Abstract
The paper considers extensions of a normal bimodal logic $\textbf{KL}$ in which the two necessity operators are mutual one-sided inverses. A continuum of maximal consistent normal extensions of $\textbf{KL}$ is constructed, each of which has infinitely many quasi-normal Post complete extensions. Some syntactic properties of maximal consistent normal bimodal logics and in particular of such extensions of $\textbf{KL}$ are investigated.
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