Do I Know Ω? An Axiomatic Model of Awareness and Knowledge

In modeling game and decision theory situations, it has been usual to start by considering Ω, the set of conceivable states of the world. I wish to propose a more fundamental view. I do not assume that the agent knows Ω. Instead the agent is assumed to derive for herself a representation of the universe. Given her knowledge and her ability to reason about it the agent deduces a set of conceivable states of the world and a set of possible states of the world. The epistemic model considered in this paper uses a propositional framework. The model distinguishes between the knowledge of the existence of a proposition, which I call awareness, and the knowledge of the truth or the falsity of the proposition. Depending upon whether one assumes that the agent is aware or not of all the propositions, she will or will not have a “complete model” of the world. When the agent is not aware of all the propositions, the states of the world and the possibility correspondence imaginable by her are coarser than the modeler's. The agent has an incomplete knowledge of both the states of the world and the information structure. In addition, I extend the model with “incompleteness” to a dynamic setting. Under the assumption that the agent's knowledge is non-decreasing over time, I show that the set of states of the world conceivable by the agent and her possibility correspondence get finer over time.