Abstract
The article begins by describing two longstanding problems associated with direct inference. One problem concerns the role of uninformative frequency statements in inferring probabilities by direct inference. A second problem concerns the role of frequency statements with gerrymandered reference classes. I show that past approaches to the problem associated with uninformative frequency statements yield the wrong conclusions in some cases. I propose a modification of Kyburg’s approach to the problem that yields the right conclusions. Past theories of direct inference have postponed treatment of the problem associated with gerrymandered reference classes by appealing to an unexplicated notion of projectability. I address the lacuna in past theories by introducing criteria for being a relevant statistic. The prescription that only relevant statistics play a role in direct inference corresponds to the sort of projectability constraints envisioned by past theories.
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Notes
I will assume that the conclusions of direct inference are single-case probability statements whose truth conditions (or acceptability conditions) are implicitly relativized to the epistemic situation of respective agents. The proposed account of direct inference could also be formulated so that the conclusions of direct inference are tantamount to defeasible prescriptions about what an agent’s degrees of belief in given propositions should be.
The Problem of the Reference Class is often presented as a decisive objection to the objectivity of direct inference. But arguments for the claim that the problem is decisive typically go no further than describing the problem as I have here (cf. Fitelson et al. 2005; Rhee 2007). See also (Hájek 2007, pp. 568–569), where skepticism about direct inference vis-à-vis the Problem of the Reference Class is premised on idiosyncratic features of Reichenbach’s account of direct inference. A presentation of the problem by means of an interesting example can be found in (Colyvan and Regan 2001) and (Colyvan et al. 2007), though the theory of direct inference presented here, and the theory presented in (Pollock 1990), have the resources to adequately address the example.
The key elements of Reichenbach’s account of direct inference are also present in (Venn 1866).
Almost all accounts of direct inference adopt some form of this prescription, which is closely related to the principle of specificity which is an element of many approaches to defeasible reasoning in the field of artificial intelligence (cf. Horty et al. 1990; Kraus et al. 1990; Geffner and Pearl 1992). Even Salmon’s proposal that direct inferences be based on the broadest homogeneous reference class is similar to Reichenbach’s proposal inasmuch as: (1) the non-homogeneity of a proposed reference class compels one to reason from statistics for a proper subset of that reference class (if a direct inference is possible), and (2) the possession of statistics for a homogeneous subset of a proposed reference class demonstrates the non-homogeneity of the proposed reference class, if the statistics for the two sets differ (Salmon 1971).
According to Reichenbach’s official view, the statistical statements that may serve as premises for direct inference are statements of frequency in the limit. Roughly: limiting frequencies are defined relative to an infinite sequence, R, and the limiting frequency of T among R is defined as the frequency of elements of T among the first n elements of R as n approaches ∞ (provided that the frequency of T among R goes to a limit as n approaches ∞). In what follows, I will mostly ignore this detail of Reichenbach’s theory, and acknowledge Reichenbach’s official view only at relevant points.
Kyburg also showed that gerrymandered target classes are sufficient to lead to unreasonable direct inferences (Kyburg 1974). My approach to the ‘projectability’ problems associated with direct inference applies equally to problems generated by gerrymandered target classes.
Although Salmon does not discuss the problem of gerrymandered reference classes, his account of direct inference (Salmon 1971, 1977, and 1984) is adequate to address some of the projectability problems associated with direct inference. However, there is no obvious way to modify Salmon’s restrictive paradigm, which identified reference classes with infinite sequences of temporally ordered of events, in order to apply Salmon’s account of direct inference to typical cases where we would like to use frequency information about a population that is not temporally ordered to make a direct inference regarding one of its members.
Hempel’s closely related account of inductive-statistical explanation also appeals to an unexplicated notion of law-like statistical generalizations as a proxy for projectability constraints (Hempel 1968).
In (Pollock 2007) and other unpublished essays, Pollock developed an approach to direct inference that appears not to rely on projectability constraints in the same manner as his earlier approach (Pollock 1990). However, Pollock’s more recent approach still relies, at the foundational level, on a variety of direct inference (what Pollock calls “the statistical syllogism”) that invokes an unexplicated notion of projectability. Pollock’s recent work also bears similarities to the random-worlds approach to inferring single-case probabilities, as proposed in (Bacchus et al. 1996) and (Halpern 2003), inasmuch as the outputs of Pollock’s approach are sensitive to the manner in which inputs to the theory are represented. The indifference principles presented in (Pollock unpublished) are particularly suggestive of this problem.
To generate a corresponding problem for Reichenbach’s official view (which identifies reference classes with infinite sequences), we need only consider the limiting frequency for a given target property among a reference class defined by an exhaustive listing of the known properties of the individual about which we wish to make a direct inference. For such reference classes, we are rarely in a position to make an informative judgment regarding the value of the relevant limiting frequency. See (Fetzer 1977), for a discussion of the present problem in connection with a close reading of Reichenbach’s official view.
It is understood that the range of values for the broader class and for the narrower class are interval-valued.
The workability of the doctrines of Pollock and Bacchus regarding the sort of statistical statements that are appropriate to serve as premises for direct inference is similar to the doctrine of Kyburg. 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.
The example presented here is adapted from (Stone 1987).
One exception is the theory of Isaac Levi (1982). Levi proposed, roughly, that correct instances of direct inference presuppose that the object of interest, c, is presented to us as a trial of a stochastic process that generates varying results with certain chances, and that the probability we assign to c having a respective target property be identical to the chance of a trial of the respective sort (i.e., a trial which has the relevant reference property) having the respective target property. Since these preconditions are not satisfied in the ACME Urn case, Levi’s theory will not permit a direct inference. Levi’s theory sets an extremely high threshold to surpass before one is permitted to make a direct inference. I share the view of other advocates of direct inference that Levi sets the bar too high.
According to Kyburg’s theory, the direct inference based on the set of all balls held in ACME Urns goes through, since one’s frequency information regarding the set of balls held in U b (namely, that the frequency of red balls among U b is in [0, 0.51]) is less precise. The theories of Pollock and Bacchus, respectively, permit us to draw the conclusion that the nomic probability and the expected frequency of red balls among the set of all balls held in ACME Urns is 0.51. Since the theories of Pollock and Bacchus do not permit a direct inference based on the set of balls held in U b that contradicts the conclusion that the probability that b is red is 0.51, the conclusion that the probability that b is red is 0.51 goes through, according to their theories.
Suppose, for example, that “R*” is introduced as a name for the first element of some ordering, ρ, of the subsets of R that contain |R′| elements, where our agent has no information regarding the principle according to which the elements of ρ were ordered.
Here as elsewhere, I will say that c is a random element of R, if c was selected from among the elements of R, by a process that was equally likely to yield each element of R.
The example is adapted from Pollock (1990, p. 84). A second example, from Pollock, that I will not discuss concerns the inference to the conclusion that a given bird with a broken wing is likely to be able to swim the English Channel, by appeal to the statistic that most birds can fly or swim the English Channel (which is true in virtue of the fact that most birds can fly). The condition that is used to address the example presented here applies equally to the case of the bird with the broken wing.
The best way to see this point is to imagine the situation as a two-tiered lottery, where, first, the frequency of elements of T among R is selected and, next, an element of R is selected at random.
This feature is also common to all of the examples that I have been able to concoct. Many of these examples are more insidious than the examples found in the literature.
Assuming that chains of inference resemble proofs in a formal language, it is intended that membership in a set, R′, be characterized by the satisfaction of a first order formula, φ(x), with a single free variable x. We may then regard R′ as the set of objects that satisfy φ(x).
In determining the applicability of the present definition, it is assumed that beliefs about the value of an expected frequency cannot be justified on the basis of testimony (or similar means) in cases where one is aware of the basis upon which the testifier formed her judgment. In other words, the more fundamental evidence must be given priority when that evidence is available. (The present proviso is made for the purposes of the definition of relevance, and is not proposed as essential to the concept of justification.)
Because (DH∩~L12)∪{Flint} and {Flint} are informativeness incompatible, it may appear that the application of the definition of informativeness incompatibility (as a criterion for subset defeat) is sufficient to address the Problem of Relevant Statistics. In fact, applications of that definition are not sufficient to address the problem. For one, we must restrict the application of informativeness incompatibility, as a criterion for subset defeat, to cases where an agent’s statistics for a subset of a proposed reference class are relevant.
I leave the treatment of reference classes corresponding to reference properties of differing arity for another day (cf. Pust 2011; Thorn forthcoming).
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Acknowledgments
This work was supported by the LogiCCC EUROCORES program of the ESF and DFG. For helpful comments on earlier presentations of this paper, I thank audiences at the University of Arizona, the University of Düsseldorf, and the Third Formal Epistemology Festival at the University of Toronto. I also thank Terry Horgan, Shaughan Lavine, Gerhard Schurz, and especially John Pollock and two anonymous referees for Erkenntnis for helpful comments on earlier drafts of the paper.
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Thorn, P.D. Two Problems of Direct Inference. Erkenn 76, 299–318 (2012). https://doi.org/10.1007/s10670-011-9319-6
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DOI: https://doi.org/10.1007/s10670-011-9319-6