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 (phi -> psi) ->
(Kw phi -> Kw psi). Knowing whether logic cannot define many common frame
properties, and its expressive power 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.
