Density Elimination and Rational Completeness for First-Order Logics

  • Agata Ciabattoni
  • George Metcalfe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4514)


Density elimination by substitutions is introduced as a uniform method for removing applications of the Takeuti-Titani density rule from proofs in first-order hypersequent calculi. For a large class of calculi, density elimination by this method is guaranteed by known sufficient conditions for cut-elimination. Moreover, adding the density rule to any axiomatic extension of a simple first-order logic gives a logic that is rational complete; i.e., complete with respect to linearly and densely ordered algebras: a precursor to showing that it is a fuzzy logic (complete for algebras with a real unit interval lattice reduct). Hence the sufficient conditions for cut-elimination guarantee rational completeness for a large class of first-order substructural logics.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avron, A.: A constructive analysis of RM. Journal of Symbolic Logic 52(4), 939–951 (1987)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Baaz, M., Zach, R.: Hypersequents and the proof theory of intuitionistic fuzzy logic. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 187–201. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Ciabattoni, A.: Automated generation of analytic calculi for logics with linearity. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 503–517. Springer, Heidelberg (2004)Google Scholar
  4. 4.
    Ciabattoni, A., Esteva, F., Godo, L.: T-norm based logics with n-contraction. Neural Network World 12(5), 441–453 (2002)Google Scholar
  5. 5.
    Cintula, P., Hájek, P.: On theories and models in fuzzy predicate logics. Journal of Symbolic Logic 71(3), 832–863 (2006)Google Scholar
  6. 6.
    Esteva, F., Gispert, J., Godo, L., Montagna, F.: On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. Studia Logica 71(2), 199–226 (2002)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124, 271–288 (2001)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  9. 9.
    Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s MTL logic. Studia Logica 70(2), 183–192 (2002)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Metcalfe, G., Montagna, F.: Substructural fuzzy logics. To appear in Journal of Symbolic Logic,
  11. 11.
    Montagna, F., Ono, H.: Kripke semantics, undecidability and standard completeness for Esteva and Godo’s logic MTL∀. Studia Logica 71(2), 227–245 (2002)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Ono, H., Komori, Y.: Logic without the contraction rule. Journal of Symbolic Logic 50, 169–201 (1985)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Takano, M.: Another proof of the strong completeness of the intuitionistic fuzzy logic. Tsukuba J. Math. 11, 851–866 (1984)MathSciNetGoogle Scholar
  14. 14.
    Takeuti, G., Titani, T.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. Journal of Symbolic Logic 49(3), 851–866 (1984)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Agata Ciabattoni
    • 1
  • George Metcalfe
    • 2
  1. 1.Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040 WienAustria
  2. 2.Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240USA

Personalised recommendations