Modal Logics and Intermediate Logics Motivated by an Open Problem on c.c.c. Forcing

Abstract

We investigate three intermediate logics and their modal counterparts that play an important role in the study of modal logics of forcing classes. Hamkins, Leibman and Löwe have conjectured that S4.tBAis an upper bound of the modal logic of c.c.c. forcing; Inamdar has proved that S4.sBAis such an up-per bound; in Chapter 3 of our thesis, we prove under additional assumptions that S4.FPFA, a modal logic that is contained in the intersection of the two other logics, is an upper bound. We do not know whether our additional assumptions are true; if so, our result proves the conjecture by Hamkins, Leibman and Löwe.

The remaining chapters of the thesis study the six mentioned logics in more detail. In Chapter 4 we prove two conjectures by Nick Bezhanishvili on generalized Medvedev logics; in Chapter 5 we connect the concept of nerve to Medvedev logic; in Chapter 6 we prove that S4.SBAis not finitely axiomatisable over Cheq; and finally, in Chapter 7 we prove that Cheqis not axiomatisable with five or six variables.