Abstract
Public announcements are used in dynamic epistemic logic to model certain kinds of information change. A formula \(\left<\psi\right>\varphi \) 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 \(\lozenge\) can be added to an epistemic logic with public announcements. Ideally a formula \(\lozenge \varphi \) would hold if and only if there is a formula ψ such that \(\left<\psi\right>\varphi \). However, in order to avoid circularity the \(\lozenge\) operator can only quantify over those ψ that are \(\lozenge\)-free. So \(\lozenge \varphi \) holds if and only if there is a \(\lozenge\)-free ψ such that \(\left<\psi\right>\varphi \).
As a result it does not follow immediately from the definition that \(\left<\psi\right>\varphi \) implies \(\lozenge \varphi \) if ψ contains a \(\lozenge\). But the implication may still hold in some cases. In this paper I show that on finite models \(\left<\psi\right>\varphi \) implies \(\lozenge \varphi \) for every ψ, and that on finitely branching models \(\left<\psi\right>\varphi \) implies \(\lozenge \varphi \) for every ψ if ϕ is \(\lozenge\)-free. Finally I also show that there are ϕ and ψ such that \(\left<\psi\right>\varphi \) does not imply \(\lozenge \varphi \) even on a finitely branching model.
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Kuijer, L.B. (2014). How Arbitrary Are Arbitrary Public Announcements?. In: Colinet, M., Katrenko, S., Rendsvig, R.K. (eds) Pristine Perspectives on Logic, Language, and Computation. ESSLLI ESSLLI 2013 2012. Lecture Notes in Computer Science, vol 8607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44116-9_8
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