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

## TL;DR

This paper characterizes the modal logic governing $\sigma$-centered forcing extensions in set theory, establishing it as exactly the logic $ extsf{S4.2}$, and extends the result to related forcing classes.

## Contribution

It proves that the ZFC-provable principles of $\sigma$-centered forcing correspond precisely to the modal logic $ extsf{S4.2}$, and generalizes this to other forcing classes.

## Key findings

- Modal logic of $\sigma$-centered forcing is $ extsf{S4.2}$.
- Results extend to other classes of forcing.
- Provides a modal logic framework for forcing principles.

## 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.