Re-interpreting the Modal -calculus

We reexamine the modal-calculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with xed points. We propose interpreting formulas into a eld of subsets of states instead of the full power set lattice used by Kozen. Under this interpretation we relate image compact modal frames with Scott continuity of the box modality, m-saturated transition systems and descriptive modal frames. Also, it is shown that the class of image compact modal frames satisses the Hennessy-Milner property. We conclude by showing that for descriptive modal-frames the standard interpretation coincides with the one we proposed.