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From Semantic Games to Provability: The Case of Gödel Logic


We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al. (in: M.-J. Lesot, S. Vieira, M.Z. Reformat, J.P. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of \(\text{ SeqGZL } \) (A. Ciabattoni, and T. Vetterlein, Fuzzy Sets and Systems 161(14):1941–1958, 2010) and in a sequent-of-relations calculus (M. Baaz, and Ch.G. Fermüller, in: N.V. Murray, (ed.), Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively.


  1. Avron, A., Hypersequents, logical consequence and intermediate logics for concurrency, Annals of Mathematics and Artificial Intelligence 4:225–248, 1991.

    Article  Google Scholar 

  2. Baaz, M., Infinite-valued gödel logics with 0-1-projections and relativizations, in P. Hajek, (ed.), Gödel’96: Logical foundations of mathematics, computer science and physics, vol. 6 of Lecture Notes in Logic, Association for Symbolic Logic, 1996, pp. 23–33.

  3. Baaz, M., A. Ciabattoni, and Ch.G. Fermüller, Cut-elimination in a sequents-ofrelations calculus for gödel logic, in ISMVL ’01: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, IEEE Computer Society, 2001, pp. 181–186.

  4. Baaz, M., A. Ciabattoni, and Ch.G. Fermüller, Hypersequent calculi for gödel logics - a survey, Journal of Logic and Computation 13(6):835–861, 2003.

    Article  Google Scholar 

  5. Baaz, M., A. Ciabattoni, and Ch.G. Fermüller, Sequent of relations calculi: A framework for analytic deduction in many-valued logics, in M. Fitting, and E. Orłowska, (eds.), Beyond Two: Theory and Applications of Multiple-Valued Logic, vol. 114 of Studies in Fuzziness and Soft Computing, Physica-Verlag Heidelberg, 2003, pp. 157–180.

  6. Baaz, M., and Ch.G. Fermüller, Analytic calculi for projective logics, in N.V. Murray, (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Springer, Berlin, Heidelberg, 1999, pp. 36–51.

  7. Baaz, M., and H. Veith, Interpolation in fuzzy logic, Archive for Mathematical Logic 38(7):461–489, 1999.

  8. Baaz, M., and R. Zach, Hypersequents and the proof theory of intuitionistic fuzzy logic, in P.G. Clote, and H. Schwichtenberg, (eds.), International Workshop on Computer Science Logic, vol. 1862 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2000, pp. 187–201.

  9. Bhargava, A.K., Fuzzy set theory fuzzy logic and their applications, S. Chand Publishing, 2013.

  10. Ciabattoni, A., and T. Vetterlein, On the (fuzzy) logical content of cadiag-2, Fuzzy Sets and Systems 161(14):1941–1958, 2010.

  11. Dummett, M., A propositional calculus with denumerable matrix, The Journal of Symbolic Logic 24(2):97–106, 1959.

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

  13. Fermüller Ch.G., T. Lang, and A. Pavlova, From truth degree comparison games to sequents-of-relations calculi for gödel logic, in M.-J. Lesot, S. Vieira, M.Z. Reformat, J.P. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems, Springer International Publishing, Cham, 2020, pp. 257–270.

  14. Fermüller, Ch.G., and G. Metcalfe, Giles’s game and the proof theory of Łukasiewicz logic, Studia Logica 92(1):27–61, 2009.

  15. Fermüller, Ch.G., and N. Preining, A dialogue game for intuitionistic fuzzy logic based on comparisons of degrees of truth, in Proceedings of InTech’03, 2003, pp. 142–151.

  16. Gödel, K., Zum intuitionistischen aussagenkalkül (1932) (reprint), Journal of Symbolic Logic 55(1):344–344, 1990.

  17. Jeavons, P., D. Cohen, and M. Gyssens, Closure properties of constraints, Journal of ACM 44(4):527–548, 1997.

  18. Mann, A.L., G. Sandu, and M. Sevenster, Independence-friendly logic: A game-theoretic approach, Cambridge University Press, 2011.

  19. Sola, H., P. Burillo, and F. Soria, Automorphisms, negations and implication operators, Fuzzy Sets and Systems 134:209–229, 2003.

  20. Takeuti, G., and S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, The Journal of Symbolic Logic 49(3):851–866, 1984

  21. Trillas, E., Sobre funciones de negación la teoria de conjuntos difusos, Stochastica 3(1):47–60, 1979 (english translation in: S. Barro, A. Bugarin, and A. Sobrino, (eds.), Advances in Fuzzy Logic, Public University of Santiago de Compostela, Spain, 1998, pp. 31–45).

  22. van Benthem, J., Logic in Games, MIT Press, 2014.

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A. Pavlova were supported by FWF project 793 W1255-N23. R. Freiman and T. Lang were supported by FWF project P 794 32684. The authors are grateful to the anonymous referees for comments.


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Correspondence to Alexandra Pavlova.

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Pavlova, A., Freiman, R. & Lang, T. From Semantic Games to Provability: The Case of Gödel Logic. Stud Logica 110, 429–456 (2022).

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  • Gödel logic
  • Fuzzy logic
  • Semantic games
  • Provability game
  • Analytic calculus