Cylindric Modal Logic

The formalism of cylindric modal logic can be motivated from two directions. In its own right, it forms an interesting bridge over the gap between propositional formalisms and first-order logic, in that it formalizes first-order logic as if it were a modal formalism: The assignments of first-order variables can be seen as states or possible worlds of the modal formalism, and the quantifiers ∃ and ∀ may be studied as special cases of the modal operators ♢ and ☐, respectively. Elaborating this idea, we find that from this modal viewpoint, the standard semantics of first-order logic corresponds to just one of many possible classes of Kripke frames, and that other classes might be of interest as well.