 Different choices of axioms governing belief, knowledge and their interaction give rise to different epistemic logics. A significant amount of work in epistemic logic characterizes these systems and proves various surprising consequences of the axioms. Kripke models are the basis of the most widely used semantics for epistemic logic. (See Goldblatt (2006) for a discussion of whether or not these models are properly attributed to Kripke.) A Kripke model equips the set of possible worlds \(W\) with a binary indistinguishibility relation over \(W\). The basic idea is that two worlds \(w,w^{\prime}\) stand in the relation \(w R_a w'\) if, were \(w\) the actual world, agent \(a\) could not rule out that the actual world is \(w'\). Then, the model is connected with the formal language via the following definition: \[ w \vDash {\bf B}_a\phi \text{ iff } w'\vDash \phi \text{ for all } w' \text{ such that } w R_a w^\prime. \] In other words: it is true in \(w\) that \(a\) believes \(\phi\) iff \(\phi\) is true in all the worlds \(w'\) which \(a \) cannot distinguish from \(w.\) Equipped with this semantics, it is possible to prove various elegant correspondences between epistemic principles and structural properties of the indistinguishibility relation. For example, the principle \( {\bf B}\phi \rightarrow {\bf BB} \phi \) is validated whenever the indistinguishibility relation is transitive. The suspicious principle \({\bf B}\phi \rightarrow \phi\) is validated whenever the indistinguishibility relation is reflexive. A significant amount of theoretical activity is devoted to the interplay between intuitive epistemic principles expressed in the language of modal logic and elegant structural conditions on indistinguishibility relations. For an excellent introduction to this subject see the entry on epistemic logic, or the more general entry on modal logic. For more on the relationship between the logic of knowledge and belief, see the entry on philosophical aspects of multi-modal logic. More recently, alternative topological semantics for epistemic logic have seen significant theoretical activity (for example: Bjorndahl and Özgün, forthcoming).