Abstract
Various types of semantics games for deductive fuzzy logics, most prominently for Łukasiewicz logic, have been proposed in the literature. These games deviate from Hintikka’s original game for evaluating classical first-order formulas by either introducing an explicit reference to a truth value from the unit interval at each game state (as in [4]) or by generalizing to multisets of formulas to be considered at any state (as, e.g., in [12,9,7,10]). We explore to which extent Hintikka’s game theoretical semantics for classical logic can be generalized to a many-valued setting without sacrificing the simple structure of Hintikka’s original game. We show that rules that instantiate a certain scheme abstracted from Hintikka’s game do not lead to logics beyond the rather inexpressive, but widely applied Kleene-Zadeh logic, also known as ‘weak Łukasiewicz logic’ or even simply as ‘fuzzy logic’ [27]. To obtain stronger logics we consider propositional as well as quantifier rules that allow for random choices. We show how not only various extensions of Kleene-Zadeh logic, but also proper extensions Łukasiewicz logic arise in this manner.
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References
Aguzzoli, S., Gerla, B., Marra, V.: Algebras of fuzzy sets in logics based on continuous triangular norms. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS, vol. 5590, pp. 875–886. Springer, Heidelberg (2009)
Bennett, A.D.C., Paris, J.B., Vencovska, A.: A new criterion for comparing fuzzy logics for uncertain reasoning. Journal of Logic, Language and Information 9(1), 31–63 (2000)
Cintula, P., Hájek, P., Noguera, C. (eds.): Handbook of Mathematical Fuzzy Logic. College Publications (2011)
Cintula, P., Majer, O.: Towards evaluation games for fuzzy logics. In: Majer, O., Pietarinen, A.-V., Tulenheimo, T. (eds.) Games: Unifying Logic, Language, and Philosophy, pp. 117–138. Springer (2009)
Fermüller, C.G.: Revisiting Giles’s game. In: Majer, O., Pietarinen, A.-V., Tulenheimo, T. (eds.) Games: Unifying Logic, Language, and Philosophy, Logic, Epistemology, and the Unity of Science, pp. 209–227. Springer (2009)
Fermüller, C.G.: On matrices, Nmatrices and games. Journal of Logic and Computation (2013) (page to appear)
Fermüller, C.G., Metcalfe, G.: Giles’s game and the proof theory of Łukasiewicz logic. Studia Logica 92(1), 27–61 (2009)
Fermüller, C.G., Roschger, C.: Randomized game semantics for semi-fuzzy quantifiers. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part IV. CCIS, vol. 300, pp. 632–641. Springer, Heidelberg (2012)
Fermüller, C.G., Roschger, C.: From games to truth functions: A generalization of Giles’s game. Studia Logica (2013) (to appear)
Fermüller, C.G., Roschger, C.: Randomized game semantics for semi-fuzzy quantifiers. Logic Journal of the IGPL (to appear)
Gerla, B.: Rational Łukasiewicz logic and DMV-algebras. Neural Networks World 11, 579–584 (2001)
Giles, R.: A non-classical logic for physics. Studia Logica 33(4), 397–415 (1974)
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)
Giles, R.: Semantics for fuzzy reasoning. International Journal of Man-Machine Studies 17(4), 401–415 (1982)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers (2001)
Hájek, P.: Why fuzzy logic? In: Jacquette, D. (ed.) Blackwell Companion to Philosophical Logic, pp. 596–606. Wiley (2002)
Hájek, P.: On witnessed models in fuzzy logic. Mathematical Logic Quarterly 53(1), 66–77 (2007)
Hintikka, J.: Language-games for quantifiers. In: Rescher, N. (ed.) Studies in Logical Theory, pp. 46–72. Blackwell, Oxford (1968); Reprinted in [19]
Hintikka, J.: Logic, language-games and information: Kantian themes in the philosophy of logic. Clarendon Press Oxford (1973)
Hintikka, J., Sandu, G.: Game-theoretical semantics. In: Handbook of Logic and Language. Elsevier (2010)
Hisdal, E.: Are grades of membership probabilities? Fuzzy Sets and Systems 25(3), 325–348 (1988)
Lawry, J.: A voting mechanism for fuzzy logic. International Journal of Approximate Reasoning 19(3-4), 315–333 (1998)
Lorenzen, P.: Logik und Agon. In: Atti Congr. Internaz. di Filosofia, Venezia, Settembre 12-18, vol. IV, Sansoni (1960)
Lorenzen, P.: Dialogspiele als semantische Grundlage von Logikkalkülen. Archiv Für Mathemathische Logik und Grundlagenforschung 11, 32–55, 73–100 (1968)
Mann, A.L., Sandu, G., Sevenster, M.: Independence-friendly logic: A game-theoretic approach. Cambridge University Press (2011)
McNaughton, R.: A theorem about infinite-valued sentential logic. Journal of Symbolic Logic 16(1), 1–13 (1951)
Nguyêñ, H.T., Walker, E.A.: A first course in fuzzy logic. CRC Press (2006)
Paris, J.B.: A semantics for fuzzy logic. Soft Computing 1(3), 143–147 (1997)
Paris, J.B.: Semantics for fuzzy logic supporting truth functionality. In: Novák, V., Perfilieva, I. (eds.) Discovering the World with Fuzzy Logic, pp. 82–104. Physica-Verlag (2000)
Peters, S., Westerståhl, D.: Quantifiers in language and logic. Oxford University Press, USA (2006)
Scarpellini, B.: Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz. Journal of Symbolic Logic 27(2), 159–170 (1962)
Sevenster, M., Sandu, G.: Equilibrium semantics of languages of imperfect information. Annals of Pure and Applied Logic 161(5), 618–631 (2010)
Wen, X., Ju, S.: Semantic games with chance moves revisited: from IF logic to partial logic. Synthese 190(9), 1605–1620 (2013)
Zadeh, L.A.: Fuzzy logic. IEEE: Computer 21(4), 83–93 (1988)
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Fermüller, C.G. (2014). Hintikka-Style Semantic Games for Fuzzy Logics. In: Beierle, C., Meghini, C. (eds) Foundations of Information and Knowledge Systems. FoIKS 2014. Lecture Notes in Computer Science, vol 8367. Springer, Cham. https://doi.org/10.1007/978-3-319-04939-7_9
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DOI: https://doi.org/10.1007/978-3-319-04939-7_9
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