Knowing Whether

Knowing whether a proposition is true means knowing that it is true or knowing that it is false. In this paper, we study logics with a modal operator Kw for knowing whether but without a modal operator K for knowing that. This logic is not a normal modal logic, because we do not have Kw(φ → ψ) → (Kwφ → Kwψ). Knowing whether logic cannot define many common frame properties, and its expressive power is less than that of basic modal logic over classes of models without reflexivity. These features make axiomatizing knowing whether logics non-trivial. We axiomatize knowing whether logic over various frame classes. We also present an extension of knowing whether logic with public announcement operators and we give corresponding reduction axioms for that. We compare our work in detail to two recent similar proposals.