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 \(\textrm{RCA}_0\) , and that, over \(\textrm{RCA}_0\) , \(\textrm{ACA}_0\) 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 \(\textrm{WKL}_0\) over \(\textrm{RCA}_0\) .