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Triviality Pursuit

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Abstract

The thesis that probabilities of conditionals are conditional probabilities has putatively been refuted many times by so-called ‘triviality results’, although it has also enjoyed a number of resurrections. In this paper I assault it yet again with a new such result. I begin by motivating the thesis and discussing some of the philosophical ramifications of its fluctuating fortunes. I will canvas various reasons, old and new, why the thesis seems plausible, and why we should care about its fate. I will look at some objections to Lewis’s famous triviality results, and thus some reasons for the pursuit of further triviality results. I will generalize Lewis’s results in ways that meet the objections. I will conclude with some reflections on the demise of the thesis—or otherwise.

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Notes

  1. Page references are to Harper et al. (1981) for all articles that are cited as appearing there.

  2. I owe this point to Ned Hall.

  3. I will speak interchangeably of ‘propositions’ and ‘sentences’, and I will use both set-theoretic operations and sentential connectives, in a way that I hope is perspicuous.

  4. At the end of the paper I will briefly discuss ‘no truth value’ accounts of indicative conditionals, which seemingly evade the triviality results, suggesting that even these accounts may not evade them altogether after all.

  5. One wonders what the conditional becomes when it appears in the scope of two or more probability functions—when, for example, P assigns probabilities to various hypotheses about another function P*’s assignments! (Shades here of the contextualist and relativist literature on reported speech and eavesdropping cases.).

  6. I don't actually need the full strength of the assumption of boldness in the proof. All I need is that, for any non-trivial probability function P, and propositions A and B as described in the proof, the rule can revise P so as to give probability 1 to (A & B) − (A□ → □B), or so as to give probability one to (A & B). That is certainly weaker than boldness, but I don't know a neat way of characterizing it.

  7. As I hope is obvious, P − Q is shorthand for P & ¬Q.

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Acknowledgments

I am indebted to Ned Hall, Richard Jeffrey, David Lewis, Bas van Fraassen, and Lyle Zynda for very helpful comments on a short, early precursor to this paper. I am grateful to participants in a reading group in Leuven for discussion of a resuscitated version of it—especially Jake Chandler, Igor Douven, and David Etlin. I have recently substantially revised and expanded it. For further valuable comments on this transmogrified version, I thank especially Rachael Briggs, John Cusbert, Daniel Greco, Aidan Lyon, Daniel Nolan, Wolfgang Schwarz, and Dan Singer. Thanks also to Brett Calcott and Ralph Miles for their help.

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Hájek, A. Triviality Pursuit. Topoi 30, 3–15 (2011). https://doi.org/10.1007/s11245-010-9083-2

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