In this thesis we investigate various fragments of second-order logic that arise naturally in considerations related to modal logic. The focus is on questions related to expressive power. The results in the thesis are reported in four independent but related chapters (Chapters 2, 3, 4 and 5).

In Chapter 2 we study second-order propositional modal logic, which is the system obtained by extending ordinary modal logic with second-order quantification of proposition symbols. We show that the alternation hierarchy of this logic is infinite, thereby solving an open problem from the related literature. In Chapter 3 we investigate the expressivity of a range of modal logics extended with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the chapter is that the resulting extension of (a version of) Boolean modal logic can be effectively translated into existential monadic second-order logic. As a corollary we obtain decidability results for multimodal logics over various classes of frames with built-in relations.

In Chapter 4 we study the equality-free fragment of existential second-order logic with second-order quantification of function symbols. We show that over directed graphs, the expressivity of the fragment is incomparable with that of first-order logic. We also show that over finite models with a unary relational vocabulary, the fragment is weaker in expressivity than first-order logic.

In Chapter 5 we study the extension of polyadic modal logic with unrestricted quantification of accessibility relations and proposition symbols. We obtain a range of results related to various natural fragments of the system. Finally, we establish that this extension of modal logic exactly captures the expressivity of second-order logic.