  In a seminal work~\cite{Fine:2018}, Fine classifies several forms of
ignorance, among which are Fitchean ignorance, first-order ignorance, Rumsfeld
ignorance, and second-order ignorance. It is shown that there is interesting
relationship among some of them, which includes that in ${\bf S4}$, all
higher-order ignorance are reduced to second-order ignorance. This is thought
of as a bad consequence by some researchers. It is then natural to ask how to
avoid this consequence. We deal with this issue in a much more general
framework. In detail, we treat the forms of Fitchean ignorance and first-order
ignorance as primitive modalities and study them as first-class citizens under
neighborhood semantics, in which Rumsfeld ignorance and second-order ignorance
are definable. The main contributions include model-theoretical results such as
expressivity and frame definability, and axiomatizations. Last but not least,
by updating the neighborhood models via the intersection semantics, we extend
the results to the dynamic case of public announcements, which gives us some
applications to successful formulas.
