The modal logic of forcing

@article{Hamkins2005TheML,
  title={The modal logic of forcing},
  author={Joel David Hamkins and Benedikt Loewe},
  journal={Journal of Immunology},
  year={2005}
}
A set theoretical assertion psi is forceable or possible, written lozenge psi, if psi holds in some forcing extension, and necessary, written square psi, if psi holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2. 

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