Unifying Probability and Logic for Learning
Marcus Hütter, Kee Siong Ng, John Lloyd, William Uther · ANU Open Research (Australian National University) · 2013
1 Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values, but so far a completely satisfactory integration of logic and probability has been lacking. In particular the inability of confirming universal hypotheses has plagued most if not all systems so far. We address this problem head on. The main technical prob-lem to be discussed is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among oth-ers, is that the probability distribution (i) is consis-tent with the knowledge base, (ii) allows for a con-sistent inference procedure and in particular (iii) re-duces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive rea-soning and (v) learning in the limit and in partic-ular (vi) allows confirmation of universally quanti-fied hypotheses/sentences. We show that probabil-ities satisfying (i)-(vi) exist, and present necessary and sufficient conditions (Gaifman and Cournot). The theory is a step towards a globally consistent and empirically satisfactory unification of probabil-ity and logic.