Fitchean Ignorance and First-order Ignorance: A Neighborhood Look

TL;DR

This paper studies different forms of ignorance using neighborhood semantics, providing new axiomatizations, expressivity results, and dynamic extensions with public announcements, to better understand and avoid the reduction of higher-order ignorance.

Contribution

It introduces a neighborhood semantics framework treating Fitchean and first-order ignorance as primitive, with new model-theoretic results and dynamic extensions.

Findings

Neighborhood semantics for ignorance forms

Axiomatizations and expressivity results

Dynamic updates with public announcements

Abstract

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.