Fitchean Ignorance and First-order Ignorance: A Neighborhood Look
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In a seminal work <cit.>, 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 the notions, which includes that in 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. We can see that even under neighborhood semantics, there is also interesting relationship among the aforementioned four notions. The main contributions include axiomatizations and model-theoretical results such as expressivity and frame definability.
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