Modal logics of conjunctively closed provability predicates

We investigate non-normal modal logics corresponding to provability predicates ${\mathrm{P}\mathrm{r}}_T(x)$ satisfying the derivability condition $\mathbf{C}$: $T⊢{\mathrm{P}\mathrm{r}}_T(\phi )\wedge {\mathrm{P}\mathrm{r}}_T(\psi )\to {\mathrm{P}\mathrm{r}}_T(\phi \wedge \psi )$. The modal counterpart of this condition is the axiom scheme $\mathsf{C}$: $\mathrm{\square }A\wedge \mathrm{\square }B\to \mathrm{\square }(A\wedge B)$. First, we introduce a new semantics based on closure operators for non-normal modal logics including logics adopting $\mathsf{C}$ as an axiom scheme. We prove modal completeness for several non-normal modal logics studied in this paper with respect to this semantics. Second, we prove the arithmetical completeness theorems for the logics $\mathsf{C}\mathsf{N}$, $\mathsf{C}\mathsf{N}\mathsf{P}$, $\mathsf{C}\mathsf{N}\mathsf{F}$, $\mathsf{C}\mathsf{N}\mathsf{P}\mathsf{F}$, and $\mathsf{C}\mathsf{N}\mathsf{D}$ by using our new semantics.