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Generalizing probabilistic material implication and Bayesian conditionals
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DOI：10.1016/j.ijar.2023.109021.png)
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Abstract
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Conditional statements in natural language of the form if A then B have multiple interpretations that require different logical treatment. In this paper, we focus on probabilistic if A then B rules that are given either a Bayesian interpretation via conditional probabilities P(B | A), or couched as probabilistic material implication. While some have argued that Bayesian conditionals are the correct way to think about such rules, 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 implication, in particular, various bounds as well as properties of inference chains in graphs. And most importantly we provide a novel and natural interpretation of the generalized implication's main parameter. (c) 2023 Elsevier Inc. All rights reserved.
Keywords:
Probabilistic logic
Interval probabilities
Inferences and bounds
Bayesian conditional and material
implication
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