A probabilistic approach to default reasoning

A logic is defined which in addition to propositional calculus contains several types of probabilistic operators which are applied only to propositional formulas. For every s ∈ S, where S is the unit interval of a recursive nonarchimedean field, an unary operator P≥s(α) and binary operators CP=s(α, β) and CP≥s(α, β) (with the intended meaning ”the probability of α is at least s”, ”the conditional probability of α given β is s, and ”the conditional probability of α given β is at least s”, respectively) are introduced. Since S is a non-archimedean field, we can also introduce a binary operator CP≈1(α, β) with the intended meaning ”probabilities of α ∧ β and β are infinitely close”. Possible-world semantics with a probability measure on an algebra of subsets of the set of all possible worlds is provided. A simple set of axioms is given but some of the rules of inference are infinitary. As a result we can prove the strong completeness theorem for our logic. Formulas of the form CP≈1(α, β) can be used to model default statements. We discuss some properties of the corresponding default entailment. We show that, if the language of defaults is considered, for every finite default base our default consequence relation coincides with the system P. If we allow arbitrary default bases, our system is more expressive than P. Finally, we analyze properties of the default consequence relation when we consider the full logic with negated defaults, imprecise observation etc.