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(∼h) when P(∼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-chance theory of the apparent fine-tuning 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 Hosiasson-Lindenbaum ‘free-standing’ conditional probabilities are provided. The extended on-line version of this article (URL: http://www.scar.utoronto.ca/~sobel/UNCERTAINEVID.pdf) 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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References
Armendt B. (1980) Is there a Dutch Book Argument for probability kinematics. Philosophy of Science 47, 583–588
Bradley R. (2005), Radical Probabilism and Bayesian conditioning. Philosophy of Science 72, 342–364
Hailperin T. (1996), Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications. Lehigh University Press, Bethlehem
Henle, J.M. and Kleinberg, E.M. (1979), Infinitesimal Calculus, Cambridge, Mass.
Hosiasson-Lindenbaum, J. (1940), On Confirmation, Journal of Symbolic Logic 5, 133–148.
Jeffrey R.C. (1990) The Logic of Decision: Second Edition (paperback edition). University of Chicago Press, Chicago
Jeffrey, R. (1992), Probability and the Art of Judgment, University Press, Cambridge. (References are to “Conditioning, Kinematics, and Exchangeability,” reprinted with corrections from Causation, Chance, and Credence, W. Harper and B. Skyrms, editors, Volume 1, 1988, Kluwer.)
Jeffrey R. (2004). Subjective Probability: The Real Thing. University Press, Cambridge
Shimony A. (1999). Can the fundamental laws of nature be the results of evolution?. In: Butterfield J., Pagonis C. (eds). From Physics to Philosophy. Cambridge, Cambridge University Press.
Smolin L. (1997). The Life of the Cosmos. Oxford University Press, Oxford
Sobel J.H. (1987). Self-doubts and Dutch Strategies. Australasian Journal of Philosophy 65, 56–81 (Linked to http://www.scar.utoronto.ca/~sobel/.)
Sobel J.H. (1997) On the significance of conditional probabilities. Syntheses 109, 311–344 (Linked to http://www.scar.utoronto.ca/~sobel/.)
Sobel J.H. (2003), The design inference: Eliminating chance through small probabilities, by W.A. Dembski. Mind 112, 521–525
Sobel J.H. (2004) Logic and Theism: Arguments For and Against Beliefs in God. University Press, Cambridge
Sober E. (2002), Intelligent Design and Probability Reasoning. International Journal for Philosophy of Religion 52, 65–80
Wagner C.G. (1992), Generalized probability kinematics. Erkenntnis 36, 245–257
Wagner C.G. (2002), Probability kinematics and commutativity. Philosophy of Science 69, 266–278
Wagner C.G. (2004) Modus Tollens Probabilized. British Journal of Philosophy of Science 55: 747–753
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Sobel, J.H. Modus Ponens and Modus Tollens for Conditional Probabilities, and Updating on Uncertain Evidence. Theory Decis 66, 103–148 (2009). https://doi.org/10.1007/s11238-007-9072-0
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DOI: https://doi.org/10.1007/s11238-007-9072-0