The Finite Frame Property of Some Extensions of the Pure Logic of Necessitation

We study the finite frame property of some extensions of Fitting, Marek, and Truszczyński’s pure logic of necessitation \(\textbf{N}\) $\mathbf{N}$ . For any natural numbers m, n, we introduce the logic \(\textbf{N}^+\textbf{A}_{m,n}\) ${\mathbf{N}}^{+}{\mathbf{A}}_{m,n}$ by adding the single axiom scheme \(\Box ^n \varphi \rightarrow \Box ^m \varphi \) ${\square }^n\phi \to {\square }^m\phi$ and the rule \(\dfrac{\lnot \Box \varphi }{\lnot \Box \Box \varphi }\) ${\displaystyle \frac{\neg \square \phi }{\neg \square \square \phi }}$ ( \({\text {Ros}}^\Box \) ${\text{Ros}}^{\square }$ ) into \(\textbf{N}\) $\mathbf{N}$ . We prove the finite frame property of \(\textbf{N}^+\textbf{A}_{m, n}\) ${\mathbf{N}}^{+}{\mathbf{A}}_{m,n}$ with respect to Fitting, Marek, and Truszczyński’s relational semantics. We also prove that for \(n \ge 2\) $n\ge 2$ , the logic obtained by removing the rule \({\text {Ros}}^\Box \) ${\text{Ros}}^{\square }$ from \(\textbf{N}^+\textbf{A}_{0, n}\) ${\mathbf{N}}^{+}{\mathbf{A}}_{0,n}$ is incomplete with respect to that semantics.