Reply to Crupi et al.'s ‘Bayesian Confirmation by Uncertain Evidence’ ([2008])

Crupi et al. ([2008]) propose a generalization of Bayesian confirmation theory that they claim to adequately deal with confirmation by uncertain evidence. Consider a series of points of time t0,..., ti,..., tn such that the agent’s subjective probability for an atomic proposition E changes from Pr0(E) att0 to... to Pri(E) atti to... to Prn(E) attn. It is understood that the agent’s subjective probabilities change for E and no logically stronger proposition, and that the agent updates her subjective probabilities by Jeffrey conditionalization. For this specific scenario the authors propose to take the difference between Pr0(H) andPri(H) as the degree to which E confirms H for the agent at time ti (relative to time t0), C0,i(H, E). This proposal is claimed to be adequate, because C0,i (H, E) < C0,n(H, E) if both Pr0(E) < Pri (E) < Prn(E) andPr0(H) < Pr0(H|E). The authors show the last proposition to hold for all ‘Pr-incremental ’ measures of confirmation C0,i(H, E), that is, all functions that depend only on Pr0(H) andPri(H) and that are increasing in Pri(H) and non-decreasing in Pr0(H). Examples include the distance measure, the ratio measure, the odds or log-likelihood ratio measure and the normalized distance measure (Crupi et al. [2008], Section 2). I agree that, from a Bayesian point of view, the authors ’ proposal adequately deals with confirmation by uncertain evidence. In fact, for the specific scenario described above, this is what I claim myself in section 11, p. 111ff, of my ([2005]) 1, even though I arrive at this conclusion in a somewhat different way. However, the account by Crupi et al. ([2008]) is more general than my stance on 1 There is an unfortunate typographical error at the bottom of p. 112, where the probability measure Pri should not be conditional on E. However, this is clear from what I say on the pages following that paragraph and does not seem to have misled the authors.

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• I.—studies in the Logic of Confirmation (I.)

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