The Modal Logic of Generic Multiverses

In this thesis, we investigate the modal logic of forcing and the modal logic of grounds of generic multiverses. Hamkins and Löwe showed that the ZFC-provable modal principles of forcing, as well as of grounds, are exactly the theorems of the modal logic S4.2 (see [16],[17]). We prove that the modal logic of forcing of any generic multiverse is also exactly S4.2 by showing that any model of ZFC has a ground whose modal logic of forcing is S4.2. Moreover, we show that the modal logic of grounds of any generic multiverse is contained in S4.2Top. In particular, this implies that the modal logic of grounds of any generic multiverse with a bedrock is exactly S4.2Top. Furthermore, we show that the modal logic of any generic multiverse obtained by forcing with a progressively closed class product satisfying certain de nability conditions the only method known to us to produce multiverses without a bedrock is contained in S5.