Uniform Rules and Dialogue Games for Fuzzy Logics

  • Agata Ciabattoni
  • Christian G. Fermüller
  • George Metcalfe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3452)


We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental t-norm based fuzzy logics i.e., Łukasiewicz logic, Gödel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequents-of-relations. Such a framework can be interpreted via a particular class of dialogue games combined with bets, where the rules reflect possible moves in the game. The problem of determining the validity of atomic relational hypersequents is shown to be polynomial for each logic, allowing us to develop Co-NP calculi. We also present calculi with very simple initial relational hypersequents that vary only in the structural rules for the logics.


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  1. 1.
    Aguzzoli, S.: Uniform description of calculi for all t-norm logics. In: Henkin, L., et al. (eds.) Proceedings of 34th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2004), pp. 38–43 (2004)Google Scholar
  2. 2.
    Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence 4(3–4), 225–248 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Avron, A., Konikowska, B.: Decomposition Proof Systems for Gödel-Dummett Logics. Studia Logica 69(2), 197–219 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baaz, M., Ciabattoni, A., Fermüller, C.: Cut-elimination in a sequents-of-relations calculus for Gödel logic. In: International Symposium on Multiple Valued Logic (ISMVL 2001), pp. 181–186. IEEE, Los Alamitos (2001)Google Scholar
  5. 5.
    Baaz, M., Fermüller, C.: Analytic calculi for projective logics. In: Murray, N.V. (ed.) TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 36–50. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Fermüller, C., Preining, N.: A dialogue game for intuitionistic fuzzy logic based on comparison of degrees of truth. In: Proceedings of In Tech 2003 (2003)Google Scholar
  7. 7.
    Giles, R.: A non-classical logic for physics. Studia Logica 4(33), 399–417 (1974)MathSciNetGoogle Scholar
  8. 8.
    Giles, R.: A non-classical logic for physics. In: Wojcicki, R., Malinkowski, G. (eds.) Selected Papers on Łukasiewicz Sentential Calculi, pp. 13–51. Polish Academy of Sciences (1977)Google Scholar
  9. 9.
    Hähnle, R.: Automated Deduction in Multiple-Valued Logics. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
  10. 10.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)zbMATHCrossRefGoogle Scholar
  11. 11.
    Jeavons, P.G., Cohen, D.A., Gyssens, M.: Closure properties of constraints. The Journal of the ACM 44, 527–548 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lorenzen, P.: Logik und Agon. In: tti Congr. Internaz. di Filosofia, pp. 187–194. Sansoni (1960)Google Scholar
  13. 13.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Goal-directed calculi for Gödel-Dummett logics. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 413–426. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Analytic proof calculi for product logics. Archive for Mathematical Logic 43(7), 859–889 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Goal-directed methods for Łukasiewicz logics. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 85–99. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Sequent and hypersequent calculi for abelian and Łukasiewicz logics. To appear in ACM TOCL (2005)Google Scholar
  17. 17.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, Chichester (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Agata Ciabattoni
    • 1
  • Christian G. Fermüller
    • 1
  • George Metcalfe
    • 1
  1. 1.Technische Universität WienViennaAustria

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