A Logic with Conditional Probabilities

Abstract

The paper presents a logic which enriches propositional calculus with three classes of probabilistic operators which are applied to propositional formulas: P _≥ s(α), 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. Possible-world semantics with a 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 nonarchimedean field, making it possible to define another probabilistic operator CP _≈ 1(α, β) with the intended meaning ”probabilities of α ∧ β and β are almost the same”. This last operator may be used to model default reasoning.