Axiomatic First-Order Probability

Most languages for the Semantic Web have their logical basis in some fragment of first-order logic. Thus, integrating first-order logic with probability is fundamental for representing and reasoning with uncertainty in the semantic web. Defining semantics for probability logics presents a dilemma: a logic that assigns a real-valued probability to any first-order sentence cannot be axiomatized and lacks a complete proof theory. This paper develops a first-order axiomatic theory of probability in which probability is formalized as a function mapping Gödel numbers to elements of a real closed field. The resulting logic is fully first-order and recursively axiomatizable, and therefore has a complete proof theory. This gives rise to a plausible reasoning logic with a number of desirable properties: the logic can represent arbitrarily fine-grained degrees of plausibility intermediate between proof and disproof; all mathematical and logical assumptions can be explicitly represented as finite computational structures accessible to automated reasoners; contradictions can be discovered in finite time; and the logic supports learning from observation.