A Material Theory of Induction

December 25, 2002; rev.,

February 12, 2003

A Material Theory of Induction

John D. Norton[1]

Department of History and Philosophy of Science

University of Pittsburgh

In this material theory of induction, the inductive inferences of science are not licensed by universal schemas. They are grounded in matters of fact that hold only in particular domains, so that all inductive inference is local. While every schema of the present literature can be applied in the appropriate domain, the material theory suggests that universal schemas can never do full justice to the variety of inductive inferences and inductive inference schemas must ultimately be treated as individuals peculiar to their domains.

1. Introduction

There is a long standing, unsolved problem associated with inductive inference as it is practiced in science. It is not The Problem of Induction, the problem of finding a justification of induction. The problem to be addressed here is that, after two millennia of efforts, we have been unable to agree on the correct systematization of induction. There have always been many contenders. Some variant of Bayesianism probably now enjoys the leading position, although other schemes, such as inference to the best explanation, retain a considerable following. All this can change. In the late 19th century, some one hundred years after Bayes made his formula known, the leading systematization was the methods catalogued by Bacon, Herschel and, most precisely, Mill. This instability stands in strong contrast to deductive logic. The deductive syllogisms identified by Aristotle remain paradigms of deduction, with their very dreariness a mark of their unchallenged security. A comparable ancient contribution to inductive logic, induction by simple enumeration, has been a favored target of vilification for millennia.

The problem is deepened for philosophers of science by the extraordinary success of science at learning about our world through inductive inquiry. How is this success to be reconciled with our continued failure to agree on an explicit systematization of inductive inference? The contrast with deduction is striking again. A comparable enterprise, based on deductive inference, is mathematics. Our near universal agreement that deduction is well systematized by symbolic logic makes profound results commonplace: for example, it is impossible to capture all the truths of arithmetic as theorems deductively inferred from a finite axiom system in a first order language.

It is high time for us to recognize that our failure to agree on a single systemization of inductive inference is not merely a temporary lacuna. It is here to stay. In this paper I will propose that we have failed, not because of lack of effort or imagination, but because we seek a goal that in principle cannot be found. My purpose is to develop an account of induction in which the failure becomes explicable and inevitable; and it will do this without denying the legitimacy of inductive inference. We have been misled, I believe, by the model of deductive logic into seeking an account of induction based on universally applicable schemas. In its place I will develop an account of induction with no universal schemas. Instead inductive inferences will be seen as deriving their license from facts. These facts are the material of the inductions; hence it is a "material theory of induction." An old tradition has sought to do this by seeking universal facts to underwrite induction. The best know of these is Mill's principle of the uniformity nature. Instead I will suggest that there are no universal facts that can do this. Particular facts in each domain license the inductive inferences admissible in that domain—hence the slogan: "All induction is local." My purpose is not to advocate any particular system of inductive inference. Indeed I will suggest that the competition between the well established systems is futile and that each can be used in the right domain.

In Section 2, I will lay out the basic notions of a material theory of induction. Theories of induction must address an irresolvable tension between the universality and the successful functioning of some formal account of induction. The present literature favors universality over function. I urge that we can secure successful functioning by forgoing universality and that this is achieved in a local, material theory of induction. According to this material theory, there can be no universal inductive inference schemas. So, in Section 3., I will review briefly the principal approaches to induction in the present literature in order to show how all depend ultimately on local matters of fact. In Section 4., I will illustrate how the imperfections of fit between existing schemas and actual inductions become greater as the relevant domain becomes narrower and suggest that some inductive inferences are best understood as individuals peculiar to a particular domain. In Section 5., I will review how, according to the material theory, inductive success in science does not require a command of an extensive repertoire of inductive inference schemas. Merely knowing more matters of fact can extend our inductive reach. In Section 6., I will argue that a material theory of induction eludes The Problem of Induction, in so far as the simple considerations that visit the problem on a formal theory fail to generate comparable difficulties for a material theory. Finally Section 7. contains concluding remarks.

2. The Material View

Universal Schemas of Deduction and Induction

All deductive logic, from the simplest syllogistic logic to the most sophisticated symbolic system, is based on the idea of universal schemas. They function as templates into which content can be freely inserted to produce the valid inferences licensed by the logic. The simple syllogism with the medieval mnemonic "bocardo" illustrates this as well as any:

Some A's are not B's.

All A's are C's.

Therefore, some C's are not B's.

A, B and C can be replaced by any terms we please and we are assured of recovery of a valid deductive inference. Superficially, things seem the same with inductive inference. Take the scheme of enumerative induction:

Some A's are B's.

Therefore, all A's are B's.

For better or worse, we substitute terms for A and B to recover enumerative inductions:

The first example is somehow quite different from the second. The first is so secure that chemistry texts routinely report the melting points of elements on the basis of measurements on a few samples, relying on inductive inferences of exactly this type. The second is quite fragile. The evidence drawn from some samples of wax at most supplies weak support for the generalization. Our hesitation derives from our knowledge that there are all sorts of waxes and the suspicion that it was just some sort of accident that led the few samples reported to have the same melting point. Our worry is not just that enumerative induction is inductive so that there is always some risk of failure. There is a systematic difference between the two so that the first is quite secure but there is something elusively suspect about the second.[2].Here is another example of two different instantiations—one licit, one suspect—of an inductive inference scheme that I will call "projection":

Mill's "Problem of Induction"

Mill (1872, pp. 205-206) sees this as the most severe of problems:

When a chemist announces the existence and properties of a newly-discovered substance, if we confide in his accuracy, we feel assured that the conclusions he has arrived at will hold universally, though the induction be founded but on a single instance. We do not withhold our assent, waiting for a repetition of the experiment; or if we do, it is from a doubt whether the one experiment was properly made, not whether if properly made it would be conclusive. Here, then, is a general law of nature inferred without hesitation from a single instance, a universal proposition from a singular one. Now mark another case, and contrast it with this. Not all the instances which have been observed since the beginning of the world in support of the general proposition that all crows are black would be deemed a sufficient presumption of the truth of the proposition to outweigh the testimony of one unexceptionable witness who should affirm that, in some region of the earth not fully explored, he had caught and examined a crow and found it to be gray.

Why is a single instance, in some cases, sufficient for a complete induction, while in others, myriads of concurring instances, without a single exception known or presumed, go such a very little way towards establishing a universal proposition? Whoever can answer this question knows more of the philosophy of logic than the wisest of the ancients and has solved the problem of induction. [my emphasis]

What solution might we envisage? The natural reaction is to say that the original schemas are elliptic and that extra conditions must be added to block misapplications. So we might say that enumerative induction can only be carried out on A's that belong to a uniform totality; or that projection can only be applied to properties that are projectible. The goal is to augment the schemas to restore their successful functioning, while allowing them to remain universally applicable. As we shall see in examples below, such efforts eventually prove to be self defeating, falling to a fatal tension between universality and successful functioning.

Augmented schemas could certainly function well if they were allowed to mention the specific facts that underpin our judgments of the varying success of the inferences. They would note that all samples of bismuth are uniform just in the property that determines their melting point, their elemental nature, but may well not be uniform in irrelevant properties such as their shapes or locations; or that wax samples lack this uniformity in the relevant property, since wax is the generic name for various mixtures of hydrocarbons. Yet if they are to be universal, the schemas must rise above such specific facts. Stripped of these specifics, there seems to be no general formula that gives a viable, independent meaning to "uniform" or "projectible." We are reduced to making them synonyms for "properties for which the enumerative induction (or projection) schema works." So the assertion that projection is licit only when used with projectible properties would reduce to the circularity that it is licit only when it works.

Formal and Material Theories of Inference

This tension of universality and successful functioning is routinely addressed by insisting on universality and then seeking unsuccessfully to secure proper functioning. If we forgo universality, functioning can be quite readily secured. But we must reconceive the fundamental nature of inductive inference. Our present theories of induction are what I shall call formal theories. The admissibility of an induction is ultimately traced back to the form of the inference; it is valid if it is built from a licit schema. The alternative is what I shall call a material theory, reflecting the difference between the form and the matter of an induction. In it, the admissibility of an induction is ultimately traced back to a matter of fact. That is, the successful functioning of the logic depends ultimately on matters of fact. We are licensed to infer from the melting point of some samples of an element to the melting point of all samples by a fact about elements. It is that elements are generally uniform in their physical properties. So if we know the physical properties of one sample of the element, we have a license to infer that other samples will most likely have the same properties. The license does not come from the form of the inference, that we proceed from a "some…" to an "all…". It comes from a fact relevant to the material of the induction.

In advocating a material theory of induction in this paper, my principal contention is that all induction is like this. All inductions ultimately derive their licenses from facts pertinent to the matter of the induction. I shall call these licensing facts the material postulate of the induction.

The characters of the induction in a material theory are determined by the material postulates. They may certainly be truth conducive, as opposed to being merely pragmatically or instrumentally useful, as long as the material postulates are strong enough to support it. How each induction will be truth conducive will also depend on the material postulate and may well suffer a vagueness inherited from the present induction literature. Chemical elements are generally uniform in their physical properties, so the conclusion of the above induction is most likely true. In this case, "most likely" denotes high frequency of truth among many cases of elements tested. Such frequentist readings will not always be possible.

All Induction is Local

There has been a long history of attempts to identify facts about the world that could underwrite induction. Best known is Mill's (1872, Book III, Ch. III) "axiom of the uniformity of the course of nature." Russell (1912, Ch.6) defined the principle of induction in terms of the probability of continuation of a repeated association between "a thing of a certain sort A" and "a thing of certain sort B." Russell's later (1948, Part 6, Ch. IX, pp. 490-91) expansion has five postulates that include a quite specific "postulate of spatio-temporal continuity in causal lines" which "den[ies] 'action at a distance'."[3]

All these efforts fall to the problem already seen, an irresolvable tension between universality and successful functioning. On the one hand, if they are general enough to be universal and still true, the axioms or principles become vague, vacuous or circular. A principle of uniformity must limit the extent of the uniformity posited. For the world is simply not uniform in all but a few specially selected aspects and those uniformities are generally distinguished as laws of nature. So, unless it introduces these specific facts or laws, an effort to formulate the limitation can only gesture vaguely that such uniformities exist. Any attempt to characterize them further would require introducing specific facts or laws that would violate universality. On the other hand, if the axiom or principle is to serve its function of licensing induction, it must introduce these specific facts and forfeit universality. So Russell ends up denying action at a distance. If his account is to cover all induction, we must conclude that induction is impossible in any universe hosting action at a distance.

We should not conclude from our long history of failure to find universal facts that can underwrite induction that the quest is fundamentally misdirected. We are right to seek facts to underwrite induction. Our error has been to seek universal facts. Instead I propose that we look to facts that obtain only in specific domains; that is, facts that obtain "locally." As a result, inductive inference schemas will only ever be licensed locally. For example, enumerative induction on the physical properties of elements is licensed locally in chemistry by facts about elements. In another domain, inference to the best causal explanation might be licensed by our specific knowledge of the causes prevailing in the domain. In a controlled study, we know that the only systematic difference between the test and control group is the application of the treatment and we have some belief that the treatment has causal powers. So we infer that the only viable explanation of the difference between the two groups is the causal powers of the treatment. In yet another domain, probabilistic inference is called for. We believe that genetic markers in DNA are distributed independently in the population. So when two blood samples agree in many of the markers, we are licensed to conclude at a high level of certainty that the two samples came from the same individual. In all these cases, the facts that license the inductions—their material postulates—obtain only locally. So the inductions themselves are only licensed locally.

Solution of Mill's Problem: The Particularity of Very Local Inductions

The solution of Mill's problem is now evident. Why are enumerative inductions on bismuth and wax of such differing strength? It is because they are essentially different inductions. One is ultimately underwritten by facts about elements that license strong support from a few instances. The other has a much weaker underwriting in facts about waxes that license little or no support from a few instances to the generalization. That the two inductions have the same general form is a distraction. They do not derive their license from this similarity of form.

This last example shows that there are significant limits in seeking to characterize particular inductive inferences within the system of schemes in the present literature. Since these schemes have been proposed as universal, in the end there is always some imperfection in the fit between them and actual inductive inference. As the relevant domain becomes smaller and the resulting inductions stronger, the inferences derive less of their strength from any recognizable inductive inference form and more directly from the material facts that underwrite them. So the fit becomes worse. Thus we should expect cases of inductive inferences that prove to be too hard to characterize, while at the same time we are quite sure of their strength from merely inspecting the particular case. See Section 4 below for some candidate examples.