Advertisement

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)

Abstract

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.

Keywords

Fuzzy Logic Rational Completeness Propositional Variable Sequent Calculus Substructural Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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, http://www.math.vanderbilt.edu/people/metcalfe/publications
  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