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.

中文翻译：

---

中心强迫和相关强迫类别的模态逻辑

我们考虑“$\varphi $在每一处都是真实的$\西格玛$-居中的强迫延伸，”表示$\方\varphi $, 及其对偶“$\varphi $在某些情况下是正确的$\西格玛$-居中的强迫延伸，”表示$\菱形\varphi $（在哪里$\varphi $是集合论中的一个陈述），它产生了一个概念原则$\西格玛$-居中强迫. 我们证明如果 ZFC 是一致的，那么模态逻辑$\西格玛$- 中心强迫，即 ZFC 可证明的原则$\西格玛$- 中心强迫，正是$\mathsf {S4.2}$. 我们还将这个结果推广到其他相关的强迫类别。