Dialogue games for Dishkant’s quantum modal logic

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V.L. Vasyukov

Abstract

Recently some elaborations were made concerning the game theoretic semantic of $\L_{\aleph_0}$ and its extension. In the paper this kind of semantics is developed for Dishkant’s quantum modal logic $\L$Q which is also, in fact, the specific extension of $\L_{\aleph_0}$. As a starting point some game theoretic interpretation for the S$\L$ system (extending both $\L$ukasiewicz logic $\L_{\aleph_0}$ and modal logic S5) was exploited which has been proposed in 2006 by C. Fermuller and R. Kosik. They, in turn, based on ideas already introduced by Robin Giles in the 1970th to obtain a characterization of $\L_{\aleph_0}$ in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion.

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Статьи

References

Dishkant, H., An Extension of the Lukasiewicz Logic to the Modal Logic of Quantum Mechanics, Studia Logica 37(2):149–155, 1976.
Fermuller, C.G., and R. Kosik, Combining supervaluation and degree based reasoning under vagueness, Proceedings of LPAR 2006, volume 4246 of LNAI. Springer, 2006, pp. 212–226.
Fermuller, 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.
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Giles, R., A non-classical logic for physics, in R. Wojcicki, G. Malinowski (eds.), Selected Papers on Lukasiewicz Sentential Calculi, Polish Academy of Sciences, 1977, pp. 13–51.
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Vasyukov, V.L., The Completeness of the Factor Semantics for Lukasiewicz’s Infinite-valued Logics, Studia Logica 52(1):143–167, 1993.
Vasyukov, V.L., Quantum Logics, Moscow: Per Se, 2005 (in Russian).

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