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First, let us fix a countable model of ZFC plus V=L, say, and consider it in the context of all its outer models. Since we can force so as to destroy any stationary set in this model, while preserving others, we see that there is an infinite family of independent buttons, "the $n^{th}$ stationary set in the $L$-least partition is no longer stationary." This can be made true in an outer model (by forcing) and once true, remains true in all further outer models; and the statements can be controlled independently. And since we also have a family of independent switches arising from the GCH patterns, it follows by the main modal logic analysis (as in Structural connections...), it follows that the modal logic of outer-models is contained within S4.2. And it certainly contains S4, because it is reflexive and transitive.

First, let us fix a countable model of ZFC plus V=L, say, and consider it in the context of all its outer models. Since we can force so as to destroy any stationary set in this model, while preserving others, we see that there is an infinite family of independent buttons, "the $n^{th}$ stationary set in the $L$-least partition of $\omega_1^L$ is no longer stationary." This can be made true in an outer model (by forcing) and once true, remains true in all further outer models; and the statements can be controlled independently. And since we also have a family of independent switches arising from the GCH patterns, it follows by the main modal logic analysis (as in Structural connections...), it follows that the modal logic of outer-models is contained within S4.2. And it certainly contains S4, because it is reflexive and transitive.

In that paper, we consider several different notions of set-theoretic accessibility, as below, from forcing accessibility to Grothendieck-Zermlo potentialism to rank extensions or transitive extensions or submodel potentialism and others.

If you hope to prove that only S4 is valid, then you could follow some of the recent work on the universal algorithm and universal finite sets. The general consequence of the existence of these finite sequences with the universal extension property is that they cause the existence of railyard labelings, which then cause the modal logic to be contained in S4.

  I don't know if there is any universal finite sequence phenomenon for outer models, and this also is an interesting question.

So I believe that the exact modal logic is an open question.

In that paper, we consider several different notions of set-theoretic accessibility, as below, from forcing accessibility to Grothendieck-Zermelo potentialism to rank extensions or transitive extensions or submodel potentialism and others.

If you hope to prove that only S4 is valid, then you could follow some of the recent work on the universal algorithm and universal finite sets. The general consequence of the existence of these finite sequences with the universal extension property is that they cause the existence of railyard labelings, which then cause the modal logic to be contained in S4. I don't know if there is any universal finite sequence phenomenon for outer models, and this also is an interesting question.

So I believe that the exact modal logic of outer-model potentialism is an open question.