In this paper we introduce Second-order Propositional Announcement Logic (SOPAL): a language to express arbitrary announcements in Public Announcement Logic, by means of propositional quantification. We present SOPAL within a multi-agent context, and show that it is rich enough to express complex notions such as preservation under arbitrary announcements, knowability, and successfulness. We analyse the model theory of SOPAL and prove that it is strictly more expressive than Arbitrary PAL [2], and as expressive as Second-order Propositional Epistemic Logic [4], even though exponentially more succinct than the latter. These results points to a rich logic, with nice computational properties nonetheless, such as a decidable model checking problem and a complete axiomatisation.