Logical omniscience vs. logical ignorance on a dilemma of epistemic logic

Abstract

We propose in the paper a new solution to the so-called Logical Omniscience Problem of epistemic logic. Almost all attempts in the literature to solve this problem consist in weakening the standard epistemic systems: weaker sytems are considered where the agents do not possess the full reasoning capacities of an ideal reasoner. We shall argue that this solution is not satisfactory: in this way omniscience can be avoided, but many intuitions about the concepts of knowledge and belief get lost. We shall show that axioms for epistemic logics must have the following form: if the agent knows all premises of a valid inference rule, and if she thinks hard enough, then she will know the conclusion. To formalize such an idea, we propose to “temporalize” epistemic logic, that is, to introduce a temporal component into the language. We develop a logic based on this idea and show that it is suitable for formalizing the notion of explicit belief.

I thank my teachers Peter Steinacker and Heinrich Herre for their invaluable help during the last several years. I have learned much from Werner Stelzner's works on non-modal epistemic logic. I had the chance to discuss preliminary versions of this paper with several people at different occasions. I thank all of them for their comments which have helped me very much to improve the presentation of the paper.