Deduction from conditional knowledge

Abstract

We propose a Bayesian second-order probability approach to the problem of making inferences from conditional probability information. We present examples that, on the surface, appear to indicate a “disconnect” between logic and probability. These examples seem to indicate that conclusions that can be deduced when premises are completely reliable can no longer be deduced when the reliability of premises (as measured by their probability) is anything less than perfect. To overcome the resulting “deductive paralysis”, we present probabilistic forms of logic in which the goal is not to reach conclusions that are certain to be true, but rather to reach conclusions that are true “nearly always” or “on average”. In so doing, we reveal a hidden smooth connection between logic and probability. The use of fuzzy logic to model linguistic probabilities is also discussed.