Abstract

This paper poses a problem for Lewis’ Principal Principle in a subjective Bayesian framework: we show that, where chances inform degrees of belief, subjective Bayesianism fails to validate normal informal standards of what is reasonable. This problem points to a tension between the Principal Principle and the claim that conditional degrees of belief are conditional probabilities. However, one version of objective Bayesianism has a straightforward resolution to this problem, because it avoids this latter claim. The problem, then, offers some support to this version of objective Bayesianism.

Highlights

  • This paper poses a problem for Lewis’ Principal Principle in a subjective Bayesian framework: we show that, where chances inform degrees of belief, subjective Bayesianism fails to validate normal informal standards of what is reasonable

  • That C1–4 are inconsistent poses a problem for subjective Bayesianism, which is unable to capture normal informal standards of what is reasonable when applying the Principal Principle

  • This is because the problem can be attributed to CBCP, which is a principle that subjective Bayesianism endorses but objective Bayesianism need not endorse

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Summary

The problem

Let the belief function B specify the degrees of belief of a particular agent: BE(A) is the degree to which she believes A, supposing only E, for all propositions A and E. In the context of CBCP and Bayesian Conditionalisation, the Principal Principle implies that at time t, if one’s evidence includes the proposition that the current chance of A is x one should believe A to degree x, as long as one’s other evidence E does not include anything that defeats this ascription of rational belief. It is at least permissible that: C3: P (A|XEF ) = 0.7 This is a small step from C1, we do not have enough information to tell whether the Principal Principle forces a credence of 0.7 in A because we do not know whether EF is admissible with respect to proposition A. To maintain otherwise—i.e., to maintain that one’s degree of belief in rain in Abergwyngregyn ought to be influenced at least as much by one’s prior degree of belief

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An objective Bayesian resolution
Objective
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Any way out?
Attack the conditions
Advocate a conditional principal principle
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Move to imprecise probability
Conclusion
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