On Definability in Multimodal Logic II. Defining Knowledge in Terms of Belief∗

The question of whether knowledge is definable in terms of belief, which has played an important role in epistemology for the last fifty years, is studied here in the framework of epistemic and doxastic logics. Three notions of definability are considered: explicit definability, implicit definability, and reducibility, where explicit definability is equivalent to the combination of implicit definability and reducibility. It is shown that if knowledge satisfies any set of axioms contained in S5, then it cannot be explicitly defined in terms of belief. S5 knowledge can be implicitly defined by belief, but not reduced to it. On the other hand, S4.4 knowledge and weaker notions of knowledge cannot be implicitly defined by belief, but can be reduced to it by defining knowledge as true belief. It is also shown that S5 knowledge cannot be reduced to belief and justification, provided that there are no axioms that involve both belief and justification. ∗We thank Samson Abramsky, Arnon Avron, Johan van Benthem, Sergiu Hart, Martin Meier, and Jouku Väänänen for helpful discussions. †Supported in part by NSF under grants TR-0325453 and IIS-0534064, by FOSR under grant FA9550-05-1-0055. ‡Support by the Israeli Science Foundation under grant 891/04 is acknowledged.