Bayesian Networks for Logical Reasoning

By identifying and pursuing analogies between causal and logical in uence I show how the Bayesian network formalism can be applied to reasoning about logical deductions. Despite the fact that logic itself is about certainty, logical reasoning takes place in a context of very little certainty. In fact the very search for a proof of a proposition is usually a search for certainty: we are unsure about the proposition and want to become sure by nding a proof or a refutation. Even the search for a better proof takes place under uncertainty: we are sure of the conclusion but not of the alternative premises or lemmas. Uncertainty is rife in mathematics, for instance. A good mathematician is one who can assess which conjectures are likely to be true, and from where a proof of a conjecture is likely to emerge | which hypotheses, intermediary steps and proof techniques are likely to be required and are most plausible in themselves. Mathematics is not a list of theorems but a web of beliefs, and mathematical propositions are constantly being evaluated on the basis of the mathematical and physical evidence available at the time. 1 Of course logical reasoning has many other applications, notably throughout the eld of articial intelligence. Planning a decision, parsing a sentence, checking a computer program, maintaining consistency of a knowledge base and deriving predictions from a model are only few of the tasks that can be considered theorem-proving problems. Finding a proof is rarely an easy matter, thus automated theorem proving and automated proof planning are important areas of active research. 2 However, current systems do not tackle uncertainty in any fundamental way. I will argue in this paper that Bayesian networks are particularly suited as a formalism for logical reasoning under uncertainty, just as they are for causal reasoning