December 25, 2002; rev.,
February 12, 2003
A Material Theory of Induction
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
jdnorton@pitt.edu

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:

Some samples of the element bismuth melt at 271oC.
Therefore, all samples of the element bismuth melt at 271oC.	Some samples of wax melt at 91oC.
Therefore, all samples of wax melt at 91oC.	

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..Here is another example of two different instantiations—one licit, one suspect—of an inductive inference scheme that I will call "projection":

Every case of infantile Tay-Sachs disease so far has been fatal.
Therefore, the next case of infantile Tay-Sachs disease will be fatal.	Every straw added so far has not harmed this camel's back.
Therefore, the next straw added will not harm this camel's back.	

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'."
	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.
3. A Little Survey of Induction
	If all induction is local, then there can be no universal inductive inference scheme. My purpose in this section is to review some of the more important systems of inductive inference and show their dependence on contingent matters of fact. While it will be impossible for me to address every system in the literature, I believe I can give reasonable coverage by recognizing that virtually all the systems can be fitted fairly well into one of three broad families, with all members of the same family in some way exploiting the same inductive principle. I will seek to show that
• each viable inductive inference scheme requires matters of fact to licence it;
• the matters of fact are local in character, illustrating how successful functioning requires the sacrifice of universality; and that
• the particular material postulates discussed are sufficient to license the relevant induction.
Inductive Generalization
	All members of this family are based on the principle that an instance confirms the generalization. The best known and original form is enumerative induction, in which one instance of an A that is B confirms that all A's are B's. Hempel's (1945) satisfaction criterion of confirmation provides a logically robust definition of an instance in the context of first order predicate logic, while still exploiting the same idea that the instance confirms the generalization. A serious shortcoming of enumerative induction is that it merely licenses inferences from "some…" to "all…" and that is far too narrow for many applications in science. Augmented forms couple enumerative induction to schemes for introducing theoretical terms. The most important of these arise in Mill's (1872, Book III, Ch. 7) methods. For example, Mill's "joint method of agreement and disagreement" begins with our finding cases in which an antecedent always contains A if the consequent has B (necessity) and the antecedent always fails to have A if the consequent fails to have B (sufficiency). The joint method uses enumerative induction to infer that this necessity and sufficiency of A for B obtains not just in the cases investigated, but in all cases; and it licenses us to interpret this necessity and sufficiency as causation, so that A is the cause of B or an indispensable part of it. Glymour's (1980) bootstrap confirmation relation allows far greater liberty in the rules that can be used to introduce theoretical terms. The bootstrap confirmation relation requires that an instance of the hypothesis being supported is to be deduced from the evidence. In deducing it, we are free to use any part of the theory in question to introduce theoretical terms. In the context of Newtonian mechanics, we may use force = mass x acceleration to replace acceleration terms by force terms, so that evidence concerning accelerations can yield an instance of hypothesis pertaining to forces. In both these extensions, the basic inductive principle remains the same: a "some…" is replaced by an "all…".
	We have already seen through the examples of the melting points of wax and bismuth how the applicability of enumerative induction depends upon certain facts relevant to the example. These facts decide how strongly an enumerative induction supports its conclusion and even whether the scheme is applicable at all. Might we find a universal characterization of what these facts add and thereby restore the successful functioning of enumerative induction while saving its universality? The question has been investigated fairly extensively in the literature on Goodman's (1983) problem of "grue". (see Stalker, 1994) In that problem, examining an emerald and finding it is green provides an instance of the hypothesis that all emeralds are green; and it is also an instance of the hypothesis that all emeralds are "grue": green if examined prior to some future time T and blue otherwise. We are quite happy to have the instance confirm the first hypothesis but admit little or no support for the second. One of the more satisfying resolutions distinguishes green as a natural kind term from grue, which fails to be. (Quine, 1970) So the first hypothesis only is confirmed by its instances, since only it is formulated in terms of natural kinds. Might we retain the universality of enumerative induction if we augment it with the requirement that it be used only with natural kind terms? The solution is incomplete for two reasons. First, it presumes a robust account of natural kinds. But we do not have one. Second, it still does not allow us to distinguish the very strong from the very weak enumerative inductions. Both "bismuth" and "wax" are natural kinds yet they enter into enumerative inductions of very different strengths. More fundamentally, even if we resolve these problems, if enumerative induction requires natural kinds, then we must presume that enumerative induction can only work in a universe with sufficiently stable regularities to admit natural kinds. That ours is such a universe is a material postulate.
Hypothetical Induction
	Under the rubric of hypothetico-deductive confirmation, the basic principle of this family is that the ability of a theory, possibly with auxiliary assumptions, to entail the evidence is a mark of its truth. Its use is venerable, tracing back at least to the "saving of phenomena" in ancient astronomy. The basic principle by itself is actually rarely used alone because it is quite indiscriminate. According to it a trivial piece of evidence, that there is a triangle whose angles sum to 180o, is not just evidence for the Euclidean geometry that entails it but for the entire edifice of the science that may contain the geometry, including all of modern biology, geology and astrophysics.
	Expanding on such problems, we can quickly see that anything but an utterly trivial version of hypothetico-deductive confirmation requires additional restrictions that are factual. For example it allows that A&B is confirmed by A. The relation is non-trivial just in so far as it accords support to B, the second conjunct of A&B. But it should only do that if there is some relation of dependence between A and B. So A might be the report that we have measured the sum of the angles of a particular triangle and found it to be 180o; and B might be Freudian psychology. We would certainly not expect this A to lend support to this B, even though A&B does entail B. But it would support B if B were Euclid's fifth postulate of the parallels. For then A and B are related in that they are both parts of Euclidean geometry. More carefully, the relevance might be recovered from the assumption that the geometry is homogeneous, that is, of constant curvature. The presumption of a homogenous geometry functions as a material postulate in so far as it licenses inference from the value reported for the sum of the angles of a triangle, to the particular geometry prevailing, and then to the properties of parallel lines. Since we lack a general, formal account of the relation of dependence between A and B, a material fact decides whether the relation obtains and whether A confirms B.
	There have been many attempts to tame the indiscriminateness of hypothetico-deductivism. They provide an interesting study in the tension between universality and successful functioning. In so far as they succeed in bringing proper functioning to the scheme, they do so by making the scheme dependent on particular facts, so that universality is compromised. These attempts can be grouped according to their broad strategy.
Exclusionary Accounts
	In hypothetical induction, we know that some hypothesis deductively entails the evidence. But there are infinitely many competing hypotheses that may also be able to entail the evidence. The most direct way to exclude them is to demonstrate additionally that the falsity of the hypothesis entails the falsity of the evidence. That is rarely possible, since this additional result is very strong. It is equivalent (by contraposition) to the evidence deductively entailing the hypothesis. However our inductive worries would be essentially eliminated if we could show that the falsity of the hypothesis makes very likely the falsity of the evidence; or more briefly, that if the hypothesis weren't true we very likely wouldn't have gotten the evidence. Many accounts seek to augment the hypothetico-deductive scheme by demonstrating this additional result. As we shall see in the examples to follow, the demonstration of the additional results depends upon particular facts prevailing in the relevant domain. These facts function as the material postulate that licenses the induction in accord with the material theory.
	The most straightforward of these accounts is modeled after traditional error statistical analysis. See Giere (1983) and, for a more thorough account, Mayo (1996). In a controlled study, any systematic difference in outcome between test and control group is attributed to the treatment applied to the test group. Typical study designs randomize over the two groups so that the only systematic difference in composition between the two groups is, most probably, the treatment which is then most likely responsible for differing outcomes. Generalizing these sorts of canonical examples, Mayo (1996, Ch.6) calls a test severe if the procedure is very unlikely to be passed if the hypothesis is false. Clearly passing a severe test is a strong license for an hypothesis. The license derives directly from the facts that make the test severe. These can be facts about the randomizing of test and control group, so that the "very likely" will be given through the physical probabilities of the randomizer. Or they may be vaguer. Replacing the bulb is a good test of the hypothesis that my lamp fails to operate because of a burnt out bulb, since it is very unlikely for another defect to be remedied by replacing the bulb. In this case, the probability judgments are vaguer, but still grounded in common experience of the world. That experience licenses the inference.
	Facts that are far more difficult to identify underwrite another version of this augmentation. In arguments to a common cause (Salmon, 1984, Ch. 8) or in common origin inferences (Janssen, manuscript) the hypothesis is inferred from the evidence on the strength of our conviction that it would be an astonishing coincidence if the evidence obtained without the hypothesis also being true. So early last century Perrin showed that there were roughly a dozen different methods of experimentally measuring Avogadro's number N, which gives the size of atoms. That they all gave approximately the same value would be astonishing if there weren't atoms. Similarly, in developing his special theory of relativity, Einstein found it remarkable that our physical theories required material processes to contract, dilate and more in just the perfect concert that made impossible any determination of the aether's presumed absolute state of rest. The astonishing coincidence could be eradicated if we conclude that all these processes are responding to the same background space and time that lacked an absolute state of rest.
	What underwrites these inferences are elusive but widely shared judgments over what would happen if the relevant hypotheses were false. If there weren't atoms, we would expect some sort of continuum theory to prevail and we generally agree that these sorts of theories would not reproduce Perrin's experimental results. Similarly if the structure of space and time did harbor an absolute state of rest after all, we would expect that some physical process in space and time would reveal it. These are factual presumptions about the realm of possibility and they are the material postulates that underwrite the inferences.
Simplicity
	The most obvious and perennially popular augmentation of the hypothetico-deductive scheme uses the notion of simplicity. (See for example, Foster and Martin, 1966, Part III)  While many hypotheses or theories may deductively entail the evidence, in the augmented scheme we are licensed to accord inductive support only to the simplest of them. In my view, our decisions as to what is simple or what is simpler in these cases depend essentially upon the facts or laws that we believe to prevail. These facts dictate which theoretical structures may be used and the appeal to simplicity is really an attempt to avoid introducing theoretical structures unsuited to the physical reality governed by those facts or laws. That is, these facts or laws function as material postulates that license what we may or may not infer. Appeals to simplicity in the context of confirmation in science are really indirect appeals to facts presumed prevailing in the relevant domain.
	This can be seen quite clearly in the most popular example of the use of simplicity in confirmation theory, curve fitting. When the evidence is presented as finitely many points on a sheet of graph paper and we find that a linear and a quadratic equation can be made to fit pretty much equally well, we routinely infer to the linear equation on the grounds that it is simpler. Our choice is licensed by facts tacitly assumed to obtain in the relevant domain.
	To see this, first note that the applicability of the procedure depends upon getting certain assumptions right. An equation expressing a law is linear only with certain choices of variables. We can create an equation for the law that uses any function that strikes our fancy by the simple expedient of rescaling one of the original variables by the inverse of that function. Moreover there is nothing especially hallowed about the common choice of linear, quadratic, cubic, quartic, etc. as the natural sequence of functions of increasing complexity. We use it if we choose to concern ourselves with the class of polynomials. But if we choose to describe the equations to be fitted in terms of the differential equations that generate them, we would be quickly drawn to a quite different class of simple functions. Both exponential and trigonometric functions (sine, cosine) would appear quite early since they are the functions that are proportional to their own first and second derivatives.
	So picking the simplest curve can only make sense evidentially if we make the right choices above. And we make those choices correctly if we think that the variables and function hierarchy selected somehow map onto the physical reality at hand. So we describe motions in space in terms of distances covered and times elapsed since our basic physics tells us they are the basic quantities. If our system is one describing growth, we would quickly look to fitting exponential functions since they are functions that figure in the laws governing growth. In that case, we would certainly prefer an exponential curve over, say, a quintic curve that fitted as well, even though the exponential curve corresponds to a polynomial with infinitely many terms.  Similarly if our system represents any sort of cyclic process, we would quickly look to sine and cosine functions, whose polynomial expansions have infinitely many terms, even though a 25th order polynomial might give us an equation with as many hills and valleys.
	In short, we have no universal scheme or universal formal rules that define what is simpler or simplest. In so far as we are able, we choose the functions appropriate to the facts that we believe prevail. These beliefs over pertinent matters of fact determine our expectations over which functions, expressed in appropriate variables, are likely to be encountered in the relevant domain. In the resulting hierarchy, the simpler functions are deemed the more likely. So these beliefs function as material postulates that license inference to the simplest curve adequate to the data as the most likely curve.
Abduction: Inference to the Best Explanation
	In this approach, we do not just require that the evidence be entailed by the hypothesis or theory it confirms; the evidence must also be explained by it. (Harman,1965; Lipton 1991) The practice of abduction is straightforward. In each domain we are supplied with or may even create a repertoire of explanatory resources. We then choose among them for the one that explains the evidence best. So Hubble in the late 1920s observed that light from distant galaxies is shifted to the red, with the shift proportional to the distance. Galactic physics supplies several candidate explanations. The red shift may arise from a velocity of recession; or it may arise from the gravitational slowing of the distant galaxy's temporal processes, in accord with the general theory of relativity. The explanatory repertoire is limited. One might have expected it to include a process in which an interstellar medium reddens the light, just as a red filter might color a spotlight. Such a resource is not in the repertoire since chemistry supplies no medium that uniformly slides the spectral lines of the galactic light along the frequency scale. We routinely choose the velocity of recession as the explanation, since it explains the best, accommodating the linear dependence of red shift on distance to a uniform motion of expansion of the galaxies.
	Examples in which the facts are uncertain underscore how the prevailing facts determine what may be inferred. Consider, for example, a controlled study of the healing efficacy of prayer. A theist would readily accept divine intervention as the best explanation of a positive result in the study. An atheist however would conjecture some as yet unnoticed flaw in the experimental design as the best explanation and may even deny that divine intervention would count as an explanation at all. The difference depends fully on their differences over the facts that prevail. An analogous example would be the differing appraisal of controlled studies on telepathy by parapsychologists and skeptics. Or in another example, astronomers and astrologers will differ markedly on how celestial bodies may figure as explanations of terrestrial phenomena. And modern day astronomers would no doubt not avail themselves of Newton's explanatory repertoire. He proposed (1692-3, first letter) that the planets have just the right velocities to produce stable orbits since they were "impressed by an intelligent Agent."
	The important point is that the facts prevailing in the relevant domain dictate what can count as an explanation for the evidence. Indeed that something explains at all is itself recovered by direct inspection of the case at hand and not by demonstrating conformity to some externally supplied template for what counts as a good explanation. We recognized that a velocity of recession is the best explanation of the galactic red shift without any thought to precisely what we might mean by saying that it explains the red shift. Did it explain by displaying causes, by subsumption under a covering law, by unifying, or by displaying statistically relevant factors?
	So inferences to the best explanation are licensed by facts pertinent to the local domain that supply us explanatory resources. These facts express our expectations about the processes prevailing with the better explanations deemed more likely. The clearest case is when we take explanations to be displaying causes. The pertinent facts state what causes we expect to be active. We generally deem the better explanation to be the one that invokes the more likely case of fewer independent causes. So we are licensed to infer to the best explanation as the most likely.
Reliabilism
	In a reliabilist account, we are licensed to believe hypotheses or theories produced by a reliable method. We routinely accept, for example, the diagnoses of expert car mechanics or physicians on the principle that the method they have used to arrive at their diagnoses are reliable, although we may not understand the method. Reliabilists propose that science, properly practiced, uses reliable methods and that is why we can believe its products. I will consider such accounts as providing the augmentation needed to tame the indiscriminateness of hypothetico-deductivism, for that is a common use.
	We can see immediately that the material postulate that underwrites our inference in accepting these results is just our belief that the method is reliable. That belief is not only about the method. It is also about the world. For it incorporates the belief that the world is such that these methods can work reliably. In an uncooperative world, no method can succeed. We would properly hold little hope for a method touted as reliably finding a system to beat the house in a casino, for the casino sets up its games to make this impossible. Or we should have little hope in the prognostications of an entrail reader, no matter how expertly the best methods of entrail reading have been followed. The real world does not admit prediction through reading of entrails, except perhaps for the health of the flock from which the sacrified animal was drawn.
	The method used may be quite explicit, in which case its reliability admits further analysis, either through other accounts of induction or directly through the facts that underwrite them. Or the reliability may be the practical end of the analysis, as is the case with expert diagnosticians. They do use explicit methods, codified in manuals. However these are supplemented by the experts' experience and in ways that may not admit explicit analysis.
	The best know reliabilist account is Popper's (1959) falsificationism. According to it, scientists subject present theories and hypotheses to severe test by comparing their predictions with experience. If they are falsified, scientists are led to conjecture new theories or hypotheses and the latest products of this process are regarded as "well corroborated." While Popper has vigorously insisted (e.g. p.33) that this notion of corroboration is not confirmation, I follow the tradition that holds that Popper's account simply fails to bear close enough resemblance to scientific practice if corroboration does not contain a license for belief, with better corroboration yielding a stronger license. (See Salmon, 1981.) Popper does not give much account of the details of the method. The process of conceiving the new hypothesis is explicitly relegated to psychology and the inclination to take any philosophical interest in it disavowed as "psychologism." (pp.31-32). Lakatos (1970) gives a more elaborate accounting of a falsification driven method in his “methodology of scientific research programs.” Decisions as to which research program a scientist ought to follow are dictated by guidelines deriving from such considerations as success at novel prediction. That we have any warrant to believe the products of such methods is licensed by factual matters: Is the method reliable? Are scientists actually following those methods properly? And more deeply: Is the world such that these methods can eventually succeed if followed?
	The methods described are incomplete—they fall far short of an algorithm that can be followed mechanically. This once again reflects the tension of universality and successful functioning in the facts that underwrite induction. Perhaps we can eventually construct a full and complete account of a reliable method that works universally. Yet well before this extraordinarily optimistic goal is achieved, if we are so lucky as to have any complete methods at all, they will be narrowly specialized to quite particular domains, sacrificing universality for successful functioning.
Probabilistic Accounts
	In these accounts, degrees of belief are represented by magnitudes and the import of new evidence is to rearrange these magnitudes according to a definite calculus, usually the probability calculus. The inspiration for the family came from the discovery that the stochastic properties of physical systems could be represented by a calculus of probabilities. In Bayesian confirmation theory (Howson and Urbach, 1989), this calculus is used to represent and update beliefs. Since it posits that our degrees of belief should conform to the same calculus as governs stochastic processes in the physical world, Bayesianism's least troublesome application arises when our beliefs pertain to stochastic processes. So, if a coin toss has  a physical probability of 1/2 of yielding a head (henceforth a "chance" of 1/2), then our degree of belief in the outcome of a head ought to be that same 1/2. In this case, the material facts that license the ascribing of probabilistic degrees of belief are obvious: they are simply the physical properties of the coin toss system that generated the chances. They readily license truth conducive induction. The chance of at least one head in a run of ten independent tosses is more than 0.99. So if our degrees of belief conform to the chances, we will assign probability of least 0.99 to what will be the true outcome in over 99% of runs. 
	What makes matters more complicated is that Bayesianism asserts that the same notion of degrees of belief and the same calculus should be applied in all cases, including those in which no stochastic processes deliver convenient chances. So our prior beliefs in big bang and steady state cosmologies are probabilities measured on the same scale as our belief that a coin will come up heads or tails and they obey the same dynamics.
	It is not too hard to conceive physical systems whose factual properties would preclude degrees of belief conforming to the probability calculus. The steady state theory of cosmology posits that hydrogen atoms materialize randomly and uniformly in an infinite space. How are we to conform our beliefs to that fact for some hydrogen atom? We require a uniform probability distribution over all space. Traditionally that is achieved with an improper prior that assigns equal finite probability to equal finite volumes of space, at the cost of giving up the axiom of probability theory that requires the sum of all probabilities to be unity. In this case the sum is infinite. (See Jeffreys, 1961, §3.1) In a slightly more complicated example, we might imagine the hydrogen atom constrained to materialize along some unit interval of space, [0,1]. The chances of the appearance in any subinterval would be given by the natural measure that, for example, assigns to the interval [1/4, 3/4] chance 1/2 = 3/4 – 1/4. Consider any subset of points in [0,1]. It is a reasonable question to ask what our belief is that the hydrogen atom appears at a point in the subset. However, if we require that our degrees of belief are probabilities conforming to the chances, then there will be infinitely many subsets for which no probability can be defined. These are the non measurable of measure theory (Halmos, 1950, §16). 
	There is another way in which we can see how material facts condition the content of Bayesianism and it does not call up esoterica like non measurable sets. Bayesianism is vacuous until we ascribe some meaning to the probabilities central to it. Until then, they are just mathematical parameters. Each way of ascribing meaning brings factual presumptions with it. One popular way is to interpret the probabilities in terms of actions and recover their properties from our preferences. The best known of these is the Dutch book argument. (De Finetti, 1937) It is presumed that a degree of belief p in some outcome A makes us indifferent to which side of a bet we take in which winning $(1-p) is gained if A obtains and $p is lost if A fails. Given that we accept all bets of this form, it is quickly shown that we would accept a combination of bets (a "Dutch book") that force a sure loss, unless our degrees of belief p conform to the probability calculus. The relevant material facts are the rules for how you choose to convert degrees of belief into acceptable wagers, the existence of a framework for realizing the wagers and our preference for avoiding Dutch books. It is easy to see that these material facts would obtain in some contexts, such as if we are making wagers on horses at a racetrack. Even then, the details of the calculus that our degrees of belief would be constrained to follow are be quite sensitive to the details of the rules. For example, if we decide that we are never indifferent to which side of a wager we accept, the avoidance of a Dutch book can no longer force additivity of our degrees of belief. There will be other contexts in which it is hard to see how these material facts could be made to obtain. How might our beliefs in big bang versus steady state cosmology be converted into the relevant sorts of wagers? Perhaps if one is a scientist choosing a research career, that choice might have the flavor of a wager. But one can still have beliefs on the theories when one has no personal stake at all in which is true and no interest in betting on it.
	To the extent that the facts prevailing in a domain do not support realization of degrees of belief in bets, then avoidance of a Dutch book fails to constrain those degrees of belief. The obtaining of these facts, however, does not comprise a material postulate strong enough to support truth conducive induction (unlike the earlier case of material facts pertaining to stochastic processes). Rather they merely license inductions that are pragmatically useful in so far as they generate beliefs that will not support actions that may bring sure losses.
	Nevertheless Bayesian confirmation theory seems to be the most general of all the systems currently in favor. The reason is that it is a rather weak system. We get very little from it until we specify what might need to be a quite large number of conditional probabilities (the likelihoods) and these are determined by the factual properties of the relevant system. Each such specification yields a mini-inductive logic adapted to the facts of the relevant domain. So we might ask if the hypothesis H that all swans are white is confirmed by the evidence E that this swan is white in the natural sense that P(H|E) > P(H). Even though we already know the likelihood P(E|H)=1, we cannot answer at all until we specify the likelihood P(E|-H), which requires us to judge how likely it is to find a white swan if not all swans are white. These likelihoods, which determine the mini-inductive logic, are in turn fixed by our prior probability distribution, since, for example, P(E|-H) = P(E&-H)/P(-H). So we have the curious result that a mythical prior probability distribution, formed in advance of the incorporation of any evidence, decides how any evidence we may subsequently encounter will alter our beliefs. 
4. Inductions Too Local to Categorize
	In the little survey above, I have tried to show how the inductive inference schemas recognized in the literature depend on material facts. I have already suggested in Section 2 above that there will always be some imperfection in the fit between actual inductions and the schemes intended to work universally; and that the fit will worsen as the domain narrows and the inductions become more dependent on narrower material postulates for their increasing strength.
	The clearest examples of such inductions arise when a major advance in the science brings a major enhancement to our inductive powers. In such cases, the inductive inference schemas supplied by the methodology literature remain essentially unchanged. The enhancement in our inductive reach must be attributed to the new material postulates made available in the relevant domain.
	Prior to Lavoisier and the establishment of modern chemistry in the late 18th and early 19th century, it was quite hard to know what sorts of properties of substances were likely to be the same across all samples. Learning which substances were elements and compounds and which mixtures dramatically improved our inferential abilities. To know that something was an element or compound brought a license to infer that all its samples were most likely alike in physical properties. We also learned when to infer that our failure to bring about a transformation might merely be due to our failure to find the right methods or when it could be generalized to an impossibility. From a chemical perspective there were no barriers to transforming inorganic matter into organic matter; it was a case of the former. But our failure to transform lead into gold was a case of the latter. While these all fit the form of enumerative inductions (with some failing) they have become so completely modified by the particular chemical facts in the domain that the characterization has almost no practical value.
	More examples: Once we learned in the 1920s that the nature and properties of the elements are due to the quantum properties of electrons trapped by atomic nuclei, we had a much stronger basis for knowing which unfilled spaces in the periodic table might really coincide with undiscovered chemical elements, where the table might be expanded and where no such expansion would be possible; and we secured a greater ability to decide when a new substance with certain stable properties might be a new element. After Newton showed us the gravitational forces that act between celestial bodies, we were given a new prescription for inferring to causes. All we needed to show was that some effect could be generated within the repertoire of Newton's system and we could infer to its reality. So Newton himself showed that the moon's gravitational attraction caused our tides and that comets were deflected in their motion about the sun by the force of gravity from the sun. The scheme even licensed inferences to new bodies. The planet Neptune was discovered in the 19th century by working back to the location of an undiscovered body that could cause perturbations in the planet Uranus' motion.
	In these cases, the added inferential power that comes from knowing more does not come from delivery of some new schema. In the cases above, it is even hard to know what to call the schemas. The inference to the moon's gravity as cause of the tides or to a new planet is not just simply finding an hypothesis that saves the phenomena. It has to do it in the right way. One might be tempted to talk of best explanations, common causes or consiliences. None quite capture the strength of the inference; some inferences to best explanations or common causes can be weak and some strong. The clearest explication of what that right way amounts to is just a local fact: the hypotheses do it in accord with the repertoire of Newtonian gravitation theory. Our confidence in Newton's theory underwrites the strength of the induction.
5. The Control of Inductive Risk
Strategies in a Formal and Material Theory
	In inductive inference we take an inductive risk: the danger that we may accept or accord high belief to a result that turns out to be false. In science we seek to reduce this inductive risk as much as possible. The strategies for controlling inductive risk turn out to be rather different according to a formal theory of induction or a material theory.
	According to a formal theory, we approach the problem in two ways. First we seek to amass as much evidence as possible. The better the evidence the stronger will be our inductive inferences. Second we seek to expand the inductive inference schemas available to us. While this second approach might seem promising, with the notable exception of continuing work in statistics, there has been relatively little work by scientists devoted to expanding our repertoire of inductive methods.
	According to the material theory of this paper, this lacuna is not so surprising. The two approaches to controlling inductive risk cannot be separated. We reduce our exposure to inductive risk by collecting more evidence. At the very same time, exactly because we learn more from the new evidence, we also augment our inductive schemas. For according to the material theory, all these schemas obtain only locally and are ultimately anchored in the facts of the domain. Crudely—the more we know, the better we can infer inductively. The result is that scientists do not need to pay so much attention explicitly to inductive inference. As we saw in the examples of Section 4., with each major advance in science has come a major advance in our inductive powers. The mere fact of learning more will augment their inductive powers automatically.
The Portability and Localization of Inductive Risk
	The above examples also illustrate a common dynamic in our efforts to control inductive risk. We start with an induction that uses some fairly general scheme—enumerative induction or hypothetical induction. The inferences are risky because the generic forms of the schemes are known to be unreliable. We localize the induction to a particular domain, whose material postulates licenses the induction far more securely. In the examples, with some risk, we generalized the physical properties of one sample of a substance to all. When we recognize that substance is an element, we now have recourse to the known constancy of elemental properties to underwrite the inference securely. Because of the correlation of tides with the position of the moon, we hypothesize it as the cause of the tides. Localizing the process to gravitation theory and drawing on the resources of Newton's theory, we become much more confident that the hypothesis is correct.
	This dynamic is a transporting of inductive risk from a schema to a fact, the relevant material postulate. This portability affords an important means of assessing and controlling inductive risk. As long as the inductive risk resides within the schema, we must assess it through a highly problematic judgment of the overall reliability of the relevant schema. We have little chance of coming to a clear judgment let alone determining how to reduce the risk. However once the risk is relocated in a material postulate in some local domain, our assessment of the inductive risk will depend in large measure on our confidence in the material postulate. If the inductive risk is great, we now also have a program for reducing it. We should seek more evidence relevant to the material postulate and perhaps even modify the material postulate in the light of the evidence. The result will be a more secure induction. 
	In short, we can control inductive risk by converting schematic risk into presumptive risk, since the latter can be more accurately assessed and reduced.
Demonstrative Induction as a Limiting Case
	If we can reduce inductive risk by transporting it from the schemas into the material postulates, might we eliminate it entirely? We can, in a limiting case in which the material postulate and the evidence taken together deductively entail the hypothesis at issue. There is no trickery in this limiting case. In effect we are just discovering that we have already taken the relevant inductive risk elsewhere in our investigations when we accepted the material postulate. As a result we do not need to take it again.
	This form of inference has entered the literature under many names, such as demonstrative induction, eliminative induction or Newtonian deduction from the phenomena. The latter term is appropriate since demonstrative inductions arise quite naturally in Newton's analysis of planetary motions. (See Smith, 2002; Harper, 2002) For example, he knew that the supposition that planets are attracted by an inverse square law to the sun is sufficient to generate the ellipses of the planetary orbits. But that the hypothesis saves the phenomena is not decisive. Might there be other force laws that also yield ellipses? Newton showed that this was not possible. In Propositions 44 and 45 of his Principia, he showed that alternatives would fail to yield the stable ellipses observed. Any force law in 1/rn (with r the distance to the sun) would yield a rotation of the axis of the planetary ellipse from which the value of n could be read. The evidence of the fixity of the ellipse conjoined with the relevant propositions deductively entails that n=2 and that force law is an inverse square law.
	There are numerous other examples in the literature. For example, following Planck's 1900 work, it was recognized that the hypothesis of quantization of energy would save the phenomena of the distribution of energy over different frequencies in heat radiation. But is that enough to force us to accept this hypothesis so fundamentally at odds with classical physics? Shortly after, in the early 1910s, Ehrenfest and Poincaré showed that we had to accept it. The extra presumptions needed to make the inference deductive were ones already accepted: essentially that thermal systems are really just systems with many degrees of freedom acting in the most probable way. From the relevant phenomena, they now showed one could deduce the quantization of energy. (See Norton, 1993; and for an example in Bohr's work, Norton, 2000)
6. The Problem of Induction Eluded?
	According to the material theory, each inductive inference is licensed by matters of fact. So we can only learn by induction if we already know something. Does a circularity or harmful regress lurk here? Let us look more closely. The material theory requires the facts that license inductions to be local. That means they hold only in particular domains, so they cannot be the universally applicable truths of logic. They must be contingent. Perhaps they are just brute facts of experience. But aren't such facts too lean in content to license interesting inductions? Suitably rich contingent facts must in turn be learned by inductive inferences that are in turn licensed by further contingent facts. As we trace back the chain of licensing facts, do we not end in circularity; or in an infinite regress; or in a fact that itself has no license? In short, must we not eventually fail in our efforts to license each inductive inference?
	One will quickly recognize this as the analog of a familiar problem for formal theories of induction re-expressed in the framework of a material theory of induction. It is just The Problem of Induction, that most celebrated of philosophical problems traditionally attributed to Hume. In the usual context, it asserts (Salmon, 1967, p.11) that there can be no justification of induction. A deductive justification would violate its inductive character; an inductive justification would either be circular or trigger an infinite regress. Analogously in the material theory, would not a grounding of induction in a universal truth known a priori violate the locality of induction?  And would not a grounding of a local material postulate in local material postulates either be circular or trigger an infinite regress?
	I think not. In my view, when we transport the argumentation used to set up the problem of induction to the material theory, it no longer forces the same sort of difficulty. In formal theories of induction, the vicious character of the circularity is obvious. Crudely, we are supposed to infer by induction on past successes of induction that induction will continue to work. Or we are to infer the continuing success of induction from meta-inductions on successful inductions; and they are justified by meta-meta-inductions on meta-induction; and so on, ever more fancifully. In a material theory, however, we rarely end up with circularity in seeking the material facts that license an induction, as the examples of this paper show. Instead we trace the warrant for each matter of fact back through inductions to further matters of fact, and so on, without triggering any fanciful ascent to meta- and meta-meta-levels. We certainly cannot rule out the possibility that the resulting chains (or, more likely, branching trees) do terminate in brute facts of experience that do not need further justification Or, more realistically, they may terminate in such brute facts of experience augmented by prosaic facts whose acceptance lies outside the concerns of philosophy of science—for example, that our experiences are not fabricated after all by a malicious, deceiving demon. There is no obvious harm in such termination. While a single such brute fact may not be rich enough to license a substantial induction, why should we doubt that enough of them, careful woven together through many smaller induction may not eventually license something grander? In any case, as we trace back the justifications of justifications of inductions, we are simply engaged in the repeated exercise of displaying the reasons for why we believe this or that fact within our sciences. If this process could somehow be prolonged into an infinite regress, why would it be harmful?
	Is the fear that the same brute facts of experience could be incorporated into a very different hierarchy that would somehow give license to a quite different science? If you fear this, the burden of proof is definitely on you to show how such an alternative is possible. Aside from often dubious, local underdeterminations whose importance I believe to be vastly overrated, actual inductive practice in science shows us systematically eliminating indefinitenesses and arriving at a single definite result. There is only one periodic table of the elements and the earth irrevocably orbits the sun, not vice versa. One might be tempted to generate these alternative hierarchies on the cheap by positing Cartesian deceiving demons or mad scientists using powerful computers to feed vivid delusions to our brains kept alive in vats of nutrients. If ours was such a world, our inductions would not be truth conducive. But they would fail at the hands of a blanket skepticism that doubts even the veracity of all sense experience, not because of any weakness in induction specifically.
	In sum, the simple considerations that visit The Problem of Induction on formal theories fail to generate a comparable difficulty for a material theory.
7. Conclusion
	My purpose in this paper has not been to advocate any particular scheme of inductive inference from the many that compete in the literature of philosophy of science. Rather I want to suggest that they are all admissible in the right context and to try to explain why we have such a proliferation of them in enduring conflict. I have urged that we resolve the intractable tension between the universality and the successful functioning of an inductive inference schema by forgoing universality and adopting a material theory of induction. In such a theory, the facts that prevail in each local domain in science license inductive inference schemas that are peculiar to that domain. We justify the inductive inferences of the domain by reference to these facts and not by passing through them to universal inductive inference schemas. I have tried to show how the existing schemas for inductive inference all require some local facts for their justification. Finally I have suggested that any schema with pretensions of universality will fit actual inductions imperfectly and that the fit will become worse as we proceed to narrower domains and the facts licensing the inductions become more specialized. This, I believe, explains a curious phenomenon in science. We can be quite sure of a result in science as long as we look at the particulars of the result and the evidence that supports it, but we often end up struggling to explain by means of standard inductive inference schemas how the evidence can yield that strength of support.
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