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11787-020-00241-6

Keywords

Mathematics Subject Classification

Search

Navigation