Introduction

One quite surprising development in modal logic has been the development of systems in which there are possible worlds in which some sentences are both true and false. This development has been greeted with incredulous stares of such intensity that the stares directed at Lewis's claims become shy glances. This development has been extended to the actual world, and there is a group of logicians who claim that there are some contradictions that are true, as well, of course, as being false. Note that they are not claiming that all contradictions are true, only that some are. These logicians are the dialethic logicians.

In this chapter we will consider some of these impossible worlds claims, and see what sense can be made of them. Impossible worlds appeared when doxastic logic was being discussed. There is also a group of logicians, the relevant logicians, who focused their attention on conditionals. On some relevant accounts of conditionals there are worlds in which sentences are both true and false. Conditionals have been the inspiration for many unusual things in logic, and have led to many strange uses of possible worlds. We begin with epistemic logic, move on to relevant logics, and then turn to dialethic systems of contradiction tolerant logics.

Omniscience

One of the sources of controversy in epistemic S4 is that the system describes a knower who is both logically and deductively omniscient.