Lecture Notes on Noncorrespondence 15-816: Modal Logic

In lecture 7, we have seen how axiomatics and semantics of modal logic fit together in soundness proofs and correspondence proofs. We have seen several examples of classes of Kripke frames that are characterized by formulas of propositional modal logic. These were several special cases. But we are looking for a general correspondence result. Can we find a full correspondence result? For any formula of propositional modal logic, associate a class of Kripke frames that it characterizes? And for any class of Kripke frames, associate a formula of propositional modal logic that characterizes it? Certainly not! Even for Kripke frames with countable sets of worlds, classes of Kripke frames are not countable, and cannot possibly be matched with the countable set of formulas. Let us try a more modest general correspondence between propositional modal logic and first-order definable classes of Kripke frames. For any class of Kripke frames given by a first-order formula on the frame, associate a formula of propositional modal logic that characterizes it? And for any formula of propositional modal logic, associate a class of Kripke frames—defined by a formula in first-order logic—that it characterizes? It turns out that even that is impossible!