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On Semantic Games for Łukasiewicz Logic

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Jaakko Hintikka on Knowledge and Game-Theoretical Semantics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 12))

Abstract

We explore different ways to generalize Hintikka’s classic game theoretic semantics to a many-valued setting, where the unit interval is taken as the set of truth values. In this manner a plethora of characterizations of Łukasiewicz logic arise. Among the described semantic games is Giles’s dialogue and betting game, presented in a manner that makes the relation to Hintikka’s game more transparent. Moreover, we explain a so-called explicit evaluation game and a ‘bargaining game’ variant of it. We also describe a recently introduced backtracking game as well as a game with random choices for Łukasiewicz logic.

We wish to thank Gabriel Sandu for valuable comments on a version of this paper. (Christian G. Fermüller) Supported by FWF grant P25417-G15. (Ondrej Majer) Supported by GACR grant P402/12/1309.

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Notes

  1. 1.

    The game can straightforwardly be generalized to formulas with free variables and to languages, where there may not be a constant for every domain element, by explicitly augmenting formulas by assignments. However we find it more convenient to stick with sentences and to dispense with extra notation for assignments.

  2. 2.

    Hintikka uses Nature and Myself as names for the players and Verfier and Falisifer for the two roles. To emphasize our interest in the connection to Giles’s game (see Sect. 10.4) we use Giles’s names for the players (I/You) and Lorenzen’s corresponding role names (\(\mathbf P \)/\(\mathbf O \)) throughout the paper.

  3. 3.

    Hintikka’s game, like the explicit evaluation game of Sect. 10.3, is a win/lose game where no payoff values are specified; rather it is sufficient to say that one player wins and the other player loses the game. This can be considered a special case of constant-sum by identifying winning with payoff 1 and losing with payoff 0.

  4. 4.

    One can also find the name ‘Kripke-Zadeh logic’ for this fragment of Ł in the literature see, e.g., [1]. The well—known textbook [20] even simply speaks of ‘fuzzy logic’. We will focus on Łukasiewicz logic in this paper and thus prefer to use the name Ł \(^w\).

  5. 5.

    Remember that we identify constants with domain elements and that every interpretation contains an assignment of domain elements to variables. This induces the mentioned assignment of truth values to atomic formulas. Conversely, every assignment of truth values to atomic formulas uniquely determines a function of type \(D^n \rightarrow [0,1]\) for each n-ary predicate symbol. We may therefore, without loss of generality, identify an interpretation \(\mathcal J\) with an assignment of truth values to atomic formulas.

  6. 6.

    Throughout this paper we will deal only with perfect information games and therefore never have to refer to mixed strategies.

  7. 7.

    It turns out that the powers of the players of a \(\mathcal G\)-game are not depended on the manner in which the current formula is picked at any state. Still, a more formal presentation of \(\mathcal{G}\)-games will employ the concepts of a regulation and of so-called internal states in formalizing state transitions. We refer to [5] for details.

  8. 8.

    Giles [9, 10] in fact only sketched a proof for the language without strong conjunction. For a detailed proof of the propositional case, where the game includes a rule for strong conjunction, we refer to [5].

  9. 9.

    The term ‘bargaining game’ has a different meaning in game theory see, e.g., [21]. We do not want to allude to those types of (not logic-related) games here.

References

  1. Aguzzoli S, Gerla B, Marra V (2009) Algebras of fuzzy sets in logics based on continuous triangular norms. In: Sossai C, Chemello G (eds) Proceedings of the 10th European conference on symbolic and quantitative approaches to reasoning with uncertainty (ECSQARU 2009). Springer, pp 875–886

    Google Scholar 

  2. Cintula, P, Hájek, P, Noguera C (eds) (2011) Handbook of mathematical fuzzy logic. College Publications

    Google Scholar 

  3. Cintula P, Majer O (2009) Towards evaluation games for fuzzy logics. In: Majer O, Pietarinen A-V, Tulenheimo T (eds) Games: unifying logic, language, and philosophy. Springer, pp 117–138

    Google Scholar 

  4. Fermüller CG (2009) Revisiting Giles’s game. In: Majer O, Pietarinen A -V, Tulenheimo T (eds) Games: unifying logic, language, and philosophy. Springer, pp 209–227

    Google Scholar 

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

    Google Scholar 

  6. Fermüller CG (2014) Hintikka-style semantic games for fuzzy logics. In: Beierle C, Meghini C (eds) Foundations of information and knowledge systems, proceedings of the 8th international symposium FoIKS 2014, Bordeaux 2014, vol 8367. Springer, pp 193–210

    Google Scholar 

  7. Fermüller CG (2014) Semantic games with backtracking for fuzzy logics. In: 44th international symposium on multiple-valued logic (ISMVL 2014), pp 38–43

    Google Scholar 

  8. Fermüller CG, Roschger C (2014) Randomized game semantics for semifuzzy quantiiers. Logic J Interest Group Pure Appl Logic 22(3):413–439

    Google Scholar 

  9. Giles R (1974) A non-classical logic for physics. Studia Logica 33(4):397–415

    Google Scholar 

  10. Giles R (1977) A non-classical logic for physics. In: Wojcicki R, Malinkowski G (eds) Selected papers on lukasiewicz sentential alculi. Polish Academy of Sciences, pp 13–51

    Google Scholar 

  11. Giles R (1982) Semantics for fuzzy reasoning. Int J Man- Mach Stud 17(4):401–415

    Google Scholar 

  12. Hájek P (2007) On witnessed models in fuzzy logic. Math Logic Q 53(1):66–77

    Google Scholar 

  13. Hintikka J (1968) Language-games for quantifiers. In: Rescher N (ed) Studies in logical theory, Blackwell, Oxford, pp 46–72 (Reprinted in (Hintikka, 1973))

    Google Scholar 

  14. Hintikka J (1973) Logic, language-games and information: Kantian themes in the philosophy of logic. Clarendon Press, Oxford

    Google Scholar 

  15. Hintikka J, Sandu G (2010) Game-theoretical semantics, In: Handbook of logic and language. Elsevier

    Google Scholar 

  16. Leyton-Brown K, Shoham Y (2008) Essentials of game theory: a concise multidisciplinary introduction. Morgan & Claypool Publishers

    Google Scholar 

  17. Lorenzen P (1960) Logik und Agon. In: Atti congr. internaz. di filosofia (Venezia, 12–18 settembre 1958), vol iv, pp 187–194, Sansoni

    Google Scholar 

  18. Lorenzen P (1968) Dialogspiele als semantische Grundlage von Logikkalkülen. Archiv für mathemathische Logik und Grundlagenforschung 11(32–55):73–100

    Google Scholar 

  19. Mann A, Sandu G, Sevenster M (2011) Independence-friendly logic: a game-theoretic approach. Cambridge University Press

    Google Scholar 

  20. Nguyen H, Walker E (2006) A first course in fuzzy logic. CRC Press

    Google Scholar 

  21. Osborne M, Rubinstein A (1994) A course in game theory. MIT Press

    Google Scholar 

  22. Sevenster M, Sandu G (2010) Equilibrium semantics of languages of imperfect information. Annl Pure Appl Logic 161(5):618–631

    Google Scholar 

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Author information

Authors and Affiliations

  1. Technical University of Vienna, Vienna, Austria

    Christian G. Fermüller

  2. Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic

    Ondrej Majer

Authors
  1. Christian G. Fermüller
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  2. Ondrej Majer
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Correspondence to Christian G. Fermüller .

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Editors and Affiliations

  1. LORIA, CNRS, Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Hans van Ditmarsch

  2. Department of Philosophy, University of Helsinki, Helsinki, Finland

    Gabriel Sandu

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Fermüller, C.G., Majer, O. (2018). On Semantic Games for Łukasiewicz Logic. In: van Ditmarsch, H., Sandu, G. (eds) Jaakko Hintikka on Knowledge and Game-Theoretical Semantics. Outstanding Contributions to Logic, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-62864-6_10

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