Modus Ponens and Modus Tollens for Conditional Probabilities, and Updating on Uncertain Evidence

Theory and Decision 66 (2):103 - 148 (2009)
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
Keywords sentential probability logic  Jeffrey-conditioning  hyperreal probabilities  fine-tuning
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DOI 10.1007/s11238-007-9072-0
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Elliott Sober (2002). Intelligent Design and Probability Reasoning. International Journal for Philosophy of Religion 52 (2):65-80.

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