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Measure-Free Conditioning, Probability and Non-Monotonic Reasoning

Recent results in the foundations of probability theory indicate that a conditional probability can be viewed as a probability attached to a mathematical entity called a measure-free conditional. Such a measure-free conditional can receive a semantics in terms of a trivalent logic and logical operations are defined on conditionals in terms of truth-tables. It is shown that these results can be useful to justify Cox's axiomatic framework for probability as well as its application to other theories of uncertainty (Shafer's plausibility functions and Zadeh's possibility measures). Moreover it is shown that measure-free conditionals have the properties of well-behaved non-monotonic inference rules.