Arithmetical completeness for some extensions of the pure logic of necessitation

TL;DR

This paper explores the arithmetical completeness of certain extensions of the pure logic of necessitation, establishing that these logics correspond to specific provability predicates in arithmetic.

Contribution

It proves that for all positive integers m,n, the logic NA_{m,n} is a provability logic, linking modal principles to arithmetic provability.

Findings

NA_{m,n} is a provability logic for all m,n ≥ 1

Established correspondence between modal logic extensions and arithmetic provability

Extended the understanding of the pure logic of necessitation's arithmetical completeness

Abstract

We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczy\'{n}ski's pure logic of necessitation $\mathbf{N}$. For $m,n \in \omega$, let $\mathbf{NA}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^n A \to \Box^m A$. In this paper, among other things, we prove that for each $m,n \geq 1$, the logic $\mathbf{NA}_{m,n}$ becomes a provability logic, that is, there exists a provability predicate $\mathrm{Pr}_T(x)$ of $T$ whose $T$-verifiable modal principles are exactly the logic $\mathbf{NA}_{m,n}$.