Probability of Implication , Logical

| 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 deene the probability of an implication \A if B". There exist two diierent approaches to deening this probability, and these approaches lead to diierent probabilistic inference rules: We may interpret the probability of an implication as the conditional probability P(A j B), in which case we get Bayesian inference. We may also interpret this probability as the probability of the material implication A _ :B, in which case we get diierent inference rules. In this paper, 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 terms which is normally encountered in fuzzy logic. I. INTRODUCTION Intuitively, when we say that an implication \A if B" (A B) is true, we mean that whenever B is true, we can therefore conclude that A is true as well. In other words, implication is what enables us to perform logical inference. In many practical situations, we have some conndence in B, but we are not 100% conndent that B is true. Similarly, we may not be 100% sure that the implication A B is true. In such situations, we can estimate the probability P(B) that B is true, and the probability P(A B) that an implication A B is true. How can we perform logical inference in such situations? Intuitively, we expect to be able to conclude that in this case, A should also be true with a certain probability; this probability should tend to 1 as the probabilities P(B) and P(A B) tend to 1. How can we extend logical implication to the probabilis-tic case? Depending on how we interpret the probability of an implication, we get two diierent There are two known answers to this question, and these answers are diierent because they use diierent formaliza-tions of the probability of implication. The rst answer from Bayesian approach, in which P(A B) is interpreted as the conditional probability P(A j B); see, e.g., 7]. The second answer comes from logical reasoning (see, e.g., 5]), where the probability P(A B) is interpreted as the probability of the corresponding \material implica-tion", i.e., the probability P(A_:B) …