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.

翻译：在开创性工作~\cite{Fine:2018}中，Fine对多种无知形式进行了分类，其中包括费希特无知、一阶无知、拉姆斯菲尔德无知和二阶无知。研究表明，这些无知形式之间存在有趣的关系，例如在${\bf S4}$中，所有高阶无知都归结为二阶无知。一些研究者认为这一结论具有不良后果。因此，自然要思考如何避免这种后果。我们在更为通用的框架中处理该问题。具体而言，我们将费希特无知和一阶无知视为原始模态，并在邻域语义学中将其作为核心研究对象加以分析（在该框架下，拉姆斯菲尔德无知和二阶无知是可定义的）。主要贡献包括模型论结果（如表达力与框架可定义性）以及公理化系统。最后同样重要的是，通过基于交集语义对邻域模型进行更新，我们将研究结果扩展到公开宣告的动态情形，从而为成功公式提供了一些应用。