How Arbitrary Are Arbitrary Public Announcements?

Public announcements are used in dynamic epistemic logic to model certain kinds of information change. A formula 〈ψ〉φ represents the statement that after ψ is publicly announced φ will be the case. Sometimes we want to reason about whether it is possible for φ to become true after some announcement. In order to do this an arbitrary public announcement operator ♦ can be added to an epistemic logic with public announcements. Ideally a formula ♦φ would hold if and only if there is a formula ψ such that 〈ψ〉φ. However, in order to avoid circularity the ♦ operator can only quantify over those ψ that are ♦-free. So ♦φ holds if and only if there is a ♦-free ψ such that 〈ψ〉φ. As a result it does not follow immediately from the definition that 〈ψ〉φ implies ♦φ if ψ contains a ♦. But the implication may still hold in some cases. In this paper I show that on finite models 〈ψ〉φ implies ♦φ for every ψ, and that on finitely branching models 〈ψ〉φ implies ♦φ for every ψ if φ is ♦-free. Finally I also show that there are φ and ψ such that 〈ψ〉φ does not imply ♦φ even on a finitely branching model.