Second-order propositional modal logic and monadic alternation hierarchies

We establish that the quantifier alternation hierarchy of formulae of secondorder propositional modal logic (SOPML) induces an infinite corresponding semantic hierarchy over the class of finite directed graphs. This solves an open problem problem of van Benthem (1985) and ten Cate (2006). We also identify modal characterizations of the expressive powers of second-order logic (SO) and monadic second-order logic (MSO) in terms of extensions of modal logic with second-order quantification.