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We prove the uniform Lyndon interpolation property (ULIP) of some extensions of the pure logic of necessitation $\mathbf{N}$. For any $m, n \in \mathbb{N}$, $\mathbf{N}^+\mathbf{A}_{m,n}$ is the logic obtained from $\mathbf{N}$ by adding a single axiom $\Box^n \varphi \to \Box^m \varphi$, which is a $\Diamond$-free modal reduction principle, and a...
We study the finite frame property of some extensions of Fitting, Marek, and Truszczyński’s pure logic of necessitation $$\textbf{N}$$ N . For any natural numbers m , n , we introduce the logic $$\textbf{N}^+\textbf{A}_{m,n}$$ N + A m , n by adding the single axiom scheme $$\Box ^n \varphi \rightarrow \Box ^m \varphi $$ □ n φ → □ m φ and the rule $...
We study the finite frame property of some extensions of Fitting, Marek, and Truszczy\'nski's pure logic of necessitation $\mathbf{N}$. For any natural numbers $m, n$, we introduce the logic $\mathbf{N}^+\mathbf{A}_{m, n}$ by adding the single axiom scheme $\Box^n \varphi \to \Box^m \varphi$ and the rule $\dfrac{\neg \Box \varphi}{\neg \Box \Box \v...
