Reflexive-insensitive logics, the box-dot translation, and the modal logic of generic absoluteness

In this paper we analyze the connection between reflexive-insensitive modal logics, logics of provability, and the modal logic of forcing. Because of the inter-definability of the ∘-operator that characterizes the reflexive-insensitive logics and Boolos’s ⊡-operator, characterization results for the reflexive-insensitive logics can be used to give a partial solution of the boxdot conjecture, and to illuminate some closely related problems. In turn, this will facilitate the application of reflexive-insensitive logics to the study of the modal logic of forcing and truth and the logic of forcing persistent sentences. We then show that the modal logic of forcing, the modal logic of forcing and truth, and the modal logic of forcing persistency give rise to the same characterizability results. We conclude by arguing that the modal logic of forcing should be better understood as the modal logic of generic absoluteness.