Skip to main content
SpringerLink
Log in

Giles’s Game and the Proof Theory of Łukasiewicz Logic

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

In the 1970s, Robin Giles introduced a game combining Lorenzen-style dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in Łukasiewicz logic. In this paper, it is shown that ‘disjunctive strategies’ for Giles’s game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In particular, such strategies mirror derivations in a hypersequent calculus developed in recent work on the proof theory of Łukasiewicz logic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamson A., Giles R.: ‘A game-based formal system for Ł∞’. Studia Logica 1(38), 49–73 (1979)

    Article  Google Scholar 

  2. Aguzzoli S., Ciabattoni A.: ‘Finiteness in infinite-valued logic’. Journal of Logic, Language and Information 9(1), 5–29 (2000)

    Article  Google Scholar 

  3. Avron A.: ‘A constructive analysis of RM’. Journal of Symbolic Logic 52(4), 939–951 (1987)

    Article  Google Scholar 

  4. Baaz, M., and G. Metcalfe, ‘Herbrand’s theorem, skolemization, and proof systems for first-order Łukasiewicz logic’. To appear in Journal of Logic and Computation.

  5. Chang C.C.: ‘Algebraic analysis of many-valued logics’. Transactions of the American Mathematical Society 88, 467–490 (1958)

    Article  Google Scholar 

  6. Ciabattoni, A., C. G. Fermüller, and G. Metcalfe, ‘Uniform Rules and Dialogue Games for Fuzzy Logics’, in Proceedings of LPAR 2004, volume 3452 of LNAI, Springer, 2005, pp. 496–510.

  7. Ciabattoni, A., and G. Metcalfe, ‘Bounded Łukasiewicz logics’, in M. Cialdea Mayer and F. Pirri (eds.), Proceedings of TABLEAUX 2003, volume 2796 of LNCS, Springer, 2003, pp. 32–48.

  8. Cignoli, R., I.M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, volume 7 of Trends in Logic, Kluwer, Dordrecht, 1999.

  9. Cintula, P., and O. Majer, ‘Evaluation games for fuzzy logics’, in O. Majer, A.V. Pietarinen, and T. Tulenheimo (eds.), Games: Unifying Logic, Language, and Philosophy, Springer, 2009, pp. 117–138.

  10. Dershowitz N., Manna Z.: ‘Proving termination with multiset orderings’. Communications of the Association for Computing Machinery 22, 465–476 (1979)

    Google Scholar 

  11. Esteva F., Godo L., Hájek P., Montagna F.: ‘Hoops and fuzzy logic’. Journal of Logic and Computation 13(4), 532–555 (2003)

    Article  Google Scholar 

  12. Fermüller, C. G., ‘Parallel dialogue games and hypersequents for intermediate logics’, in M. Cialdea Mayer and F. Pirri (eds.), Proceedings of TABLEAUX 2003, volume 2796 of LNCS, Springer, 2003, pp. 48–64.

  13. Fermüller, C. G., ‘Revisiting Giles - connecting bets, dialogue games, and fuzzy logics’, in O. Majer, A. V. Pietarinen, and T. Tulenheimo (eds.), Games: Unifying Logic, Language, and Philosophy, Springer, 2009, pp. 209–227.

  14. Fermüller, C. G., andA. Ciabattoni, ‘From intuitionistic logic to Gödel-Dummett logic via parallel dialogue games’, in Proceedings of the 33rd IEEE International Symposium on Multiple-Valued Logic, Tokyo, May 2003.

  15. Fermüller, C. G., and R. Kosik, ‘Combining supervaluation and degree based reasoning under vagueness’, in Proceedings of LPAR 2006, volume 4246 of LNAI, Springer, 2006, pp. 212–226.

  16. Giles R.: ‘A non-classical logic for physics’. Studia Logica 4(33), 399–417 (1974)

    Google Scholar 

  17. Giles R.: ‘Łukasiewicz logic and fuzzy set theory’. International Journal of Man-Machine Studies 8(3), 313–327 (1976)

    Article  Google Scholar 

  18. Giles, R., ‘A non-classical logic for physics’, in R.Wojcicki and G. Malinowski (eds.), Selected Papers on Łukasiewicz Sentential Calculi, Polish Academy of Sciences, 1977, pp. 13–51.

  19. Giles R.: ‘Semantics for fuzzy reasoning’. International Journal of Man-Machine Studies 17, 401–415 (1982)

    Article  Google Scholar 

  20. Hähnle, R., Automated Deduction in Multiple-Valued Logics, Oxford University Press, 1993.

  21. Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)

    Google Scholar 

  22. Lorenzen, P., ‘Logik und Agon’, in Atti Congr. Internaz. di Filosofia, Sansoni, 1960, pp. 187–194.

  23. Łukasiewicz, J., and A. Tarski, ‘Untersuchungen über den Aussagenkalkül’, Comptes Rendus des Séances de la Societé des Sciences et des Lettres de Varsovie, Classe III 23, 1930.

  24. Metcalfe G., Olivetti N., Gabbay D.: ‘Sequent and hypersequent calculi for abelian and Łukasiewicz logics’. ACM Transactions on Computational Logic 6(3), 578–613 (2005)

    Article  Google Scholar 

  25. Metcalfe, G., N. Olivetti, and D. Gabbay, Proof Theory for Fuzzy Logics, volume 36 of Applied Logic, Springer, 2008.

  26. Mundici, D., ‘The logic of Ulam’s game with lies’, in C. Bicchieri and M.L. Dalla Chiara (eds.), Knowledge, belief and strategic interaction, Cambridge University Press, 1992, pp. 275–284.

  27. Mundici D.: ‘Ulam’s game, Łukasiewicz logic and C*-algebras’. Fundamenta Informaticae 18, 151–161 (1993)

    Google Scholar 

  28. Mundici D., Olivetti N.: ‘Resolution and model building in the infinite-valued calculus of Łukasiewicz’. Theoretical Computer Science 200(1–2), 335–366 (1998)

    Article  Google Scholar 

  29. Olivetti N.: ‘Tableaux for Łukasiewicz infinite-valued logic’. Studia Logica 73(1), 81–111 (2003)

    Article  Google Scholar 

  30. Prijatelj A.: ‘Bounded contraction and Gentzen-style formulation of Łukasiewicz logics’. Studia Logica 57(2–3), 437–456 (1996)

    Article  Google Scholar 

  31. Ragaz, M. E., Arithmetische Klassifikation von Formelmengen der unendlichwertigen Logik, PhD thesis, ETH Zürich, 1981.

  32. Rose A., Rosser J.B.: ‘Fragments of many-valued statement calculi’. Transactions of the American Mathematical Society 87, 1–53 (1958)

    Article  Google Scholar 

  33. Scarpellini B.: ‘Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz’. Journal of Symbolic Logic 27(2), 159–170 (1962)

    Article  Google Scholar 

  34. Wagner, H., ‘A new resolution calculus for the infinite-valued propositional logic of Łukasiewicz’, in R. Caferra and G. Salzert (eds.), FTP 98, Technical Report E1852-GS-981, TU Wien, Austria, 1998, pp. 234–243.

Download references

Author information

Authors and Affiliations

  1. Institut für Computersprachen, Technische Universität Wien, Favoritenstraße 7, Vienna, Austria

    Christian G. Fermüller

  2. Mathematics Department, Vanderbilt University, 1326 Stevenson Center, Nashville, USA

    George Metcalfe

Authors
  1. Christian G. Fermüller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George Metcalfe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian G. Fermüller.

Additional information

Presented by Daniele Mundici

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fermüller, C.G., Metcalfe, G. Giles’s Game and the Proof Theory of Łukasiewicz Logic. Stud Logica 92, 27–61 (2009). https://doi.org/10.1007/s11225-009-9185-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-009-9185-2

Keywords

Search

Navigation