The Monadic Grzegorczyk Logic

We develop a semantic criterion for determining whether a given monadic modal logic axiomatizes the one-variable fragment of a predicate modal logic. We show that the criterion applies to the monadic Grzegorczyk logic $\textbf{MGrz}$, thus establishing that $\textbf{MGrz}$ axiomatizes the one-variable fragment of the predicate Grzegorczyk logic $\textbf{QGrz}$. This we do by proving the finite model property of $\textbf{MGrz}$, which is achieved by strengthening the notion of a maximal point of a descriptive $\textbf{MGrz}$-frame and by refining the existing selective filtration methods.