In deference to the occasion [the Aquinas Lecture], we have spent so long on assaying the stability of some of the bridges that recent scholars have thrown across the gap between the tradition of the Posterior Analytics and modern views on what constitutes the basic form of scientific inference that we are going to have to telescope these later views in a rather summary way. The earlier emphasis on demonstration was not entirely lost; indeed, it took daring form in Kant’s Metaphysical Foundations of Natural Science. But two other kinds of inference more or less supplanted demonstration as the paradigm of the scientist. (Demonstration remained a will-o’-the-wisp for those who took mechanics as the paradigm of science; it is all too easy to see its conceptual structure, whether in its Newtonian or more recent relativistic forms, as so luminous as also to be necessary.) One of these types of inference has a familiar label: “induction.” The other (which we shall call “retroduction”) even still has not, which is rather extraordinary. /67/

Nominalism. “Modernity” began, as everyone knows, with the via moderna of the fourteenth century, which was “modern” in part because of its rejection of the notion of demonstration central to the Aristotelian tradition it opposed. The notion of necessity that the possibility of demonstration in natural science conveyed seemed to a great many theologians, from the first introduction of the Aristotelian “natural” works in the early thirteenth century onward, to require an unacceptable restriction on God’s freedom in creating, and an equally unacceptable determinism of causal action on the part of creatures. The fourth of the 219 propositions condemned by Bishop Tempier in 1277 rejects the claim “that one should not hold anything unless it is self-evident or can be manifested from self-evident principles.”92 The idea that the human mind can, on the grounds of reflection, on what is perceived, affirm a necessary relation between cause and effect was regarded as a challenge to the notion of miracle so central to the Christian economy.

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The nominalism of Ockham, and especially the more extreme versions of that nominalism in the works of Nicholas d’Autrecourt, Jean de Mirecourt and Robert Holkot, presented an alternative to the Aristotelian scheme in which the emphasis has shifted from the universal to the particular, from demonstration to induction. /68/ Induction itself in one sense resembles the epagoge of the earlier tradition because it begins from observed regularities of co-occurrence in the sensible world. But it differs in a crucial way: instead of the intellect’s going on to grasp the nature of the cause sufficiently clearly to allow an unqualified affirmation of necessary connection between cause and particular effect to be made, induction according to Ockham rests simply on the evidence of the co-occurrences—and has the degree (and only the degree) of logical force that this conveys. It is further dependent on a principle of uniformity of nature which itself has to be regarded as hypothetical, since it rests on the ordination of God’s will to a common course of nature, which is not absolute but, in principle at least, open to exception. One thing is said to be the efficient cause of another if in the presence of the first the second follows, nothing more. A causal relation between A and B cannot, therefore, be known a priori; it can be learnt only from the repeated experience of their conjunction. Induction is thus a matter of generalization from a limited set of instances of a regularity. It is, if you will, a kind of sampling.

Nicholas goes on to draw a more skeptical conclusion than Ockham had done. Since the existence of effects cannot strictly entail the existence of corresponding causes, the best that one can /69/ aspire to in natural science is a degree of probability. The attempt to infer to essence or substance from perceived particulars must necessarily beg the question. In turning away so decisively from the ideal of demonstration, Nicholas is not especially advocating the importance in human terms of empirical investigation: what little can be learned (he says) can be learned in a short while, but it must be learned from things, not from the works of Aristotle.93 There has been a great deal of discussion among historians of later medieval thought in late years about the influence of nominalist ideas on the origins of modern science. Our concern here is with the notion of induction only. The nominalists advocated the substitution of induction (in the sense of generalization) for demonstration as the paradigm mode of inference in natural science, challenging the fundamental notion of nature on which the earlier account had rested. But they did not work this up into a formal account of method in natural science.

Bacon. That was left to Francis Bacon, and this brings us to the second significant moment in the long story of inductivism. In the New Organon (1620), Bacon proposed a new method which was to replace

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that of the Posterior Analytics. He saw the two methods as diametrically opposed: /70/

There can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgement and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried.94

Bacon tries to find a middle position between the essentialism of Aristotle and the more extreme forms of nominalism:

Though in nature nothing really exists besides individual bodies, performing pure individual acts according to a fixed law, yet in philosophy this very law, and the investigation, discovery, and explanation of it, is the foundation as well of knowledge as of operation. And it is this law with its clauses that I mean when I speak of forms, a name which I the rather adopt because it has grown into use and become familiar.95 /71/

There is a shift here from forms, understood as intrinsic to natural things, to forms understood as laws, as modes of action extrinsically imposed by a Lawgiver. These latter can still be called “eternal and immutable,” and hence the natural philosopher can still aim at certainty. But the mode of attaining it is quite different.

Bacon sets out to construct natural histories organized by tables of presence, absence and degree (from which J. S. Mill much later got his methods of Sameness, Difference, and Concomitant Variation). These tables link regularly co-occurring factors; this is what for Bacon defines induction. The evidence for causal relationship comes from finding factors either invariably linked in observation or co-varying in a significant way. Evidence against is provided by absence, when presence might have been expected. The method is thus one of generalization, with an element of testing provided by the tables of absence.96 Such a method “leaves but little to acuteness and strength of wits, but places all wits and under- standings nearly on a level.”97 (There is some disagreement as to how seriously he meant this.) And it provides “not pretty and probable conjectures, but certain and demonstrable knowledge.”98 Again this would need to be qualified. For one thing, he stresses the “dullness, incompetency and deceptions of the senses.”99 For /72/ another, he treats his “axioms” as plausible conjectures meant to be tested by the “trial by fire” that crucial experiment can afford. Still, he does assume that at the end of that sometimes laborious process, a causal link that is “sure and indissoluble” can be found.100

Though Bacon is trying very hard to separate himself from the Aristotelian tradition, one can still catch echoes of epagoge here. In

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particular, he is seeking to discover regular and reliable correlations; his “laws,” though their ontological basis is quite different, loosely correspond to the “natures” of the older tradition. The process of discovery, an “act of the intellect” left unexplained by Aristotle, is not spelled out in terms of the logical procedures to be followed. Bacon provides, then, not just a conception of what counts as science, but a general methodology to enable that goal to be attained.

If this were all, then Bacon could be presented as the inductivist par excellence, his aphorisms constituting the “Ur” text for those concerned to lay out the method of induction. But as we shall see in more detail later, a second quite different sort of inference is also hinted at in the pages of the New Organon. It is doubtful that Bacon was aware of the tension between the two methods or of the incompleteness of induction without the other mode of inference to back it up. /73/ Induction is a matter of noting correlations between observables; unless both elements related by the “law” are observable, a correlation between them obviously cannot be discovered, on the basis of sense-evidence alone. Even if one were to extend the notion of observation (and Bacon was surprisingly wary of such an extension, recommending against a dependence on the new-fangled instruments just then coming into use), it would still be true that inductive method is strictly limited to factors that are observable in some sense. How, then, is the story to be extended to unobservables? Bacon in his famous discussion of the nature of heat in Book II of the New Organon showed himself perfectly willing to assert that the “heat” of a body is to be understood in terms of the motions of imperceptibly small parts of the body. We shall, however, leave this question aside for the moment in order to complete this quick survey of significant moments in the development of ideas about induction.

Hume. Hume’s contribution was of a different sort, closer in spirit to that of Nicholas d’Autrecourt (whose name he had almost certainly never heard of ) than of Bacon. Though in the introduction to the youthful Treatise of Human Nature (1739), he had promised “a complete system of the sciences, built on a foundation almost entirely new,” in practice, he left the natural sci- /74/ ences to Newton and his heirs, content to take Newton at his word that the method of these sciences is inductive:

And although the arguments from experiments and observations by induction be no demonstration of general conclusions, yet it is the best way of arguing which the nature of things admits of, and may be looked on as so much the Stronger, by how much the induction is more general.101

Hume’s interest was not specifically in induction as it occurs in natural science. His challenge was to inductive procedure generally, whether in the routines of daily life or in the more technical pursuits of the natural philosopher. “Reasoning concerning matters of fact” (he did not use the

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term, “induction”) is, according to him, founded on the relation of cause and effect, which in turn reduces to a combination of constant conjunction, contiguity and temporal succession. The idea of necessary connection, which we also associate with the causal relationship, can be explained as an expectation brought about by association or habit. But such an expectation in no way justifies the prediction that C, which has been constantly conjoined with E in the past, will once again be followed by E on the next occasion. This skeptical undermining of the rationality of belief /75/ in the logical force of inductive inference offers no threat to daily living or to the natural sciences, according to Hume, but only because it is powerless to overcome the natural sentiments and convictions that govern daily life.

This is the famous “problem of induction” which went almost unnoticed among Hume’s first readers but which has so intrigued twentieth-century analytic philosophers.102 If one accepts Hume’s starting points, that all our ideas are derived from sense-impressions or inner feelings, and that causal relationship reduces to nothing more than the fact of constant conjunction in the past of certain classes of sense-impressions lacking any intrinsic connection, then indeed it would be difficult to warrant belief in induction. (One can never, of course, conclusively prove that on a given occasion E must follow C; Hume’s use of the phrase “demonstrative reasoning” is part of the problem here.) But then, of course, these same starting points would not enable us to distinguish between genuine laws and accidental correlations. And causal inference to underlying structure is excluded. Hume’s radical empiricism could take account neither of law nor of theory, as these had come to be understood in the natural science of the previous century. /76/

If the only form of nondeductive inference were to be the inductive one, as empiricists have always tended to believe, then Hume’s problem would still pose a troublesome challenge. It might perhaps be too easy to say that the famous “problem” is an artifact, that it vanishes if this faulty assumption be rejected. But, as we shall see, if the right form of nondeductive inference be recognized, genuine laws and accidental generalizations can be readily distinguished and the connection between cause and effect becomes something more than constant conjunction.

J. S. Mill. The fourth moment in the tale of induction can be kept a brief one. Mill’s System of Logic (1843) may be called the “Inductivist Manifesto.” There is only one type of nondeductive inference, and that is induction, understood as a straightforward procedure of generalization or sampling: “It consists in inferring from some individual instances in which a phenomenon is observed to occur, that it occurs in all instances of a certain class.”103 This method, properly used, enables the “ultimate laws of nature” to be discovered. Its validity rests upon a general principle of uniformity

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of nature. (His attempt to rest this principle in turn upon a broader induction is clearly fallacious.) Since causal relations hold only between observables, there can be no inference to unobservables. It need hardly be said that at the /77/ very time Mill was writing, the growing reliance of natural scientists on noninductive inference to and from unobservables was leading to dramatic advances in fields like optics, chemistry, and theory of gases.

Logical Positivism. The best-remembered moment in the story occurred in our own century. A brief reminder of the salient points must suffice. The logical positivists used the term, “induction,” to cover all forms of nondeductive inference. Carnap attempted, unsuccessfully, to construct a theory of logical probability, what he called an “inductive logic,” that would link hypothesis and evidence by means of some sort of “credence function.” Such a logic, he believed, would provide the rules for inductive thinking, thus enabling rational choices to be made between hypotheses. He rejected the traditional view of inductive reasoning that would make it an inference from premisses (evidence) to conclusion (usually a law), proposing instead that it is an assessment of the credibility of a particular hypothesis in the light of specific evidence, however the hypothesis be arrived at.104 The hypothesis itself might be a law, a singular prediction, or a theory. And the logic, though a logic of induction (in Carnap’s sense of the term), was deductive in form, enabling him (he hoped) to evade Hume’s challenge. /78/

Much more commonly, however, the focus of positivist concern was on how to get from the singular observation statements from which science begins to the laws of which they believed “finished” science to consist. Induction in this case would involve something like the traditional methods of Sameness, Difference, and Concomitant Variation, that Mill had taken over from the works of Bacon and John Herschel.105 It would essentially be a special kind of sampling. The “laws” arrived at in this way would, thus, be empirical generalizations. But not all laws are of this sort. Besides empirical laws, there are also “theoretical” laws, those that make use of “theoretical” terms, i.e., terms that refer to hypothetical (unobservable) entities.106 But how are these “laws” to be arrived at? Not by means of inductive generalization clearly. Carnap saw the difficulty:

How can theoretical laws be discovered? We cannot say: “Let’s just collect more and more data, then generalize beyond the empirical laws until we reach theoretical ones.” No theoretical law was ever found that way . . . . We never reach a point at which we observe a molecule . . . . For this reason, no amount of generalization from observations will ever produce a theory of molecular processes. Such a theory must /79/ arise in another way. It is stated not as a generalization of facts but as a hypothesis. The hypothesis is then tested in a manner analogous in certain ways to the testing of an empirical law.107

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“Analogous,” perhaps, but exhibiting important differences. Carnap was struggling toward a sharper distinction between theory and law, and between the processes of inference involved in each of these. But it was difficult to admit another mode of inference. An empiricist could never feel entirely easy with theoretical terms. And the “logical empiricists,” as the group preferred to be called in its later years, went to great lengths to contrive devices like “correspondence rules” to get around the hard fact that theoretical laws could simply not be derived from empirical laws, indeed that the term “law” here is close to equivocal.108 Their ambivalence toward a distinctively theoretical mode of inference led, in turn, to a famous ambivalence in regard to the existence of theoretical entities. Their instinct as positivists and empiricists was to regard theoretical terms simply as heuristic devices. But a growing appreciation of the difficulties to which this led would eventually encourage some of them, at least, to embrace a somewhat hesitant realism.109 /80/

So far, explanation has not been mentioned in this discussion of logical positivism. The well-known deductive-nomological (D-N) model proposed by Hempel and Paul Oppenheim took laws to be the primary explainers, and individual events to be the normal explananda. The apparent symmetry between explanation and prediction to which this led gave rise to problems severe enough to force the abandonment of the model. Explanation could simply not be reduced to subsumption under laws. Indeed, it appeared, laws are what have to be explained, rather than the primary explainers. The gas laws do not explain the behavior of gases. A theory of gases is needed, one that postulates an underlying structure of entities, relations, processes.110

So here, once again, something other than the product of induction, empirical laws, is needed if an adequate account is to be given of how explanation functions in science. And there is one further context where overreliance on induction also led to insuperable problems, as already noted, and that was in finding an effective way to distinguish between genuine laws and accidental generalizations. If all one has is empirical generalization, à la Hume, this crucial distinction hovers out of reach. /81/

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