We consider propositional normal unimodal pretransitive logics, i.e., logics with expressible `transitive' modality. There is a long-standing open problem about the finite model property (fmp) and decidability of pretransitive logics, in particular - for the logics K^m_n = K+^m p -> ^n p, n>m>1.
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Recently some elaborations were made concerning the game theoretic semantic of Lℵ0 and its extension. In the paper this kind of semantics is developed for Dishkant’s quantum modal logic LQ which is also, in fact, the speciﬁc extension of Lℵ0 . As a starting point some game theoretic interpretation for the S L system (extending both Lukasiewicz logic Lℵ0 and modal logic S5) was exploited which has been proposed in 2006 by C.Ferm˝uller and R.Kosik . They, in turn, based on ideas already introduced by Robin Giles in the 1970th to obtain a characterization of Lℵ0 in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion.