Probabilistic Estimation for Generalized Rough Modus Ponens and Rough Modus Tollens

We review concepts and principles of Modus Ponens and Modus Tollens in the areas of rough set theory and probabilistic inference. Based on the upper and the lower approximation of a set as well as the existing probabilistic results, we establish a generalized version of rough Modus Ponens and rough Modus Tollens with a new fact different from the premise (or the conclusion) of “if ...then ...” rule, and address the problem of computing the conditional probability of the conclusion given the new fact (or of the premise given the new fact) from the probability of the new fact and the certainty factor of the rule. The solutions come down to the corresponding interval for the conditional probabilities, which are more appropriate than the exact values in the environment full of uncertainty due to errors and inconsistency existed in measurement, judgement, management, etc., plus illustration analysis.