Bayesian networks for logical reasoning

Abstract

The Bayesian network formalism provides a robust framework for reasoning about logical deductions by identifying deep analogies between causal and logical influence. Although deductive logic traditionally addresses propositions with certain truth values, actual logical reasoning—such as mathematical conjecture evaluation and automated theorem proving—operates within contexts of significant uncertainty. Logical implication constitutes an influence relation that satisfies the same independence and irrelevance conditions as causality, justifying the application of Bayesian network algorithms to logical structures. Because logical proof graphs are typically sparse, reflecting the limited number of premises in standard inference rules, these networks offer a computationally efficient means of representing and propagating degrees of belief across a web of related propositions. This methodology enables the representation of probability distributions over clauses in logic programs and provides a structured approach to proof planning under uncertainty regarding the correct logical path. Integrating probabilistic influence into logical frameworks thus allows for the quantitative evaluation of evidence and the prioritization of search strategies in complex reasoning tasks. – AI-generated abstract.