1. Introduction

In a recent essay, John Norton presents what he terms a material theory of induction, which is intended to provide a fresh perspective on this venerable philosophical topic. The central thesis of the material theory of induction is that “All inductions ultimately derive their licenses from facts pertinent to the matter of the induction” (Norton Reference Norton2003, 650; italics in original). For instance, simple cases of enumerative induction presume that the entities with which the induction is concerned are homogenous in relevant respects (649). The facts concerning the relevant homogeneity in such a case are material facts. Thus, the material theory of induction maintains that the interesting questions regarding inductive inference will generally concern whether material facts required for the inductive inference to proceed are or are not satisfied in the case in question. One consequence of the material theory is pluralism. Since all modes of induction depend upon material facts of one sort or another and the facts may differ significantly from one case to the next, it is unreasonable to suppose that there is a single, unified schema of inductive inference that suits all cases. Rather, rival approaches can peacefully coexist: “Each can be used along with their attendant maxims on the best use of evidence, as long as we restrict their use to domains in which they are licensed by prevailing facts” (648).

In this essay, I examine the above proposal. My inquiry begins with a disambiguation of the italicized quotation cited above. This statement could be read as asserting that inductions ultimately draw their licenses from relevant material facts either wholly or in part. The latter claim, I think, is one that few would dispute. Indeed, it is hard to imagine a sensible theory of inductive inference that did not recognize that which inference may be appropriately drawn in a given instance depends in part on relevant facts of the case at hand. Thus, if only the weaker claim were intended, the material theory of induction would be rather uninteresting. I proceed, then, under the assumption that something closer to the stronger of the two readings is the correct one: the material theory of induction states that inductive inferences are justified wholly, or at least primarily, by pertinent material facts.Footnote ¹

However, I argue that this stronger reading of the material theory of induction does not withstand careful scrutiny. For which inductive inference is reasonable depends not only on material facts but also upon judgments concerning the desiderata of inductive inference. Distinct theories of inductive inference propose distinct standards of adequacy, and controversies among advocates of rival approaches typically concern the reasonableness of one set of standards vis-à-vis another. Such disputes are not about material facts but rather revolve around normative questions regarding the aims and criteria of evaluation appropriate for science. Moreover, the failure of the claim that inductive inferences are justified solely by reference to material facts is not one of relatively minor detail: fundamental disputes among leading theories of inductive inference turn on disagreements concerning desiderata, not material facts. In what follows, I elaborate the argument just sketched, and show that it significantly weakens the material theory's case for pluralism.

2. Desiderata

In contrast to the claim that inductive inferences are licensed wholly, or at least primarily, by material facts, I proposed above that they depend significantly on judgments concerning the proper standards and goals of induction, which we may for convenience call desiderata. There is one context in which the role of desiderata is immediately obvious. Since an inherent feature of inductive methods is that they are not assured of generating a correct result within a preset finite amount of time, inductive inference requires choices about how to balance the conflicting goals of informative conclusions and avoidance of error. A similar concern is the familiar issue in orthodox statistics of deciding on appropriate risks of type I and type II errors.

But if such cases constituted the extent of desiderata in inductive inference, one might still reasonably maintain that material facts provide the primary justification of inductive inferences. One might plausibly suggest that the degree of certainty or reliability demanded prior to accepting a conclusion might vary among reasonable people and that it would be asking too much of a theory of inductive inference to demand that it resolve such issues. Moreover, this concern is presented in the context of a shared approach to inductive inference. For example, the question of balancing type I and type II errors arises in the context of an elaborate theory of statistical inference and concerns only the details of its application in particular situations. If the appropriateness of that statistical theory in that context could be justified on the basis of material facts, then the material facts surely would deserve the pride of place that Norton ascribes to them. However, I argue that the role of desiderata runs far deeper than decisions concerning acceptable levels of inductive risk. Central controversies among advocates of various theories of inductive inference are disagreements concerning the right desiderata, and not disagreements about material facts. Let us consider more specifically what some the relevant desiderata are.

One appealing idea is that the guiding principle of inductive inference should be reliability: if what we aim for in science is truth, verisimilitude, or accurate predictions, then we should choose our inductive methods so as to promote those ends. Let us call this general perspective on inductive inference reliabilism. However, even at this level of generality, we face decisions about the ultimate goals of inquiry (truth, accurate predictions, etc.), which may have implications for what inductive methods would be appropriate in a given case. But let us put aside this issue, and suppose that agreement can be reached on at least a case-by-case basis regarding the ultimate goal of any given inquiry.Footnote ² Nevertheless, many questions remain concerning the appropriate standard of reliability. Since we are dealing with cases of inductive inference, it would be unreasonable to make a deductive proof the relevant standard; clearly, something less stringent is appropriate. Consider these three:

1. 1. Logical limiting reliability. The method will converge in the limit on the correct answer in every situation consistent with our background knowledge.

2. 2. Probabilistic limiting reliability. With probability 1, the method will converge in the limit on the correct answer in every situation consistent with our background knowledge.

3. 3. Finite probabilistic reliability. With a specified finite sample size, the method will generate results with acceptably low error probabilities.

Basic disputes among leading theories of inductive inference turn in large measure upon the question of which of these three standards should be demanded of science.

For example, the work of Kevin Kelly (cf. Reference Kelly1996) takes 1 as the primary standard of reliability appropriate for inductive inference.Footnote ³ Standard 1 is stronger than 2, since there are typically possibilities that must be assigned probability zero in a countably additive probability measure. Kelly's critique of Bayesian convergence theorems focuses precisely on this point: these theorems only demonstrate reliability in sense 2, where the probability measure in question is the agent's own degrees of belief (Reference Kelly1996, chapter 13). In other words, the convergence theorems show that (given the conditions of the theorem) the Bayesian agent is sure that she will converge to truth in the limit, not that she actually will so converge. Moreover, it turns out that there are some cases in which non-Bayesian methods could achieve standard 1 while no Bayesian could (cf. Kelly, Schulte, and Juhl Reference Kelly, Schulte and Juhl1997, 265).

In contrast, 3 is a standard of reliability usually demanded in orthodox statistics, as is illustrated by tests of statistical significance. Consequently, the primary critique of the Bayesian approach leveled by Deborah Mayo (cf. Reference Mayo1996, chapter 10) and other defenders of orthodox statistical methods is that, in spite of satisfying 2, Bayesian methods typically do not attain 3. An example commonly given to illustrate this critique will be described in the subsequent section. Likewise, the methods of causal inference developed by Peter Spirtes, Clark Glymour, and Richard Scheines (hereafter, SGS) fulfill standard 2, provided that certain conditions relating causality and probability obtain (cf. SGS Reference Spirtes, Glymour and Scheines2000), but one criticism of SGS's methods raised by statisticians is that they cannot be shown to be reliable in sense 3 (cf. SGS Reference Spirtes, Glymour and Scheines2000, 314–326).

So, disagreement concerning the correct standard of reliability is one source of basic differences among rival theories of inductive inference. Moreover, the controversies described above do not arise from disagreements concerning material facts. For instance, Kelly's dissatisfaction with 2 is based on the observation that a method can be reliable in that sense and yet still get the wrong result, even if all of our beliefs about material facts are correct. The argument for preferring 3 over 1 or 2 is that finite sets of data are all we will ever have, so the most important standard of reliability is one that is applicable to finite implementations of an inductive method. In response, partisans of 1 or 2 would likely reply that, though every human being is finite, science proceeds in an open-ended manner that can be reasonably modeled by approaches that countenance infinite streams of data. The source of disagreement here, therefore, is not a difference regarding what the material facts are.

In addition, the disaccord is far deeper than that illustrated by such questions as how to balance type I versus type II errors. In that case, a shared standard of reliability is presumed—namely, 3—and the issue concerns the details of its application in a particular situation. Exactly how much reliability (in the specified sense) should we insist on with regard to particular questions? But the disputes between Bayesians and orthodox statisticians, for example, arise from the more fundamental issue of the appropriateness of that standard of reliability as a criterion of justified scientific inference.

Reliabilism is not the only way to approach causal inference: intrinsically appealing principles of rationality constitute a different starting point. For example, it seems obvious that I cannot reasonably judge a hypothesis and its negation both to have a probability greater than one half. Moreover, it is natural to suppose that there is a sense in which such beliefs would be irrational even if I could somehow more effectively achieve some desired end by holding them. Let us use the term rationalism to refer to the claim that inductive inference should be governed by principles of rational belief, whose basis is of an intrinsic rather than pragmatic character.

I think that many contemporary Bayesian approaches to scientific inference can be understood in this manner. The original Dutch book arguments attempted to justify the claim that the degrees of belief of a rational person should be consistent with the axioms of probability on the grounds that a person who did otherwise would be willing to engage in bets in which he is sure to lose. However, many Bayesians find such arguments unpersuasive, since the connection between betting behavior and belief is far more complex than the argument presumes (cf. Howson and Urbach Reference Howson and Urbach1993, 89–91). As a result, several Bayesians have argued that the Dutch book argument can be reinterpreted as demonstrating an intrinsic cognitive irrationality in any degrees of belief that violate the probability axioms (cf. Howson and Urbach Reference Howson and Urbach1993, 75–86; Howson Reference Howson1997; Hellman Reference Hellman1997). Of course, such Bayesians also think that their approach can be motivated on the basis of its alleged ability to explicate commonsense norms of scientific inference and its reliability in sense 2 as shown by convergence theorems.

The Dutch book argument is not the only way to argue that rationality requires having degrees of belief consistent with the probability axioms. For example, Patrick Maher criticizes the Dutch book argument in general (Reference Maher1993, 94–104) and the “depragmatized” versions of the argument referred to above (Reference Maher1997). Instead, he advocates a representation theorem as the best means to justify the claim that a rational person should have degrees of belief that coincide with the probability calculus (cf. Maher Reference Maher1993). One begins with basic principles that place constraints on the sorts of preferences a rational person can have, and the theorem asserts that any person whose preferences satisfy these principles can be represented as having personal utility and probability functions. The basis for the principles concerning rational preference involved in the representation theorem is, in large measure, their intrinsic reasonableness. Consider Maher's statement with regard to two of these principles, transitivity and normality.

That the principles in question are generally regarded, at least upon reflection, as intrinsically reasonable is advanced as a motivation for them, to which desirable consequences or favorable applications would constitute a bonus.

The dispute between reliabilism and rationalism is of an even more fundamental sort than that between devotees of rival standards of reliability. From the perspective of reliabilism, intrinsically appealing principles of rationality are accepted only to the extent that they further, or at least do not obstruct, reliable inquiry. A staunch advocate of reliabilism regards reliability as an absolute trump over the intrinsic reasonableness of alleged principles of rationality. In contrast, one inclined to rationalism is unlikely to discount reliabilism completely, but may think that a balance should be struck between reliability and agreement with intrinsically appealing norms of rationality. For instance, such a person might be willing to opt for a less stringent standard of reliability, like 2, in order to retain his favored principles.

The above is a brief sketch of some central controversies in the current philosophical literature on inductive inference. The relevant point with regard to the material theory of induction is that these are not disagreements about what material facts do or do not obtain in given instances. Rather they concern normative issues. This calls into question the proposition that inductive inferences receive their licenses wholly, or even primarily, from material facts. For the above discussion illustrates how disputes among rival approaches to inductive inference can turn on other concerns. But since these methods sometimes sanction divergent inferences, which inductive inference should be drawn in a given case depends heavily upon what desiderata of inductive inference are chosen. In the next section, I consider an example of such a conflict.

3. Pluralism and Conflicting Desiderata

Consider an issue that divides Bayesian and orthodox statistical approaches: the relevance, or irrelevance, of stopping rules to statistical inference. Stopping rules specify when the researcher will cease collecting data and begin analyzing it. For instance, a stopping rule might specify that 100 observations will be made, and then a significance test performed to decide whether to reject or accept the null hypothesis. Although the relationship between Bayesianism and stopping rules is complex (cf. Steel Reference Steel2003), if some assumptions that most Bayesians would agree to are granted, then a Bayesian must regard the stopping rule as irrelevant to what inference should be drawn. In contrast, the stopping rule is highly significant to what conclusion may be appropriately drawn from a statistical test in orthodox Neyman-Pearson statistical theory.

Consider how this dispute plays out. Bayesians argue that intuitively bizarre inferences result from the role played by stopping rules in orthodox statistics (cf. Kyburg Reference Kyburg1974, 53–54; Howson and Urbach Reference Howson and Urbach1993, 241–243). In contrast, Mayo (Reference Mayo1996, chapter 10) charges that, as a result of disregarding stopping rules, Bayesians can be blithely unconcerned about reasoning in a way that is assured of leading to an incorrect conclusion. Joseph Kadane et al. (Reference Kadane, Schervish and Seidenfeld1996a, Reference Kadane, Schervish and Seidenfeld1996b) respond by showing that this argument works only if the Bayesian prior probability distribution violates the principle of countable additivity, which several Bayesians have argued on independent grounds is a criterion of rational degrees of belief (cf. Howson and Urbach Reference Howson and Urbach1993, 81; Maher Reference Maher1993, 198–199; Williamson Reference Williamson1999). In response, Mayo and Michael Kruse (Reference Mayo, Kruse, Corfield and Williamson2001, 393) argue that although Bayesian methods might not be assured of getting the wrong result, they are still unreliable in the sense of 3 in virtue of ignoring the stopping rule.

This controversy is largely independent of concerns about material facts. The issue of stopping rules arises in elementary examples in which all sides would be prepared to agree upon the facts of the situation, and in which there could be little serious concern regarding the applicability of either approach. For instance, the conflict can be illustrated by the textbook example of deciding whether a tossed coin is fair. Rather, the dispute turns on a disagreement about the proper desiderata of inductive inference. The Bayesian criticizes orthodox statisticians on the grounds that their approach conflicts with intrinsically appealing norms of rationality. Meanwhile, the orthodox statistician maintains that the stopping rule is necessary for reliability in the sense of 3, and adds that if attaining this standard of reliability requires rejecting some alleged norms of rationality, then so much the worse for those norms.

The dispute over stopping rules is far from being a marginal quibble; rather, it is a striking illustration of the divergence of fundamental aims and standards separating Bayesians and advocates of orthodox statistical methods. For instance, a discussion of stopping rules is the cornerstone of chapter 10 of Mayo's Error and the Growth of Experimental Knowledge, which is titled “Why You Cannot Be Just a Little Bit Bayesian.” As the chapter title indicates, Mayo argues that the Bayesian approach is fundamentally wrongheaded, thereby attempting to dissuade those who might be inclined towards a pluralistic approach combining elements of Bayesian and orthodox statistical methods. And since material facts have little, if any, relevance to the dispute, appeal to them will be of little use in defending such a pluralistic perspective.

This example illustrates how the central role of desiderata in disputes among rival theories of inductive inference undermines the material theory's case for pluralism. For imagine a person fervently committed to reliabilism, and standard 3 in particular. Such a person would happily agree that the justification for an inductive inference in any particular case depends in part on the material facts in force in that instance.Footnote ⁵ But she would certainly not see this as grounds for the ecumenical pluralism that Norton proposes. For example, from her perspective the Bayesian approach is based upon hopelessly misguided standards and goals, and hence is not merely wrong in some circumstances and right in others, but just plain wrong, period. Norton might point out that there are some situations in which material facts make reliability in the sense of 3 unattainable, and he might suggest that we should employ theories of inductive inference based on other principles when this happens. Yet a staunch supporter of 3 could respond that when finite probabilistic reliability is truly unattainable, we simply have to admit that we are unable to learn everything we want to know. As far as I can see, an emphasis on material facts reveals no incoherence in this position.

The failure of the material theory to provide a satisfactory basis for a pluralistic view of induction provides some indication of what an adequate defense of pluralism would have to look like. Such a defense would have to provide a framework for assessing the proper goals and standards of scientific inquiry, and it would also have to show how such assessments are domain specific. Whether any such an account of inductive inference can be developed is, I think, an open question.

4. Conclusion

The material theory of induction maintains that inductive inferences draw their licenses wholly, or at least primarily, from material facts, a proposition that the above discussion has endeavored to place in doubt. Yet in spite of disagreeing with Norton's material theory of induction, I find much to commend in his approach. It is valuable insofar as it takes seriously the particularistic and pluralistic nature of inductive inference in science, rather than treating it as a messy fact to be ignored or explained away. However, a philosophical account of the particularistic nature of inductive inference cannot be adequately developed and motivated without paying careful attention to desiderata in addition to material facts.