Revisiting Giles's Game

Reconciling Fuzzy Logic and Supervaluation
  • Christian G. Fermüller
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 15)


We explain Giles's characterization of Lukasiewicz logic via a dialogue game combined with bets on results of experiments that may show dispersion. The game is generalized to other fuzzy logics and linked to recent results in proof theory. We argue that these results allow one to place t-norm based fuzzy logics in a common framework with supervaluation as a theory of vagueness.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Avron, A. (1991). Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and AI, 4(3–4):225–248.Google Scholar
  2. Burns, L. C. (1991). Vagueness: An Investigation Into Natural Language and the Sorites Paradox. Kluwer, Dordrecht.Google Scholar
  3. Ciabattoni, A. and Fermüller, C. G. (2003). From intuitionistic logic to gödel-dummett logic via parallel dialogue games. In 33rd Intl. Symp. on Multiple-Valued Logic, pages 188–195. IEEE Computer Society Press, Tokyo.Google Scholar
  4. Ciabattoni, A., Fermüller, C. G., and Metcalfe, G. (2005). Uniform rules and dialogue games for fuzzy logics. In Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2004, Springer LNAI 3452, 496–510, Dordrecht.Google Scholar
  5. Cignoli, R., D'Ottaviano, I. M. L., and Mundici, D. (1999). Algebraic Foundations of Many-Valued Reasoning, volume 7 of Trends in Logic. Kluwer, Dordrecht.Google Scholar
  6. Dubois, D. and Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applications. Academic, New York.CrossRefGoogle Scholar
  7. Esteva, F., Godo, L., Hájek, P., and Montagna, F. (2003). Hoops and fuzzy logic. Journal of Logic and Computation, 13(4):532–555.CrossRefGoogle Scholar
  8. Felscher, W. (1985). Dialogues, strategies, and intuitionistic provability. Annals of Pure and Applied Logic, 28:217–254.CrossRefGoogle Scholar
  9. Fermüller, C. G. (2003a). Parallel dialogue games and hypersequents for intermediate logics. In TABLEAUX 2003, volume 2796 of LNAI, pages 48–64. Springer, Dordrecht.Google Scholar
  10. Fermüller, C. G. (2003b). Theories of vagueness versus fuzzy logic: can logicians learn from philosophers? Neural Network World Journal, 13(5):455–466.Google Scholar
  11. Fine, K. (1975). Vagueness, truth and logic. Synthèse, 30:265–300.CrossRefGoogle Scholar
  12. Giles, R. (1974). A non-classical logic for physics. Studia Logica, 4(33):399–417.Google Scholar
  13. Giles, R. (1977). A non-classical logic for physics. In Wojcicki, R. and Malinkowski, G., editors, Selected Papers on Łukasiewicz Sentential Calculi, pages 13–51. Polish Academy of Sciences, Wroclaw — Warszawa — Kraków — Gdańsk.Google Scholar
  14. Hájek, P. (1998). Metamathematics of Fuzzy Logic. Kluwer, Dordrecht.Google Scholar
  15. Hájek, P. (2002). Why fuzzy logic? In Jackquette, D., editor, A Companion to Philosophical Logic, pages 595–606. Blackwell, Oxford.Google Scholar
  16. Keefe, R. (2000). Theories of Vagueness. Cambridge University Press, Cambridge.Google Scholar
  17. Keefe, R. and Smith, P., editors (1987). Vagueness: A Reader. MIT Press, Cambridge, MA.Google Scholar
  18. Krabbe, E. C. W. (1988). Dialogue sequents and quick proofs of completeness. In Hoepelman, J. P., editor, Representation and Reasoning, pages 135–140. Max Niemeyer Verlag, Tübingen.Google Scholar
  19. Kremer, P. and Kremer, M. (2003). Some supervaluation-based consequence relations. Journal of Philosophical Logic, 32(3):225–244.CrossRefGoogle Scholar
  20. Lorenzen, P. (1960). Logik und agon. In Atti Congr. Internat. di Filosofia, volume 4, pages 187–194, Sansoni, Firenze.Google Scholar
  21. Łukasiewicz, J. (1920). Ologicetròjwartościowej. Ruch Filozoficzny, 5:169–171.Google Scholar
  22. Metcalfe, G., Olivetti, N., and Gabbay, D. M. (2004). Analytic calculi for product logics. Archive for Mathematical Logic, 43(7):859–889.CrossRefGoogle Scholar
  23. Metcalfe, G., Olivetti, N., and Gabbay, D. M. (2005). Sequent and hypersequent calculi for Abelian and Łukasiewicz logics. To appear in ACM TOCL. Available at
  24. Paris, J. (1997). A semantics for fuzzy logic. Soft Computing, 1:143–147.Google Scholar
  25. Ruspini, E. H. (1991). On the semantics of fuzzy logic. International Journal of Approximate Reasoning, 5:45–88.CrossRefGoogle Scholar
  26. Varzi, A. (2001). Vagueness, logic, and ontology. The Dialogue, 1:135–154.Google Scholar
  27. Weatherson, B. (2003). Many many problems. Philosophical Quarterly, 53:481–501.CrossRefGoogle Scholar
  28. Williamson, T. (1994). Vagueness. Routledge, London.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  • Christian G. Fermüller
    • 1
  1. 1.Technische Universität Wien FavoritenstrWienAustria

Personalised recommendations