Fitchean Ignorance and First-order Ignorance: A Neighborhood Look

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.