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We consider the modality "$\varphi$ is true in every $\sigma$-centeredforcing extension", denoted $\square\varphi$, and its dual "$\varphi$ is truein 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-provableprinciples of $\sigma$-centered forcing, is exactly $\mathsf{S4.2}$. We alsogeneralize this result to other related classes of forcing.

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