# On Semantic Gamification

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## Abstract

The purpose of this essay is to study the extent in which the semantics for different logical systems can be represented game theoretically. I will begin by considering different definitions of what it means to *gamify* a semantics, and show completeness and limitative results. In particular, I will argue that under a proper definition of gamification, all finitely algebraizable logics can be gamified, as well as some infinitely algebraizable ones (like Łukasiewicz) and some non-algebraizable (like intuitionistic and van Fraassen supervaluation logic).

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Nash### References

- 1.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. Logic, Epistemology, and the Unity of Science, vol. 15, pp. 117–138. Springer, Dordrecht (2009)CrossRefGoogle Scholar
- 2.Fermüller, C.G.: Dialogue games for many-valued logics - an overview. Stud. Logica.
**90**(1), 43–68 (2008)MathSciNetCrossRefMATHGoogle Scholar - 3.Fermüller, C.G.: On matrices, Nmatrices and Games. J. Logic Comput. (2013)Google Scholar
- 4.Hähnle, R.: Advanced many-valued logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol. 2, pp. 297–395. Springer, Dordrecht (2001)CrossRefGoogle Scholar
- 5.Hintikka, J.: Logic, Language Games, and Information. Clarendon Press, Oxford (1973)MATHGoogle Scholar
- 6.Hintikka, J., Sandu, G.: Game-Theoretical Semantics (1997)Google Scholar
- 7.Kleene, S.C.: On notation for ordinal numbers. J. Symbolic Logic
**3**(4), 150–155 (1938)MathSciNetCrossRefMATHGoogle Scholar - 8.Kleene, S.C.: Introduction to Metamathematics: Bibliotheca Mathematica. Wolters-Noordhoff, Groningen (1952)MATHGoogle Scholar
- 9.Łukasiewicz, J., Borkowski, L.: Selected Works: Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1970)Google Scholar
- 10.Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)MATHGoogle Scholar
- 11.Mundici, D.: Ulam’s games, Łucasiewicz logic, and AFC*-algebras. Fundamenta Informaticae
**18**, 151 (1993)MathSciNetGoogle Scholar - 12.Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT press, Cambridge (1994)MATHGoogle Scholar
- 13.Parikh, R.: D-structures and their semantics. Not. AMS
**19**, A329 (1972)Google Scholar - 14.Parikh, R.: The logic of games and its applications. Ann. Discrete Math.
**102**, 111–140 (1985)MathSciNetMATHGoogle Scholar - 15.Post, E.L.: Introduction to a general theory of elementary propositions. Am. J. Math.
**43**(3), 163–185 (1921)MathSciNetCrossRefMATHGoogle Scholar - 16.Urquhart, A.: Basic many-valued logic. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol. 2, pp. 249–295. Springer, Dordrecth (2001)CrossRefGoogle Scholar
- 17.van Benthem, J.: Logic games are complete for game logics. Studia Logica. Int. J. Symbolic Logic
**75**, 183–203 (2003)MathSciNetMATHGoogle Scholar - 18.van Benthem, J.: Logic in Games. MIT Press, Cambridge (2014)MATHGoogle Scholar
- 19.van Fraassen, B.C.: Presuppositions: supervaluations and free logic. In: Lambert, K. (ed.) The Logical Way of doing Things, pp. 67–92. Yale University Press (1969)Google Scholar
- 20.van Fraassen, B.C.: Singular terms, truth-value gaps, and free logic. J. Philos.
**63**(17), 481–495 (1966)CrossRefGoogle Scholar

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