# The Relationship between Knowledge, Belief and Certainty

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.

## Authors

• 74 publications
• ### Belief Induced by the Partial Knowledge of the Probabilities

We construct the belief function that quantifies the agent, beliefs abou...
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• ### Probably Approximately Knowing

Whereas deterministic protocols are typically guaranteed to obtain parti...
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• ### Syntax-based Default Reasoning as Probabilistic Model-based Diagnosis

We view the syntax-based approaches to default reasoning as a model-base...
02/27/2013 ∙ by Jerome Lang, et al. ∙ 0

• ### Towards a General-Purpose Belief Maintenance System

There currently exists a gap between the theories proposed by the probab...
03/27/2013 ∙ by Brian Falkenhainer, et al. ∙ 0

• ### A Logical Characterization of Iterated Admissibility

Brandenburger, Friedenberg, and Keisler provide an epistemic characteriz...
06/23/2009 ∙ by Joseph Y. Halpern, et al. ∙ 0

• ### A General Non-Probabilistic Theory of Inductive Reasoning

Probability theory, epistemically interpreted, provides an excellent, if...
03/27/2013 ∙ by Wolfgang Spohn, et al. ∙ 0