Fix some set of propositional atoms $\mathsf{Atoms}$ and consider a (very simple) propositional logic with a binary compatibility modality $\mathsf{C}$ as follows: $$ \phi,\phi' ::= A,B,C\in\mathsf{Atoms} \mid \phi\lor\phi' \mid \mathsf{C}(\phi,\phi') . $$
- $\mathsf{C}(\phi,\phi')$ is read as "$\phi$ and $\phi'$ are compatible".
- The intuition of $\mathsf{C}(\phi,\phi')$ is that $\phi$ and $\phi'$ might be true at the same time in some possible world (and so are "compatible").
- If the reader is familiar with modal possibility $\Diamond$, then we can think of $\mathsf{C}(\phi,\phi')$ as expressing $\Diamond(\phi\land\phi')$ (we do not need to include $\Diamond$ or $\land$ in the logic above).
This logic has intended semantics as follows:
In topological spaces, write $[[-]]$ for the denotation. Then propositional atoms denote open sets, $\lor$ denotes sets union, and $[[\mathsf{C}(\phi,\phi')]]$ is the whole space if $[[\phi]]$ and $[[\phi']]$ intersect, and is empty otherwise.
In Kripke models, $w\vDash\mathsf{C}(\phi,\phi')$ --- in words: "$\mathsf{C}(\phi,\phi)$ is valid at world $w$" --- when there exists some future world $w\to w'$ such that $w'\vDash \phi$ and $w'\vDash \phi'$.
Question: Has a modality like $\mathsf{C}$ been studied, and if so, where?
Thank you.