The modal logic of $\sigma$-centered forcing and related forcing classes

Abstract

We consider the modality "$\varphi$ is true in every $\sigma$-centered forcing extension", denoted $\square\varphi$, and its dual "$\varphi$ is true in some $\sigma$-centered forcing extension", denoted $\lozenge\varphi$ (where $\varphi$ is a statement in set theory), which give rise to the notion of a "principle of $\sigma$-centered forcing". We prove that if ZFC is consistent, then the modal logic of $\sigma$-centered forcing, i.e. the ZFC-provable principles of $\sigma$-centered forcing, is exactly $\mathsf{S4.2}$. We also generalize this result to other related classes of forcing.