Modal logics of conjunctively closed provability predicates

Abstract

We investigate non-normal modal logics corresponding to provability predicates $\mathrm{Pr}_T(x)$ satisfying the derivability condition $\mathbf{C}$: $T\vdash\mathrm{Pr}_T(\ulcorner φ\urcorner)\land\mathrm{Pr}_T(\ulcorner ψ\urcorner)\to \mathrm{Pr}_T(\ulcorner φ\landψ\urcorner)$. The modal counterpart of this condition is the axiom scheme $\mathsf{C}$: $\Box A\land\Box B\to\Box(A\land B)$. 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 $\mathsf{CN}$, $\mathsf{CNP}$, $\mathsf{CNF}$, $\mathsf{CNPF}$, and $\mathsf{CND}$ by using our new semantics.