Modus Ponens and Modus Tollens for Conditional Pro

Authors: Jordan Howard Sobel

Publish Date: 2007/09/28

Volume: 66, Issue: 2, Pages: 103-148

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Abstract

There are narrowest bounds for Ph when Pe  =  y and Ph/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 ‘Jeffreyconditioning’ 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∼h when P∼e =  y and Pe/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 randomchance theory of the apparent finetuning for life of the parameters of standard physics – merely on the ground that the theory made extremely unlikely conditions of which we are now nearly certain These theorems for bounds and updating are addressed to standard conditional probabilities defined as ratios of probabilities Adaptations for HosiassonLindenbaum ‘freestanding’ conditional probabilities are provided The extended online version of this article URL http//wwwscarutorontoca/~sobel/UNCERTAINEVIDpdf includes appendices and expansions of several notes Appendix A contains demonstrations and confirmations of elements of those adaptations Appendix B discusses and elaborates analogues of modus ponens and modus tollens for probabilities and conditional probabilities found in Elliott Sober’s “Intelligent Design and Probability Reasoning” Appendix C adds to observations made below regarding relations of Probability Kinematics and updating subject to Bounds for Probable Modus Ponens

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