Provability interpretation of non-normal modal logics having neighborhood semantics

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 $\mathsf{ECNP}$.