Uniform Lyndon interpolation for the pure logic of necessitation with a modal reduction principle

Abstract

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 $\Box^n \varphi \to \Box^m \varphi$, $\Diamond$-free modal reduction principle, together with a rule $\frac{\neg \Box \varphi}{\neg \Box \Box \varphi}$, required to make the logic complete with respect to its Kripke-like semantics. We first introduce a sequent calculus $\mathbf{GN}^+\mathbf{A}_{m,n}$ for $\mathbf{N}^+\mathbf{A}_{m,n}$ and show that it enjoys cut elimination, proving Craig and Lyndon interpolation properties as a consequence. We then introduce a general method, called propositionalization, that enables one to reduce ULIP of a logic to some weaker logic. Lastly, we construct a propositionalization of $\mathbf{N}^+\mathbf{A}_{m,n}$ into classical propositional logic $\mathbf{Cl}$, proving ULIP as a corollary. We also prove ULIP of $\mathbf{NA}_{m,n} = \mathbf{N} + \Box^n \varphi \to \Box^m \varphi$ and $\mathbf{NRA}_{m,n} = \mathbf{N} + \Box^n \varphi \to \Box^m \varphi + \frac{\neg \varphi}{\neg \Box \varphi}$ in the same manner.