A logic with approximate conditional probabilities that can model default reasoning

The paper presents the proof-theoretical approach to a probabilistic logic which allows expressions about (approximate) conditional probabilities. The logic enriches propositional calculus with probabilistic operators which are applied to propositional formulas: CP⩾s(α,β), CP⩽s(α, β) and CP≈s(α,β), with the intended meaning “the conditional probability of α given β is at least s”, “at most s” and “approximately s”, respectively. Possible-world semantics with a finitely additive probability measure on sets of worlds is defined and the corresponding strong completeness theorem is proved for a rather simple set of axioms. This is achieved at the price of allowing infinitary rules of inference. One of these rules enables us to syntactically define the range of the probability function. This range is chosen to be the unit interval of a recursive non-archimedean field, making it possible to express statements about approximate probabilities. Formulas of the form CP≈1(α, β) may be used to model defaults. The decidability of the logic is proved.