Investigating a Generalization of Probabilistic Material Implication and Bayesian Conditionals

Probabilistic "if A then B" rules are typically formalized as Bayesian conditionals P(B|A), as many (e.g., Pearl) have argued that Bayesian conditionals are the correct way to think about such rules. However, there are challenges with standard inferences such as modus ponens and modus tollens that might make probabilistic material implication a better candidate at times for rule-based systems employing forward-chaining; and arguably material implication is still suitable when information about prior or conditional probabilities is not available at all. We investigate a generalization of probabilistic material implication and Bayesian conditionals that combines the advantages of both formalisms in a systematic way and prove basic properties of the generalized rule, in particular, for inference chains in graphs.