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 contenttheoretic 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 threevalued logic introduced by Krister Segerberg \cite{Segerberg67}. Such nonclassical propositional basis allows to define a Kripkestyle 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 contenttheoretic} 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 (threevalued) 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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