Dialogue games for Dishkant’s quantum modal logic

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.