The Modal Logic S4F, the Default Logic, and the Logic Here-and-There

The modal logic S4F provides an account for the default logic of Reiter, and several modal nonmonotonic logics of knowledge and belief. In this paper we focus on a fragment of the logic S4F concerned with modal formulas called modal defaults, and on sets of modal defaults — modal default theories. We present characterizations of S4F-expansions of modal default theories, and show that strong and uniform equivalence of modal default theories can be expressed in terms of the logical equivalence in the logic S4F. We argue that the logic S4F can be viewed as the general default logic of nested defaults. We also study special modal default theories called modal programs, and show that this fragment of the logic S4F generalizes the logic here-and-there.