Abstract
Accuracy-based arguments for conditionalization and probabilism appear to have a significant advantage over their Dutch Book rivals. They rely only on the plausible epistemic norm that one should try to decrease the inaccuracy of one’s beliefs. Furthermore, conditionalization and probabilism apparently follow from a wide range of measures of inaccuracy. However, we argue that there is an under-appreciated diachronic constraint on measures of inaccuracy which limits the measures from which one can prove conditionalization, and none of the remaining measures allow one to prove probabilism. That is, among the measures in the literature, there are some from which one can prove conditionalization, others from which one can prove probabilism, but none from which one can prove both. Hence at present, the accuracy-based approach cannot underwrite both conditionalization and probabilism.
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Notes
Also, Dutch book arguments arguably rest on substantial assumptions about your credences in propositions and their negations (Hedden 2013).
We introduce the Elimination constraint in Fallis and Lewis (2016). The current paper goes beyond our prior work in assessing the consequences of Elimination for proofs of probabilism and conditionalization.
There are many different notions of symmetry in the accuracy literature; this is the one Joyce (2009, p. 274) calls “0/1-symmetry”.
However, Additivity is not beyond question, as Marxen and Rothfus (2018, p. 317) point out.
Blackwell and Drucker (2019) engage in the both kinds of critique: they argue that maximizing accuracy sometimes requires one to violate conditionalization, and that the accuracy-based derivation of conditionalization is flawed.
Pettigrew (2016) argues for a flat prior on accuracy-based grounds.
One might reasonably think that acceptable measures of accuracy should obey a stronger principle than Elimination; see Fallis and Lewis (2016).
This is trivial for the log rule, and easily proven for the spherical rule. See Fallis and Lewis (2016).
Strictly, applying these rules to a Boolean algebra requires including credences in the negations \(\lnot X_{1}\), \(\lnot X_{2}\) and \(\lnot X_{3}\), plus the tautology \(X_{1} \vee X_{2} \vee X_{3}\) and the contradiction \(\lnot (X_{1} \vee X_{2} \vee X_{3})\). But for coherent credences the inaccuracies of the tautology and the contradiction are zero, and for coherent credences and symmetric rules the inaccuracy of \(\lnot X_{i}\) is the same as that of \(X_{i}\). So the inaccuracy calculated over the entire Boolean algebra is simply twice the inaccuracy over the partition \(( X_{1}, X_{2}, X_{3}) \).
Note that Blackwell and Drucker (2019) argue that Strict Propriety is a substantive epistemic assumption; certainly its typical justification is diachronic, as it appeals to the irrationality of credence shifts in the absence of new evidence (Maher 1990, p. 112). If Blackwell and Drucker are right, then probabilism cannot be proven without substantive epistemic assumptions. If we are right about Elimination, then probabilism cannot be proven with substantive epistemic assumptions.
Dunn (2019) argues that certain weighted Brier rules are good candidates for a combined measure of inaccuracy and verisimilitude. But this measure is a hopeless measure of inaccuracy alone.
We thank James Joyce for this suggestion.
See also Fallis and Lewis (2016, p. 582).
Suppose your initial credences in the partition \((AB, A{\bar{B}}, {\bar{A}}B, {\bar{A}}{\bar{B}})\) are (0.3, 0.1, 0.3, 0.3), so your credence in A is 0.4, your credence in B is 0.6, and your credence in \((A \rightarrow B)\), interpreted as a material conditional, is 0.9. If you learn A, your credences in the partition become (0.75, 0.25, 0, 0), so your credence in A is 1, your credence in B is 0.75, and your credence in \((A \rightarrow B)\) is 0.75. This does not look misleading: although your credence in the false hypothesis B has increased a little, your credence in the true hypothesis A has increased a lot, and your credence in the false conditional \((A \rightarrow B)\) has decreased.
See Knab and Schoenfield (2015) for an argument that falsity distributions are irrelevant to accuracy. Note that the simple log rule ignores falsity distributions: accuracy is a function only of credence in the true hypothesis.
Cases in which your credence in the true hypothesis is zero are exceptions, as explained shortly.
When the fourth element is true, the inaccuracy of \((1/6,\) \(1/2,\) \(1/6,\) \(1/6,\) \(0,\) \(0,\) \(0,\) \(0)\) is 1, and the inaccuracy of \((0,\) \(3/4,\) \(0,\) \(1/4,\) \(0,\) \(0,\) \(0,\) \(0)\) is 9 / 8. Dunn (2019, p. 158) defends a weighted Brier score to combine accuracy considerations with verisimilitude considerations; in cases like this he suggests that most of the weight should fall on the atomic statements (A, B, C). Initially, your credences in (A, B, C) are \((1,\) \(2/3,\) \(1/3)\), yielding a Brier score of \(5/9 = 80/144\), and after conditionalizing they are \((1,\) \(3/4,\) \(0)\), yielding a Brier score of \(9/16 = 81/144\). So whether you use a weighted or an unweighted Brier score, the score tells you that your epistemic situation has become worse.
The simple spherical rule has this consequence. The consequence entails that the epistemic benefit of eliminating a false hypothesis scales with your credence in the truth, becoming zero when your credence in the truth is zero. This seems quite intuitive. It also entails that the epistemic cost of concentrating credence on fewer false hypotheses scales with your credence in the truth, becoming zero when your credence in the truth is zero. This is less intuitive, but seems unavoidable.
Joyce (2009, p. 274) calls this principle “0/1-symmetry”.
We would like to thank Jeffrey Dunn, Kenny Easwaran, James Joyce, Brian Knab, Graham Oddie, Richard Pettigrew, Terry Horgan, the audience at the Philosophy of Science Association Biennial Meeting in Atlanta in November 2016, and two anonymous referees for very helpful comments on earlier versions of this paper.
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Lewis, P.J., Fallis, D. Accuracy, conditionalization, and probabilism. Synthese 198, 4017–4033 (2021). https://doi.org/10.1007/s11229-019-02298-3
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DOI: https://doi.org/10.1007/s11229-019-02298-3