The Geometry of Knowledge

The most widely used attractive logical account of knowledge uses standard epistemic models, i.e., graphs whose edges are indistinguishability relations for agents. In this paper, we discuss more general topological models for a multi-agent epistemic language, whose main uses so far have been in reasoning about space. We show that this more geometrical perspective affords greater powers of distinction in the study of common knowledge, defining new collective agents, and merging information for groups of agents. 1. Epistemic logic in its standard guise 1.1. Basic epistemic logic. Epistemic logic is in wide use today as a description of knowledge and ignorance for agents in philosophy [14], computer science [13], [22], game theory [12], and other areas. In this paper, we assume familiarity with the basic language of propositional epistemic logic, interpreted over multi-agent S4 models whose accessibility relations are reflexive and transitive. Alternative model classes occur, too, such as equivalence relations for each agent in multi-agent S5–but our discussion is largely independent from such choices. The key semantic clause about an agent’s knowledge of a proposition says that Kiφ holds at a world x if and only if φ is true in all worlds y accessible for i from x. That is, the epistemic knowledge modality is really a modal box ¤iφ. For technical convenience, we will use the latter notation for knowledge in the rest of this paper. The main modern interest in epistemic logic has to do with analyzing iterated knowledge of agents about themselves and what others know, for purposes of communication and interaction. Cf. [4], [9] on systems that combine epistemic logic and dynamic logic to describe information update in groups of agents. A simple example of how the basic logic works is the model in Figure 1. The universally valid principles in our models are those of multi-agent S4. In an epistemic setting, the usual modal axioms get a special flavor. E.g., the iteration axiom ¤1φ → ¤1¤1φ now expresses ‘positive introspection’: agents who know something know that they know it. More precisely, we have S4-axioms for each separate agent, but no valid further ‘mixing axioms’ for iterated knowledge of agents,