Fuzzy Syllogisms, Numerical Square, Triangle of Contraries, Inter-bivalence

Part of the Studies in Universal Logic book series (SUL)


A new Predicate Calculus, called “Distinctive” D, is presented in the form of the Hexagon of Opposition, in which the Particular Yba (= only some b are a) is the contradictory of the Universal Uba (= all or no b are a). Y is preferred, as a primitive, to I or O, because it is more “natural” than the others. Typical inferences of the systems are the obversions: Yba=Yba′ (= only some b are not a), Uba=Uba′ (= all or no b are not a).

D-Systems include traditional Syllogisms. Polygonal and Numerical developments are being developed, including intermediate quantifiers (“the majority of”, …). Polygons are finally absorbed into the Numerical D-Square NDS. Isomorphisms are discovered between bivalent D-systems and some Non-Standard Logics by depriving the subject-class of the quantifier and transferring its (pre)numerical attribute to the truth value of the judgement. These ‘(poly-)inter-bivalent’ logics admit intermediate values between true and false. In that way we get Fuzzy Logic, and present the Fuzzy Square.


Predicate logic Quantifiers Logical diagram Square of opposition Hexagon of opposition Numerical syllogisms Non-standard logic Fuzzy syllogisms Synonyms 

Mathematics Subject Classification

03A05 03B20 03B52 03B65 03B60 03B53 05C99 


  1. 1.
    Béziau, J.-Y.: New light on the square of oppositions and its nameless corner. Log. Investig. 10, 218–232 (2003) Google Scholar
  2. 2.
    Bird, O.: Syllogistic and Its Extensions. Prentice-Hall, Englewood Cliffs (1964) Google Scholar
  3. 3.
    Blanché, R.: Structures Intellectuelles: Essai sur l’Organisation Systematique des Concepts. Vrin, Paris (1966) Google Scholar
  4. 4.
    Carnes, R.D., Peterson, P.L.: Intermediate quantifiers versus percentages. Notre Dame J. Form. Log. 32(2) (1991) Google Scholar
  5. 5.
    Cavaliere, F.: Motori di Ricerca Semantici con Antinomie Inedite e Sillogismi Sfumati. http://www.arrigoamadori.com/lezioni/AngoloDelFilosofo/AngoloDelFilosofo.htm
  6. 6.
    Dubois, D., Prade, H.: On fuzzy syllogism. Comput. Intell. 4, 171–179 (1988) CrossRefGoogle Scholar
  7. 7.
    Englebretsen, G.: The New Syllogistic. Peter Lang, New York (1987) Google Scholar
  8. 8.
    Hacker, E.A., Parry, W.T.: Pure numerical boolean syllogisms. Notre Dame J. Form. Log. 8(4), (1967) Google Scholar
  9. 9.
    Kosko, B.: Fuzzy Thinking: The New Science of Fuzzy Logic. Hyperion, New York (1993) Google Scholar
  10. 10.
    Kumova, B.I., Çakir, H.: The fuzzy syllogistic system. In: Mexican International Conference on Artificial Intelligence (MICAI’10), Pachuca. LNAI. Springer, Berlin (2010) Google Scholar
  11. 11.
    Lambert, J.H.: Neues Organon [Liepzig, 1764]. Olms, Hildesheim (1965) Google Scholar
  12. 12.
    Lindell, S.: A term logic for physically realizable models of informations. In: The Old New Logic: Essays on the Philosophy of Fred Sommers, Chap. 8. MIT Press, Cambridge (2005) Google Scholar
  13. 13.
    Moretti, A.: The geometry of logical opposition. PhD Thesis, Université de Neuchâtel, Switzerland (2009) Google Scholar
  14. 14.
    Murphree, W.A.: The numerical syllogism and existential presupposition. Notre Dame J. Form. Log. 38(1) (1997) Google Scholar
  15. 15.
    Murphree, W.A.: Numerical term logic. Notre Dame J. Form. Log. 39(3) (1998) Google Scholar
  16. 16.
    Naddeo, A., Cappetti, N., Pappalardo, M., Donnarumma, A.: Fuzzy logic application in the structural optimisation of a support plate for electrical accumulator in a motor vehicle. J. Mater. Process. Technol. 120(1–3), 303–309 (2002) CrossRefGoogle Scholar
  17. 17.
    Pfeifer, N.: Contemporary syllogistics: comparative and quantitative syllogisms. In: Kreuzbauer, G., Dorn, G. (eds.) Argumentation in Theorie und Praxis: Philosophie und Didaktik des Argumentierens, pp. 57–71. Lit, Vienna (2006) Google Scholar
  18. 18.
    Pratt-Hartmann, I.: On the complexity of the numerically definite syllogistic and related fragments. Bull. Symb. Log. 14(1), 1–28 (2008) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Sesmat, A.: Logique. Hermann, Paris (1951) Google Scholar
  20. 20.
    Seuren, P.A.M.: The natural logic of language and cognition. Pragmatics 16(1), 103–138 (2006) Google Scholar
  21. 21.
    Seuren, P.A.M.: The Logic of Language. Language from Within, vol. 2. Oxford University Press, Oxford (2010) Google Scholar
  22. 22.
    Sommers, F.: The Logic of Natural Language. Oxford University Press, Oxford (1982) Google Scholar
  23. 23.
    Sommers, F.: Predication in the logic of terms. Notre Dame J. Form. Log. 31(1) (1990) Google Scholar
  24. 24.
    Suchon, W.: Vasil’iev: what did he exactly do? Logic Logic. Philos. 7, 131–141 (1999) MathSciNetMATHGoogle Scholar
  25. 25.
    Vasiliev, N.A.: Imaginary (non Aristotelian) logic. In: Atti del V Congresso Internazionale di Filosofia, Napoli (1925) Google Scholar
  26. 26.
    Zadeh, Lofti A.: A computational approach to fuzzy quantifiers in natural languages. Comput. Math. Appl. 9(1), 149–184 (1983) MathSciNetMATHCrossRefGoogle Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.CesenaticoItaly

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