The Relationship between Knowledge, Belief and Certainty

TL;DR

This paper explores the logical relationship between knowledge, belief, and certainty, showing how different modal logics can model these concepts under various probabilistic assumptions.

Contribution

It provides a comprehensive logical framework connecting probabilistic belief and certainty, extending modal logic axiomatizations to various probabilistic structures.

Findings

KD45 logic characterizes certainty with fixed probability.

S5 logic applies when all worlds have positive probability.

Various modal logics can model different probabilistic assumptions.

Abstract

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.