FDE-based Modal Logics

In this paper, we explore modal logics over FDE from the point of view of both the Australian plan models with the Routley star operator and the American plan models with four semantic values. We show that, in general, the Australian plan modal logics are weaker, although the addition of a frame condition allows us to prove an equivalence between the two approaches. We then consider different S5 -type logics, using accessibility relations that are equivalence relations or universal relations, on the two different plans. We show that there are three different logics that result. Next, we provide sequent systems for all of the logics and show that two of the systems enjoy cut elimination. Then, we show soundness and completeness with respect to appropriate classes of models for all the logics. Finally, we discuss some features of the logics and close by highlighting two open problems.