Bonzio, Stefano and Fano, Vincenzo and Graziani, Pierluigi
(2022)
A LOGICAL MODELING OF SEVERE IGNORANCE.
[Preprint]
Abstract
In the logical context, ignorance is traditionally defined recurring to epistemic logic $S_4$ \cite{Hintikka1962}. In particular, an agent ignores a formula $\varphi$ when s/he does not know neither $\varphi$ nor its negation $\neg\varphi$: $\neg\K\varphi\land\neg\K\neg\varphi$ (where $\K$ is the epistemic operator for knowledge). In other words, ignorance is essentially interpreted as ``lack of knowledge''. \textcolor{red}{This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of $\varphi$ that stems from an unfamiliarity with its meaning.} Contrarily to this trend, in this paper, we introduce and investigate a modal logic having a primitive epistemic operator $\I$, modeling ignorance. Our modal logic is essentially constructed on the modal logics based on weak Kleene three-valued logic introduced by Krister Segerberg \cite{Segerberg67}. Such non-classical propositional basis allows to define a Kripke-style semantics with the following, very intuitive, interpretation: a formula $\varphi$ is ignored by an agent if $\varphi$ is neither true nor false in every world accessible to the agent. As a consequence of this choice, we obtain \textcolor{red}{a type of content-theoretic} notion of ignorance, which is essentially different from the traditional approach based on $S_4$. \textcolor{red}{We dub it \emph{severe ignorance}.} We axiomatize, prove completeness and decidability for the logic of reflexive (three-valued) Kripke frames, which we find the most suitable candidate for our novel proposal and, finally, compare our approach with the most traditional one.
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