Public and private communication are different: results on relative expressivity

Dynamic Epistemic Logic (DEL) is the study of how to reason about knowledge, belief, and communication. This paper studies the relative expressivity of certain fragments of the DEL language for public and private communication. It is shown that the language of public communication with common knowledge and the language of private communication with common knowledge are expressively incomparable for the class of all pointed Kripke models, which provides a formal proof that public and private communication are fundamentally different in the presence of common knowledge. It is also shown that single-recipient private communication does not add expressive power to the language of modal logic with common knowledge for any class of transitive pointed Kripke models. The latter result provides a sense in which positive introspection—believing our own beliefs—induces a kind of self-dialog.