  We consider the relation between knowledge and certainty, where a fact is
known if it is true at all worlds an agent considers possible and is certain if
it holds with probability 1. We identify certainty with probabilistic belief.
We show that if we assume one fixed probability assignment, then the logic
KD45, which has been identified as perhaps the most appropriate for belief,
provides a complete axiomatization for reasoning about certainty. Just as an
agent may believe a fact although phi is false, he may be certain that a fact
phi, is true although phi is false. However, it is easy to see that an agent
can have such false (probabilistic) beliefs only at a set of worlds of
probability 0. If we restrict attention to structures where all worlds have
positive probability, then S5 provides a complete axiomatization. If we
consider a more general setting, where there might be a different probability
assignment at each world, then by placing appropriate conditions on the support
of the probability function (the set of worlds which have non-zero
probability), we can capture many other well-known modal logics, such as T and
S4. Finally, we consider which axioms characterize structures satisfying
Miller's principle.
