Structural connections between a forcing class and its modal logic
Joel David Hamkins, George Leibman, Benedikt Löwe

Abstract:
Every denable forcing class Gamma gives rise to a corresponding
forcing modality, for which Box_Gamma phi means that phi is true in
all Gamma extensions, and the valid principles of Gamma forcing are
the modal assertions that are valid for this forcing
interpretation. For example, [9] shows that if ZFC is consistent, then
the ZFC-provably valid principles of the class of all forcing are
precisely the assertions of the modal theory S4.2. In this article, we
prove similarly that the provably valid principles of collapse
forcing, Cohen forcing and other classes are in each case exactly
S4.3; the provably valid principles of c.c.c. forcing, proper forcing,
and others are each contained within S4.3 and do not contain S4.2; the
provably valid principles of countably closed forcing, CH-preserving
forcing and others are each exactly S4.2; and the provably valid
principles of omega_1-preserving forcing are contained within
S4.tBA. All these results arise from general structural connections we
have identied between a forcing class and the modal logic of forcing
to which it gives rise.