A Logic Related to Minimal Knowledge

We introduce and study a modal logic wK4F which is closely related to the logic S4F that is important in the context of epistemic logics for representing and reasoning about an agent’s knowledge. It is obtained by adding the axiom F to the modal logic wK4, or dropping from S4F the T axiom. We show that wK4F is sound and complete with respect to the class of all minimal topological spaces i.e. topological spaces with only three open sets. We characterize the rooted frames of the logic wK4F by quadruples of natural numbers and in the same fashion we give a characterization of p-morphic images of rooted wK4F frames.