The Monadic Grzegorczyk Logic

TL;DR

This paper establishes that the monadic Grzegorczyk logic $ extbf{MGrz}$ precisely axiomatizes the one-variable fragment of the predicate Grzegorczyk logic $ extbf{QGrz}$ by developing a semantic criterion and proving the finite model property.

Contribution

It introduces a semantic criterion for monadic modal logics and proves the finite model property of $ extbf{MGrz}$, linking it to the predicate logic fragment.

Findings

$ extbf{MGrz}$ axiomatizes the one-variable fragment of $ extbf{QGrz}$

Finite model property of $ extbf{MGrz}$ established

Refined filtration methods used for proof

Abstract

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.