Modal Logic equipped with "$\overline{\Diamond}$"

Question

I am looking for some bibliography or works dealing with a modal operator $\overline{\Diamond}$ whose interpretation over a Kripke model would be $x\models\overline{\Diamond}\varphi$ iff there exists $y$ not accessible from $x$ such that it satifies $\varphi$.

Answer 1

The usual way to handle these "derived relations" is by adding additional relations to our frame, and working in a multimodal logic.

We can encode your desired behavior by adding an additional relation $\overline{R}$ to our kripke frame, and specifying that $(a,b) \in \overline{R} \iff (a,b) \not \in R$. Then we interpret $\overline{\lozenge} \varphi$ as a usual "diamond operator", where we use $\overline{R}$ instead of $R$.

Practically all of modal logic goes through in these multimodal settings. If you want a book written in this level of generality, you might consider Blackburn, de Rijke, and Venema's Modal Logic.

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I hope this helps ^_^