(This is joint work with Tim French, Fernando Velazquez, and Yi Wang)
We consider semantic structures and logics that differentiate between being uncertain about a proposition, being unaware of a proposition, becoming aware of a proposition, and getting to know the truth value of a proposition. We propose a unified setting to model this variety of static and dynamic aspects of awareness and knowledge. Various derived notions will be discussed: explicit knowledge, implicit knowledge, and also novel epistemic notions such as speculative knowledge (back to the motivation in Levesque's 'A Logic of Implicit and Explicit Belief', one can speculate over variables of which one is unaware, e.g. if you are unaware of p, then p v ~p is still speculatively known by you). A cornerstone of this framework is the notion of awareness bisimulation - this is the proper notion of structural similarity on the structures enriched with awareness of propositional variables proposed by Fagin and Halpern in 'Belief, awareness, and limited reasoning'. Just as bisimilarity preserves logical truth, awareness bisimilarity preserves the truth of which you are aware.