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Lewis’ Triviality for Quasi Probabilities

  • Eric RaidlEmail author
Article

Abstract

According to Stalnaker’s Thesis (S), the probability of a conditional is the conditional probability. Under some mild conditions, the thesis trivialises probabilities and conditionals, as initially shown by David Lewis. This article asks the following question: does (S) still lead to triviality, if the probability function in (S) is replaced by a probability-like function? The article considers plausibility functions, in the sense of Friedman and Halpern, which additionally mimic probabilistic additivity and conditionalisation. These quasi probabilities comprise Friedman–Halpern’s conditional plausibility spaces, as well as other known representations of conditional doxastic states. The paper proves Lewis’ triviality for quasi probabilities and discusses how this has implications for three other prominent strategies to avoid Lewis’ triviality: (1) Adams’ thesis, where the probability function on the left in (S) is replaced by a probability-like function, (2) abandoning conditionalisation, where probability conditionalisation on the right in (S) is replaced by another propositional update procedure and (3) the approximation thesis, where equality in (S) is replaced by approximation. The paper also shows that Lewis’ triviality result is really about ‘additiveness’ and ‘conditionality’.

Keywords

Probability of a conditional Conditional probability Triviality Plausibility measures Conditional plausibility space Stalnaker thesis Adams’ thesis Conditional valuation functions 

Notes

Acknowledgements

I would like to thank Alan Hájek for encouraging me to publish these ideas, Arno Göbel for several discussions on this topic, Niels Skovgaard-Olsen for helpful comments, the people present at the European Epistemology Workshop 2016 in Paris and colleagues from the DFG-funded ‘What-if?’ research group in Konstanz for their comments, as well as two anonymous referees who have significantly contributed to improving the quality of the paper.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of KonstanzConstanceGermany
  2. 2.Cluster of Excellence Machine LearningUniversity TübingenTübingenGermany

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