The mathematical multiverse could be viewed as a gigantic Kripke model with models of $ZFC$ as *possible worlds* connected to each other via a certain *accessibility* relation.

The modal logic associated with this multiverse has been studied in the case that the accessibility relation is *set forcing*. According to this interpretation, a statement $\varphi$ in the language of set theory is considered *possible* if it is true in *some* set forcing extension and *necessary* if holds in all of them. For more details see:

Hamkins, Joel David; Löwe, Benedikt,The modal logic of forcing, Trans. Am. Math. Soc. 360, No. 4, 1793-1817 (2008). ZBL1139.03039.

Hamkins, Joel David,The set-theoretic multiverse, Rev. Symb. Log. 5, No. 3, 416-449 (2012). ZBL1260.03103.

However, set forcing extension is just a special case of interesting and well-behaved extensions that a model of $ZFC$ might have. If $M$ is an inner model of $N$ then we call $N$ an **outer model** of $M$. Recall that any model of $ZFC$ could be considered an *inner model* of its generic extension and so a forcing extension is a special case of an *outer model*. Of course, there might be outer models of a given model of $ZFC$ which don't arise as a set forcing extension of the ground. A classical example is $L\subseteq L[0^{\sharp}]$.

Now let's ease the conditions in the above interpretations of the possibility and necessity across the multiverse so that $\varphi$ is *possible* if it is true in some *outer model* and is *necessary* if it holds in all of them. Philosophically speaking, in such an interpretation of modalities, a statement is possible if it could be realized by expanding our world-view and is necessary if it holds no matter how we expand our universe. (Here, Plato's Allegory of the Cave might be relevant where the cave or *tight universe* is our ground model).

In the view of the fact that the modal logic of set forcing multiverse is fully determined by Hamkins and Löwe to be S4.2 (under certain conditions), I wonder what could be said about the modal logic of outer model multiverse.

Question.What is the modal logic of theouter multiverse; the set-theoretic multiverse equipped with theouter modelrelation (rather than set forcing) as the accessibility bridge between the universes?