On the axiomatization of an unawareness structure from knowing-whether operators

This paper shows that, on a generalized state space model of unawareness, an agent's underlying knowledge is axiomatized from her knowing-whether operator if and only if her knowledge satisfies the Truth Axiom: whenever the agent knows an event, the event holds. The agent knows whether an event obtains if she knows it or knows its negation. Different knowledge operators lead to different knowing-whether operators if knowledge is truthful. Conversely, for any knowing-whether operator, there is a unique truthful knowledge operator that induces the given knowing-whether operator: the agent knows an event if and only if she knows whether the event holds and the event indeed holds. Qualitative or probabilistic beliefs may not be recovered from believing-whether. This paper then axiomatizes properties of knowledge and common knowledge, in terms of knowing-whether. The main contributions of the paper areas follows. First, conceptually, this paper provides a generalized-state-space model of knowledge and unawareness in which the only assumption on knowledge is the Truth Axiom. Second, practically, this paper may provide a simple way to construct a generalized-state-space model.