Kyburg, Levi, and Petersen

Abstract

In this paper I attempt to tie together a longstanding dispute between Henry Kyburg and Isaac Levi concerning statistical inferences. The debate, which centers around the example of Petersen the Swede, concerns Kyburg's and Levi's accounts of randomness and choosing reference classes. I argue that both Kyburg and Levi have missed the real significance of their dispute, that Levi's claim that Kyburg violates Confirmational Conditionalization is insufficient, and that Kyburg has failed to show that Levi's criteria for choosing reference class are problematic. Rather, the significance of the Petersen case is to show that other aspects of their respective systems are defective: for Levi his account of credal judgments other than direct inference, and for Kyburg his explanation of how indexes are associated with a body of knowledge.

Direct Inference from Imprecise Frequencies

“…The method thereby goes some distance in addressing the worry that the proposed thesis yields incredulity about the value of PROB(cT) in cases where one's information concerning the possible values of freq(T|R) is modest. Taken together, the proposed method and the proposed thesis also yield an appropriate degree of incredulity, in the face of the sort of example introduced by Stone (1987): In the example given above, the illustrated method does not entitle one to infer that PROB(cT) = 0.5, but only that PROB(cT) is close to 0.5. The exact conclusion one is permitted to draw depends on one's information concerning the size of R and the size of R.…”

“…The present conclusion is plausible. But as Stone (1987) pointed out, the doctrine (that direct inference based on a narrower reference class yields the defeat of a direct inference based on a broader class only if the conclusions of the two direct inferences are inconsistent) permits implausible conclusions in the face of examples of the following form (assuming that the second direct inference incorporates the most precise estimate for freq(T|R) that is warranted):…”

Two Problems of Direct Inference

“…For all three accounts: (1) statistical statements of the preferred sort may take on values other than one and zero in the case of unit set reference classes, and (2) the use of known frequencies in the course of direct inference is usually permitted, since point-valued frequency statements usually entail a respective preferred statistical statement of identical (or similar) value. 13 The example presented here is adapted from (Stone 1987). Now make the further assumption that one lacks any additional information about the ACME Urn Company, about the likely distributions of balls of various colors held within urns produced by the ACME Urn Company, and, generally, any information that is relevant to the probability that b is red, that is not already implicit in (1) through (5).…”

A Formal Solution to Reichenbach's Reference Class Problem

“…I will not attempt to provide formal criteria for identifying such direct inferences here, but see Pollock (1990, 84) for examples. Stone 1987). There is another possible shortcoming of [SD] and [RD], namely: It is plausible that direct inferences leading to mutually consistent conclusions can interact, thereby resulting in situations where one or both of the direct inferences are defeated (cf.…”

“…There is another possible shortcoming of [SD] and [RD], namely: It is plausible that direct inferences leading to mutually consistent conclusions can interact, thereby resulting in situations where one or both of the direct inferences are defeated (cf. Stone 1987). For example, suppose we are in a position to make direct inferences in accord with the following (generalized) instances of [DI], in the case where R′ is a proper subset of R: 4 c∈R and freq TjR ð Þ∈ 0; 0:6 ½ is a reason for concluding that PROB c∈T ð Þ∈ 0; 0:6 ½ :…”