Provability interpretation of non-normal modal logics having neighborhood semantics
Haruka Kogure
TL;DR
This work addresses arithmetical completeness for non-normal modal logics with neighborhood semantics by focusing on provability predicates satisfying $\mathbf{E}$ and embedding neighborhood semantics directly into arithmetic. It introduces two constructions, $\mathrm{Pr}_{g_0}(x)$ and $\mathrm{Pr}_{g_1}(x)$, to establish Solovay-type completeness for the logics $\mathsf{EN}$, $\mathsf{ENP}$, $\mathsf{END}$, $\mathsf{ECN}$, and $\mathsf{ECNP}$ under suitable consistency assumptions. The results show that, under $T$-consistency (e.g., $\mathrm{Con}^L_T$), these logics coincide with the corresponding provability logics $\mathsf{PL}(\mathrm{Pr}_T)$, extending the classical Solovay framework beyond Kripke semantics. The paper also outlines open questions about extensions like $\mathsf{EN4}$, $\mathsf{ENP4}$, and $\mathsf{ECN4}$, highlighting the flexibility and limits of neighborhood-based arithmetical completeness and pointing to future directions in provability logic.
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 ψ$, then $T \vdash \mathrm{Pr}_T(\ulcorner \varphi \urcorner) \leftrightarrow \mathrm{Pr}_T(\ulcorner ψ\urcorner)$. 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 $\mathsf{ECNP}$.