Reasoning about knowledge and belief: a survey

We examine a number of logics of knowledge and belief from the perspective of knowledge-based systems. We are concerned with the beliefs of a knowledge-based system, including both the system's base set of beliefs–those garnered directly from the world–and beliefs that follow from the base set. Three things to consider with such logics are the expressive power of the language of the logic, the correctness and completeness of the inferences sanctioned, and the speed with which it is possible to determine whether a given sentence is believed. The influential possible worlds approach to representing belief has the property of logical omniscience, which makes for inferences that are unacceptable in the context of belief and may take too much time to make. We examine a number of weak logics which attempt to deal with these problems. These logics divide into three categories: those that admit incomplete or inconsistent situations into their semantics, those that posit a number of distinct states for a believer which correspond roughly to frames of mind, and those that incorporate axioms or other syntactic entities directly into the semantics. As to expressive power, we consider whether belief should be represented by a predicate or a sentential operator and examine the boundary between self-referential and inconsistent systems. Finally, we consider logics of believing only , which add the assumption that a system's base set of beliefs are, in a certain sense, all that it believes.