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Game semantics allows us to look at basic logical concepts from another side.This approach to logic has a long history, there are plenty of different typesof games: provability games, semantic games, etc. And there is an interestingtype of provability games called Mezhirov's game proposed by Iliya Mezhirov forintuitionistic logic of propositions and Grzegorczyk modal logic. Mezhirov'sgame semantics for intuitionistic logic is interesting because of itssimplicity and strong connection with Kripke semantics and Kripke models. In mystudy, I try to generalize Mezhirov's result in two directions: to generalizeto intuitionistic logic of predicates (or to some its extensions) and to thecase of a connection not only between the game and tautologies of logic, butalso between the game and entailment from infinite sets of formulas.

Generalizations of Mezhirov's game semantics to predicate superintuitionistic logics and the Casari formula