Completeness theorems for modal logic in second-order arithmetic

Abstract

This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in ${\mathrm{RCA}}_0$, and that, over ${\mathrm{RCA}}_0$, ${\mathrm{ACA}}_0$ is equivalent to the strong completeness theorem for modal propositional logic using canonical models. We also consider a simpler version of the strong completeness theorem without referring to canonical models and show that it is equivalent to ${\mathrm{WKL}}_0$ over ${\mathrm{RCA}}_0$.