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 $\mathbf{C}$: $T⊢{\mathrm{Pr}}_T(┌\ddot{I}†┐)\wedge {\mathrm{Pr}}_T(┌\ddot{I}ˆ┐)\to {\mathrm{Pr}}_T(┌\ddot{I}†\wedge \ddot{I}ˆ┐)$. The modal counterpart of this condition is the axiom scheme $\mathsf{C}$: $\square A\wedge \square B\to \square (A\wedge B)$. 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 $\mathsf{CN}$, $\mathsf{CNP}$, $\mathsf{CNF}$, $\mathsf{CNPF}$, and $\mathsf{CND}$ by using our new semantics.

29 pages