Modus Ponens and Modus Tollens for Conditional Probabilities, and Updating on Uncertain Evidence
Theory and Decision 66 (2):103 - 148 (2009)
|Abstract||There are narrowest bounds for P(h) when P(e) = y and P(h/e) = x, which bounds collapse to x as y goes to 1. A theorem for these bounds -- bounds for probable modus ponens -- entails a principle for updating on possibly uncertain evidence subject to these bounds that is a generalization of the principle for updating by conditioning on certain evidence. This way of updating on possibly uncertain evidence is appropriate when updating by ’probability kinematics’ or ’Jeffrey-conditioning’ is, and apparently in countless other cases as well. A more complicated theorem due to Karl Wagner -- bounds for probable modus tollens -- registers narrowest bounds for P(not h) when P(not e) = y and P(e/h) = x. This theorem serves another principle for updating on possibly uncertain evidence that might be termed ’contraditioning’, though it is for a way of updating that seems in practice to be frequently not appropriate. It is definitely not a way of putting down a theory -- for example, a random-chanc|
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