In a recent paper, Kit Fine presents some striking results concerning the logical properties of (first-order) ignorance, second-order ignorance and Rumsfeld ignorance. However, Rumsfeld ignorance is definable in terms of ignorance, which makes some existing results and the axiomatization problem trivial. A main reason is that the accessibility relations for the implicit knowledge operator contained in the packaged operators of ignorance and Rumsfeld ignorance are the same. In this work, we assume the two accessibility relations to be different so that one of them is an arbitrary subset of the other. This will avoid the definability issue and retain most of the previous validities. The main results are axiomatizations over various proper bi-frame classes. Finally we apply our framework to analyze Fine’s results.

在最近的一篇论文中，Kit Fine 提出了一些关于（一阶）无知、二阶无知和拉姆斯菲尔德无知的逻辑属性的惊人结果。然而，拉姆斯菲尔德无知是可以用无知来定义的，这使得一些现有的结果和公理化问题变得微不足道。一个主要原因是无知和拉姆斯菲尔德无知的封装算子中包含的隐式知识算子的可达性关系是相同的。在这项工作中，我们假设两种可访问性关系不同，因此其中一个是另一个的任意子集。这将避免可定义性问题并保留大部分先前的有效性。主要结果是各种适当的双框架类的公理化。最后我们应用我们的框架来分析 Fine 的结果。