Amserdam

 If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame.  We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases.  Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing:  in this case, we are looking at the so-called "generic multiverse".