Abstract
This paper defends the claim that there is a deep tension between the principle of countable additivity and the one-third solution to the Sleeping Beauty problem. The claim that such a tension exists has recently been challenged by Brian Weatherson, who has attempted to provide a countable additivity-friendly argument for the one-third solution. This attempt is shown to be unsuccessful. And it is argued that the failure of this attempt sheds light on the status of the principle of indifference that underlies the tension between countable additivity and the one-third solution.
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Notes
Philosophers who have argued for the one-third solution include Frank Arntzenius, Dennis Dieks, Cian Dorr, Kai Draper, Adam Elga, Christopher Hitchcock, Terrance Horgan, Bradley Monton, Robert Stalnaker, Michael Titelbaum, Ruth Weintraub, as well as the 16 members of the OSCAR Seminar (2008). Philosophers who have argued for the one-half solution include Joseph Halpern, David Lewis and Christopher Meacham. For references to the relevant literature, see Ross (2010) and Pust (forthcoming).
One reason to accept CA is that it is supported by Dutch book arguments—see Williamson (1999) and Ross (2010). Another it is that it follows from the very plausible principle of conglomerability—see Schervish et al. (1984). And a third is that it plays an important role in scientific reasoning and statistical inference—see Earman (1992) and Kelly (1996).
Weatherson’s first three assumptions are not required, since the argument I will give makes no use of the post-experimental awakening.
By the ratio analysis, the previous line entails \( {{Cr_{1} \left( {h \& W\left( {\alpha ,n} \right)} \right)} \mathord{\left/ {\vphantom {{Cr_{1} \left( {h\& W\left( {\alpha ,n} \right)} \right)} {Cr_{1} \left( {\left( {h\& W\left( {\alpha ,n} \right)} \right) \vee \left( {\neg h\& W\left( {\alpha ,1} \right)} \right)} \right)}}} \right. \kern-\nulldelimiterspace} {Cr_{1} \left( {\left( {h\& W\left( {\alpha ,n} \right)} \right) \vee \left( {\neg h\& W\left( {\alpha ,1} \right)} \right)} \right)}} = Ch\left( h \right) \). And by finite additivity, the latter entails \( {{Cr_{1} \left( {h\& W\left( {\alpha ,n} \right)} \right)} \mathord{\left/ {\vphantom {{Cr_{1} \left( {h\& W\left( {\alpha ,n} \right)} \right)} {\left( {Cr_{1} \left( {h\& W\left( {\alpha ,n} \right)} \right) + Cr_{1} \left( {\neg h\& W\left( {\alpha ,1} \right)} \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {Cr_{1} \left( {h\& W\left( {\alpha ,n} \right)} \right) + Cr_{1} \left( {\neg h\& W\left( {\alpha ,1} \right)} \right)} \right)}} = Ch\left( h \right) \). Solving for \( Cr_{1} \left( {h\& W\left( {\alpha ,n} \right)} \right) \), we obtain \( Cr_{1} \left( {h\& W\left( {\alpha ,n} \right)} \right) = {{Ch\left( h \right)Cr_{1} \left( {\neg h\& W\left( {\alpha ,1} \right)} \right)} \mathord{\left/ {\vphantom {{Ch\left( h \right)Cr_{1} \left( {\neg h\& W\left( {\alpha ,1} \right)} \right)} {\left( {1 - Ch\left( h \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - Ch\left( h \right)} \right)}} \).
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Acknowledgments
Many thanks to Mark Schroeder, Robert Stalnaker and Brian Weatherson for very helpful discussions.
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Ross, J. All roads lead to violations of countable additivity. Philos Stud 161, 381–390 (2012). https://doi.org/10.1007/s11098-011-9744-z
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DOI: https://doi.org/10.1007/s11098-011-9744-z