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 ${\square }^n\phi \to {\square }^m\phi$, $◊$-free modal reduction principle, together with a rule $\frac{\neg \square \phi }{\neg \square \square \phi }$, 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 ${\mathbf{NA}}_{m,n}=\mathbf{N}+{\square }^n\phi \to {\square }^m\phi$ and ${\mathbf{NRA}}_{m,n}=\mathbf{N}+{\square }^n\phi \to {\square }^m\phi +\frac{\neg \phi }{\neg \square \phi }$ in the same manner.

20 pages