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. 1 The Paradigmatic Approach in Formal Epistemology Since Hintikka [8], the paradigmatic approach in formal epistemology has been to use modal logic to reason about the doxastic attitudes (that is, things like propositional knowledge and propositional belief) that are held by a number of individuals commonly called agents. The essence of this approach is to argue that a doxastic attitude is logically closed under certain principles that can be expressed in the language of modal logic. As an example, most authors agree that knowledge is veridical (meaning that there is no “false knowledge”: a statement φ must be true in the event agent i knows φ), and so such authors invite us to accept the modal scheme Kiφ ⊃ φ (“if agent i knows φ, then φ is true”) as a principle of logical closure that is an essential component of a correct descriptive characterization of knowledge [5]. Such argumentation generally advocates that a particular modal logic ought to be accepted as the logic of the doxastic attitude in question on the basis that it is this logic that consists of exactly those principles of logical closure that were argued to be essential to that doxastic attitude. Examples: S5 is often taken as the logic of knowledge, and KD45 is often taken as the logic of belief [5]. c ©2008 Springer Science+Business Media B.V. The original publication is available at www.springerlink.com. Citation: Bryan Renne. Public and Private Communication are Different: Results on Relative Expressivity. Synthese, 165(2):225–245, 2008.