Archive for Mathematical Logic

, Volume 46, Issue 5–6, pp 425–449 | Cite as

Fuzzy logics based on [0,1)-continuous uninorms

  • Dov Gabbay
  • George MetcalfeEmail author


Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hájek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special uninorm. A Gentzen-style hypersequent calculus is provided for CRL and used to establish co-NP completeness results for these logics.


Uninorm t-Norm Fuzzy logic Cross ratio 

Mathematics Subject Classification (2000)

03B50 03B47 03B52 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avron A. (1987). A constructive analysis of RM. J. Symb. Log. 52(4): 939–951 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Avron A. (1991). Hypersequents, logical consequence and intermediate logics for concurrency. Ann. Math. Artif. Intell. 4(3–4): 225–248 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baaz M., Hájek P., Montagna F. and Veith H. (2001). Complexity of t-tautologies. Ann. Pure Appl. Log. 113(1): 3–11 CrossRefGoogle Scholar
  4. 4.
    Cignoli R. and Torrens A. (2005). Standard completeness of Hájek basic logic and decompositions of BL-chains. Soft Comput. 9(12): 862–868 zbMATHCrossRefGoogle Scholar
  5. 5.
    Cintula, P.: Weakly implicative (fuzzy) logics I: basic properties. Arch. Math. Log. 45(6) (2007)Google Scholar
  6. 6.
    De Baets B. and Fodor J. (1999). Residual operators of uninorms. Soft Comput. 3: 89–100 Google Scholar
  7. 7.
    De Baets B. and Fodor J. (1999). Van Melle’s combining function in MYCIN is a representable uninorm: an alternative proof. Fuzzy Sets Syst. 104: 133–136 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fodor J., Yager R.R. and Rybalov A. (1997). Structure of uni-norms. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 5: 411–427 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gabbay D., Metcalfe G. and Olivetti N. (2004). Hypersequents and fuzzy logic. Revista de la Real Academia de Ciencias (RACSAM) 98(1): 113–126 zbMATHMathSciNetGoogle Scholar
  10. 10.
    Girard J. (1987). Linear logic. Theoret. Comput. Sci. 50: 1–102 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gottwald, S.: A treatise on many-valued logics. In: Studies in Logic and Computation, vol. 9. Research Studies Press, Baldock (2000)Google Scholar
  12. 12.
    Gurevich Y.S. and Kokorin A.I. (1963). Universal equivalence of ordered abelian groups (in Russian). Algebra i logika 2: 37–39 zbMATHGoogle Scholar
  13. 13.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)Google Scholar
  14. 14.
    Hájek P. and Valdés J. (1994). An analysis of MYCIN-like expert systems. Mathw. Soft Comput. 1: 45–68 Google Scholar
  15. 15.
    Hart J., Rafter L. and Tsinakis C. (2002). The structure of commutative residuated lattices. Int. J. Algebra Comput. 12(4): 509–524 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hu S. and Li Z. (2001). The structure of continuous uni-norms. Fuzzy Sets Syst. 124: 43–52 zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symb. Log. (to appear)(2007)Google Scholar
  18. 18.
    Metcalfe G., Olivetti N. and Gabbay D. (2004). Analytic proof calculi for product logics. Arch. Math. Log. 43(7): 859–889 zbMATHMathSciNetGoogle Scholar
  19. 19.
    Metcalfe G., Olivetti N. and Gabbay D. (2005). Sequent and hypersequent calculi for abelian and Łukasiewicz logics. ACM Trans. Comput. Log. 6(3): 578–613 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Silvert W. (1979). Symmetric summation: a class of operations on fuzzy sets. IEEE Trans. Man. Cybern. 9: 657–659 zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Yager R.R. (2001). Uninorms in fuzzy systems modelling. Fuzzy Sets Syst. 122: 167–175 zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Yager R.R. (2002). Defending against strategic manipulation in uninorm-based multi-agent decision making. Eur. J. Oper. Res. 141(1): 217–232 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Yager R.R. and Rybalov A. (1996). Uninorm aggregation operators. Fuzzy Sets Syst. 80: 111–120zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceKing’s College LondonLondonUK
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations