Knowledge and the Problem of Logical Omniscience

The notion of knowledge has recently acquired a great deal of importance in Computer Science, partly because of its importance in AI and expert systems, but also because of applications in distributed computing and possible, even likely relevance to cryptography. See [H2] for a representative collection of papers on knowledge. An important outstanding problem in the logic of knowledge is the problem of logical omniscience, which plagues all the current logics of knowledge that are founded on the notion of a possible world. We assume in these logics the notion of a world, or situation which is possible relative to an individual (knower) i. The individual i knows a fact B iff B is true in all the worlds which are possible for i. It follows that if i knows A and also knows A → B, then both A and A → B are true at all worlds possible for i and hence B is true at all the worlds possible for i so that i knows B. In particular, i knows all B which are logically valid. This principle, the principle of logical omniscience, is usually expressed as the formula