Probability of implication, logical version of Bayes theorem, and fuzzy logic operations

Logical inference starts with concluding that if B implies A, and B is true, then A is true as well. To describe probabilistic inference rules, we must therefore define the probability of an implication "A if B". There exist two different approaches to defining this probability, and these approaches lead to different probabilistic inference rules: we may interpret the probability of an implication as the conditional probability P(A|B), in which case we get Bayesian inference. We may also interpret this probability as the probability of the material implication A /spl or/ /spl not/ B in which case we get different inference rules. We develop a general approach to describing the probability of an implication, and we describe the corresponding general formulas, of which Bayesian and material implications are particular cases. This general approach is naturally formulated in terms of t-norms, a term which is normally encountered in fuzzy logic.