Provability interpretation of non-normal modal logics having neighborhood semantics

TL;DR

This paper explores the provability interpretation of non-normal modal logics with neighborhood semantics, introducing a new embedding method into arithmetic and establishing arithmetical completeness for several non-normal modal systems.

Contribution

It develops a novel embedding technique for neighborhood semantics into arithmetic, enabling arithmetical completeness proofs for multiple non-normal modal logics.

Findings

Proved arithmetical completeness for $ extsf{EN}$, $ extsf{ECN}$, $ extsf{ENP}$, $ extsf{END}$, and $ extsf{ECNP}$.

Introduced a new model embedding method based on neighborhood semantics.

Extended the scope of arithmetical completeness proofs to non-normal modal logics.

Abstract

We study provability predicates $\mathrm{Pr}_T(x)$ satisfying the following condition $\mathbf{E}$ from a modal logical perspective: $\mathbf{E}:$ if $ T \vdash \varphi \leftrightarrow \psi$, then $T \vdash \mathrm{Pr}_T(\ulcorner \varphi \urcorner) \leftrightarrow \mathrm{Pr}_T(\ulcorner \psi \urcorner)$. For this purpose, we develop a new method of embedding models based on neighborhood semantics into arithmetic. Our method broadens the scope of arithmetical completeness proofs. In particular, we prove the arithmetical completeness theorems for the non-normal modal logics $\mathsf{EN}$, $\mathsf{ECN}$, $\mathsf{ENP}$, $\mathsf{END}$, and $\mathsf{ECNP}$.