Logica Universalis

, Volume 10, Issue 2–3, pp 339–357 | Cite as

Syllogisms and 5-Square of Opposition with Intermediate Quantifiers in Fuzzy Natural Logic



In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic (FNL). We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.


Aristotle square of opposition fuzzy natural logic intermediate generalized quantifiers generalized Peterson’s square of opposition 

Mathematics Subject Classification

Primary 03-02 Secondary 03B15 03B50 03B52 


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  1. 1.
    Afshar M., Dartnell C., Luzeaux D., Sallantin J., Tognetti Y.: Aristotle’s square revisited to frame discovery science. J. Comput. 2, 54–66 (2007)CrossRefGoogle Scholar
  2. 2.
    Andrews P.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Kluwer, Dordrecht (2002)CrossRefMATHGoogle Scholar
  3. 3.
    Béziau J.Y.: New light on the square of oppositions and its nameless corner. Logical Invest. 10, 218–233 (2003)MathSciNetMATHGoogle Scholar
  4. 4.
    Béziau J.Y.: The power of the hexagon. Logica Universalis 6, 1–43 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Béziau, J.Y., Gan-Krzywoszyńska, K.: Handbook of abstracts of the 2nd World Congress on the Square of Opposition. Corte, Corsica, June 17–20Google Scholar
  6. 6.
    Béziau, J.Y., Gan-Krzywoszyńska, K.: Handbook of abstracts of the 3nd World Congress on the Square of Opposition. Beirut, Lebanon, June 26–30Google Scholar
  7. 7.
    Béziau, J.Y., Gan-Krzywoszyńska, K.: Handbook of abstracts of the 4nd World Congress on the Square of Opposition. Roma, Vatican, May 5–9Google Scholar
  8. 8.
    Blanché R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953)CrossRefGoogle Scholar
  9. 9.
    Brown M.: Generalized quantifiers and the square of opposition. Notre Dame J. Formal Logic 25, 303–322 (1984)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cignoli R.L.O., D’Ottaviano I.M.L., Mundici D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)CrossRefMATHGoogle Scholar
  11. 11.
    Ciucci, D., Dubois, D., Prade, H.: Oppositions in rough set theory, In: Li, T., Nguyen, H.S., Wang, G., Grzymala-Busse, J.W., Janicki, R., Hassanien, A.E., Yu H. (eds.) Proc. 7th Int. Conf. on Rough Sets and Knowledge Technology (RSKT’12), Chengdu, Aug. 17–20, LNCS, vol. 7414, pp. 504–513 (2012)Google Scholar
  12. 12.
    Ciucci, D., Dubois, D., Prade, H., The structure of oppositions in rough set theory and formal concept analysis—Toward a new bridge between the two settings. In: Beierle, C., Meghini, C. (eds.) Proc. 8th Int. Symp. on Foundations of Information and Knowledge Systems (FoIKS’14), Bordeaux, March 3–7, LNCS, vol. 8367, pp. 154–173. Springer, New York (2014)Google Scholar
  13. 13.
    Dubois D., Prade H.: From Blanch’e’s hexagonal organization of concepts to formal concept analysis and possibility theory. Logica Universalis 6, 149–169 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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, New York (2015)Google Scholar
  15. 15.
    Dvořák A., Holčapek M.: L-fuzzy quantifiers of the type \({\langle 1 \rangle}\) determined by measures. Fuzzy Sets Syst. 160, 3425–3452 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Glöckner I.: Fuzzy quantifiers: A Computational Theory. Springer, Berlin (2006)MATHGoogle Scholar
  17. 17.
    Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)CrossRefMATHGoogle Scholar
  18. 18.
    Lakoff G.: Linguistics and natural logic. Synthese 22, 151–271 (1970)CrossRefMATHGoogle Scholar
  19. 19.
    Miclet, L., Prade, H.: Analogical proportions and square of oppositions. In: Laurent, A. et al. (ed.) Proc. 15th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, July 15–19, Montpellier, CCIS, vol. 443, pp. 324–334. Springer, New York (2014)Google Scholar
  20. 20.
    Murinová P., Novák V.: A formal theory of generalized intermediate syllogisms. Fuzzy Sets Syst. 186, 47–80 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Murinová P., Novák V.: Analysis of generalized square of opposition with intermediate quantifiers. Fuzzy Sets Syst. 242, 89–113 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Murinová P., Novák V.: The structure of generalized intermediate syllogisms. Fuzzy Sets Syst. 247, 18–37 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Murinová, P.,Novák, V.: On properties of the inermediate quantifier “Many”. Fuzzy Sets Syst. submitted.Google Scholar
  24. 24.
    Novák V.: On fuzzy type theory. Fuzzy Sets Syst. 149, 235–273 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Novák, V.: Perception-Based Logical Deduction, Computational Intelligence, Theory and Applications. Springer, Berlin, pp. 237–250 (2005)Google Scholar
  26. 26.
    Novák V.: A Formal Theory of Intermediate Quantifiers. Fuzzy Sets Syst. 159, 1229–1246 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Novák V.: A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets Syst. 159(22), 2939–2969 (2008)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Novák V., Lehmke S.: Logical Structure of Fuzzy IF-THEN rules. Fuzzy Sets Syst. 157, 2003–2029 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Novák V., Perfilieva I., Močkoř J.: athematical Principles of Fuzzy Logic. Kluwer, Boston (1999)CrossRefMATHGoogle Scholar
  30. 30.
    Parsons T.: Things That are Right with the Traditional Square of Opposition. Logica Universalis 2, 3–11 (2012)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Pellissier R.: “Setting” n-opposition. Logical Universalis 2, 235–263 (2008)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Pereira-Fariña M., Díaz-Hermida F., Bugarín A.: On the analysis of set-based fuzzy quantified reasoning using classical syllogistics. Fuzzy Sets Syst. 214, 83–94 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pereira-Fariña, M., Juan, C.: Vidal and Díaz-Hermida, F. Bugarín, A.: A fuzzy syllogistic reasoning schema for generalized quantifiers. Fuzzy Sets Syst. 234, 79–96 (2014)Google Scholar
  34. 34.
    Peters, S., Westerståhl, D.: Quantifiers in Language and Logic. Claredon Press, Oxford (2006)Google Scholar
  35. 35.
    Peterson P.L.: On the logic of “Few”,“Many” and “Most”. Notre Dame J. Formal Logic 20, 155–179 (1979)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Peterson, P.P.: Intermediate Quantifiers. Logic, Linguistics, Aristotelian Semantics. Ahgate, Aldershot (2000)Google Scholar
  37. 37.
    Ross W.D.: Aristotle’s Prior and Posterior Analytics. Clarendom Press, Oxford (1949)Google Scholar
  38. 38.
    Schwartz D.G.: Dynamic reasoning with qualified syllogisms. Artif. Intell. 93, 103–167 (1997)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Thompson B.E.: Syllogisms using “few”, “many” and “most”. Notre Dame J. Formal Logic 23, 75–84 (1982)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Westerståhl D.: The traditional square of opposition and generalized quantifiers. Stud. Logic 2, 1–18 (2008)Google Scholar
  41. 41.
    Zadeh L.A.: A computational approach to fuzzy quantifiers in natural languages. Comput. Math. 9, 149–184 (1983)MathSciNetMATHGoogle Scholar
  42. 42.
    Zadeh L.A.: Syllogistic reasoning in fuzzy logic and its applications to usuality and reasoning with dispositions. IEEE Trans. Syst. Man Cybern. 5, 754–765 (1985)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institute for Research and Applications of Fuzzy Modeling, NSC IT4InnovationsUniversity of OstravaOstrava 1Czech Republic

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