A logic for inductive probabilistic reasoning

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from '70% of As are Bs" and "a is an A" infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey's rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation satisfies that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone.