Induction

"We learn either by induction or by demonstration," Aristotle writes in the Prior Analytics. "Demonstration develops from universals, induction from particulars." In the Posterior Analytics he argues that the ultimate premises of demonstration must be primary or basic truths, "immediate propositions" that have no other proposition prior to them. Since these basic premises cannot themselves be demonstrated without circularity, they must be known by another means. Aristotle's answer is that "we know the primary premises by induction," and in another place he adds, "it is by intuition that we obtain the primary premises."

The word "intuition" indicates an essential characteristic of the sort of induction which, because it is not itself a form of reasoning, can be prior to all reasoning. Reasoning is discursive, involving steps in which one proposition is drawn from another; intuition, in contrast, is immediate, like an act of seeing. When Aristotle speaks of induction as a kind of intuition, he implies that it consists in the immediate grasp of a universal truth. The role of experience in this process, discussed more fully under the idea of Experience, is to consolidate repeated perceptions into a recognition that these things are of a kind, providing the material from which the mind's intuitive act extracts the universal.

Aristotle does not think that induction in this sense can be methodically prescribed by logical rules. It is a natural act of intelligence, and though men may differ in the readiness of their native wit, induction cannot be improved or rendered more certain by following rules of inference. He does, however, distinguish scientific induction from dialectical and rhetorical induction: the latter proceeds from an enumeration of cases which may not be complete and is at best probable, whereas scientific induction rests on common experience that admits of no exceptions.

Aristotle's conception of induction as intuitive and prior to reasoning leaves open the question of how the move from particular experience to universal truth is accomplished and whether it can be rendered more reliable. Bacon's criticism of Aristotle rests in part on the complaint that induction as Aristotle conceives it is too easily satisfied, and that a systematic method of collecting and comparing instances is needed to correct the understanding's natural tendency to generalize prematurely.

Bacon's criticism of the logic of Aristotle rests on two counts: first, the over-emphasis on syllogisms, whether used dialectically or demonstratively; and second, a superficial understanding of induction. "There are and can exist but two ways of investigating and discovering truth," he writes. "The one hurries on rapidly from the senses and particulars to the most general axioms," and from them deduces the intermediate axioms. "The other constructs its axioms from the senses and particulars, by ascending continually and gradually, until it finally arrives at the most general axioms, which is the true but unattempted way."

Where Aristotle proposes that only the primary truths be established by induction while all the others are to be derived by demonstration, Bacon urges a method which mounts gradually from the least general to the most universal propositions. We should "proceed by a true scale and successive steps, without interruption or breach, from particulars to the lesser axioms, thence to the intermediate, and lastly, to the most general." The twenty-seven tables of instances set forth in the second book of the Novum Organum constitute the heart of this method: tables of presence, absence, and comparison, by which the properties of things are systematically collated and false candidates for causation progressively excluded.

Two consequences follow from these differences. In the first place, Bacon maintains that induction requires the practice of the most detailed and precise method; the collection and arrangement of particulars must be governed by explicit rules. In the second place, since genuine induction depends upon ample experiments, it belongs primarily to the physical or natural sciences. The relation between ordinary experience and planned experiment, discussed under the idea of Experience, is fundamental to Bacon's reform: where Aristotle is satisfied with common experience, Bacon insists that unusual observations must be sought out and experiments deliberately constructed.

Bacon's tables of instances bear a resemblance to what Mill would later formalize as the canons of inductive inference, though Bacon's rules are rules of tabulating the particulars from which intuitive generalizations can be formed, not rules of reasoning in the strict sense. Newton would practice a disciplined version of Baconian induction while framing its results in mathematical form.

Descartes occupies a position somewhere between Aristotle and Bacon on the question of induction. He regards arithmetic and geometry as more certain than the physical sciences because mathematics is largely developed by deduction, whereas the study of nature depends upon induction from experiments. "While our inferences from experience are frequently fallacious," he writes, "deduction, or the pure illation of one thing from another, cannot be erroneous when performed by an understanding that is in the least degree rational." In this lies, for Descartes, the superiority of mathematics.

Nevertheless, Descartes does not exclude induction as the source of the axioms of mathematics or, for that matter, of metaphysics; he only excludes the kind of induction which depends upon experiments. Such axioms as that when equals are taken from equals the remainders are equal, or that the whole is greater than any of its parts, are products of induction, as may be seen from the fact that a child can be taught these general truths only "by showing him examples in particular cases." Similarly, the metaphysical truth in the proposition "I think, therefore I exist" cannot be learned by deduction. "For our mind is so constituted by nature that general propositions are formed out of the knowledge of particulars." Descartes thus agrees with Aristotle that induction supplies the first principles from which deduction proceeds, but he holds that the inductive discovery of axioms in mathematics and metaphysics is more reliable than the experimental induction on which the physical sciences depend.

The distinction between induction by complete enumeration and induction by intuitive generalization, discussed under the idea of Reasoning, is relevant here. Descartes seems to have enumerative induction in mind when he states "that by adequate enumeration or induction is meant that method by which we attain surer conclusions than by any other type of proof, with the exception of simple intuition."

Descartes's emphasis on the certainty of deduction and the superiority of mathematics raises the question of what status inductive conclusions in the experimental sciences can claim. Newton's practice of induction in physics will face this question directly, while Hume's skeptical analysis will challenge the assumption that causal reasoning, on which experimental inference depends, can ever be rationally justified.

Newton's experiments on reflection and refraction, and the laws of optics and mechanics derived from them, seem to be of the sort from which universal laws are directly induced, much as, according to Aristotle and Descartes, the axioms of mathematics or metaphysics can be induced from simple experiences. Yet Newton does not think that the inductive establishment of such laws is as certain as demonstration. "The analytic method," he writes, "consists in making experiments and observations and in drawing general conclusions from them by induction. And although the arguing 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."

At the beginning of Book III of the Principia, Newton sets out four "Rules of Reasoning in Philosophy." We are to admit no more causes of natural things than are necessary to explain their appearances. We are to assign the same causes to the same natural effects. The qualities of bodies that cannot be increased or decreased are to be taken as universal. And propositions gathered from phenomena by induction are to be held as general, "notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur by which they may either be made more accurate, or liable to exceptions." His celebrated phrase "hypotheses non fingo" expresses the principle that whatever is not deduced from the phenomena has no place in experimental philosophy.

The conception of a well-constructed experiment, in which the operation of a universal law is exhibited in a single case, distinguishes Newton's practice from mere enumeration of instances. The experiment, if its conditions are properly controlled, makes the universal manifest in the particular, a conception related to Aristotle's account of scientific induction but given mathematical precision through the Newtonian framework. The relation between experiment and mathematical demonstration is discussed further under the ideas of Mechanics and Physics.

Newton's practice of induction in physics gave the method its greatest credibility, but it left open the philosophical questions that Hume would press: whether the move from observed regularities to universal laws can be rationally justified, and whether the causal reasoning on which experimental inference depends is itself grounded in anything more than custom.

Hume offers two reasons for the inconclusiveness and uncertainty which he thinks qualify all generalizations from experience. The first calls attention to the fact that, unlike mathematical reasoning, inferences from experience depend on the number of cases observed. "The conclusions which reason draws from considering one circle," he says, "are the same it would form upon surveying all the circles in the universe. But no man, having seen only one body move, after being impelled by another, could infer that every other body will move after a like impulse." The principle which determines the mind to form such a conclusion is, according to Hume, "Custom or Habit"; and precisely because inductive generalization is an effect of custom rather than of reasoning in the strict sense, the strength of the induction varies with the number of cases from which it arises.

To this first point Hume adds a second concerning the similarity of the cases under observation. Analogy, he says, "leads us to expect from any cause the same events, which we have observed to result from similar causes. Where the causes are entirely similar, the analogy is perfect, and the inference drawn from it is regarded as certain and conclusive. But where the objects have not so exact a similarity, the analogy is less perfect, and the inference is less conclusive." Since all the relevant cases can never be exhaustively observed, the inference from customary conjunction must always remain uncertain. The relation between cause and effect, discussed more fully under the idea of Cause, cannot on Hume's view be established by reasoning but only by repeated experience.

Hume's analysis does not counsel the abandonment of inductive practice; it requires only philosophical honesty about its epistemic status. The beliefs produced by custom may be indispensable, but they are not rationally demonstrable. The question of whether nature is uniform, on which all inductive inference depends, cannot itself be answered by induction without circularity.

Hume's challenge divides the subsequent tradition. Kant accepts it as a genuine problem and attempts to answer it by grounding the principle of causation in the structure of the understanding rather than in experience. Mill, proceeding in the opposite direction, treats the uniformity of nature as itself an inductive generalization of very great confidence, though his critics have observed that this answer begs the question Hume raised.

Kant's treatment of induction is inseparable from his response to Hume on the question of causation. Hume had argued that the principle "every event has a cause" cannot be derived from experience by induction, nor demonstrated by pure logic, and that inductive generalizations therefore rest on nothing firmer than custom. Kant accepts the force of this argument against empiricist accounts of causation, but he denies Hume's conclusion. The principle of causation, Kant holds, is synthetic a priori: it is not contained analytically in the concept of an event, yet it is necessarily and universally true because it is a condition of the possibility of experience itself. Without the concept of causation, we could not order events in time, and experience of a determinate world would be impossible. The understanding, as Kant writes, "does not draw its laws from nature, but prescribes them to nature."

This does not render induction superfluous. We still require experience to determine which particular things are causes of which particular effects. The a priori principle tells us that every event has a cause; it does not tell us which cause belongs to which event. Empirical investigation remains indispensable for that. But the general framework within which inductive inference operates is not itself an inductive generalization; it is a transcendental condition, a contribution of the mind to the structure of experience. The bearing of this on the question of necessary and contingent knowledge is discussed under the idea of Necessity and Contingency.

Kant's position represents, in a sense, a compromise between the Aristotelian view that induction yields certain first principles and the Humean view that all inference from experience is uncertain. The general principle of causation is secure, on Kant's account, but particular causal laws remain products of empirical inquiry and are therefore subject to the limitations that Hume and Newton alike acknowledged.

Whether Kant's solution succeeds depends on whether the theory of synthetic a priori knowledge is accepted. Mill, rejecting the notion that any knowledge is a priori, attempts to ground induction entirely within experience, treating even the principle of causation as an inductive generalization of very wide scope. The question of what, if anything, the mind contributes to the structure of experience prior to all observation remains one on which the empiricist and rationalist traditions continue to differ.

Mill's System of Logic represents the most sustained attempt in the empiricist tradition to formulate the rules of inductive inference. Against Kant, Mill maintains that no knowledge is a priori; even mathematical axioms are, in his view, inductive generalizations of very great confidence. Against the Aristotelian conception of induction as intuitive, Mill treats induction as a form of reasoning whose validity depends on conformity to determinate logical rules. "The business of inductive logic," he writes, "is to provide rules and models to which if inductive arguments conform, those arguments are conclusive."

His four or five methods of experimental inquiry bear a resemblance to Bacon's more numerous tables of instances, but they differ in character. Mill's methods are attempts to formulate the rules of inference for inductive reasoning: the Method of Agreement, the Method of Difference, the Joint Method, the Method of Residues, and the Method of Concomitant Variation. Each method specifies conditions under which a causal connection can be identified by comparing instances in which the phenomenon occurs with instances in which it does not. The relation of these methods to the general theory of causation is treated under the idea of Cause.

Each of Mill's methods requires a rule of inference which is itself a universal proposition about the relations of cause and effect or about the order and uniformity of nature. His critics have asked whence these universal propositions come and have pointed out that Mill cannot answer that they are themselves conclusions of inductive reasoning without begging the question. The "consilience of the results" of induction and deduction, "each corroborating and verifying the other," is, as Mill himself observes, "requisite to give to any general proposition the kind and degree of evidence which constitutes scientific proof."

The difficulty of grounding the canons of inductive inference without circularity remains, on Mill's account, unresolved. Whether induction is better understood as a form of reasoning subject to logical rules, as Mill maintains, or as an intuitive act of the intellect, as Aristotle holds, is a question on which the tradition does not reach agreement. James, approaching the problem from a different direction, will propose that the justification of inductive inference lies not in its logical form but in its practical consequences.

James approaches the problem of induction from a standpoint that differs from both the empiricist and the rationalist traditions. Rather than asking whether inductive inference can be justified by appeal to the uniformity of nature, as Mill proposes, or grounded in the a priori structure of the understanding, as Kant maintains, James asks what practical difference it makes to treat an inductive conclusion as true. On his view, truth is the name we give to ideas that work: an inductive hypothesis earns its credentials by reliably guiding successful action, and it loses them when it ceases to do so.

This does not amount to a license for arbitrary inference. Hypotheses that lead to successful predictions are retained; those that fail are abandoned. The process, as James describes it, is self-correcting without requiring any foundational justification prior to inquiry. The question of whether nature is truly uniform, which had occupied Hume, Kant, and Mill, is, on the pragmatist view, less important than the question of whether acting on the assumption of uniformity continues to yield results. The relation between belief and action, treated more fully under the ideas of Truth and Knowledge, is central to this position.

James also draws upon his psychological work to give a naturalistic account of the mind's tendency to generalize from experience. The formation of habits, both of perception and of inference, is for James a biological phenomenon: the mind evolved to detect regularities because detecting them is adaptive. The laws of thought that logicians formulate are, on this view, idealizations of processes that effective thinkers have always followed. The connection between this naturalistic account and the Aristotelian conception of induction as a natural act of intelligence, rather than a rule-governed procedure, may be closer than James himself recognized.

James's pragmatic treatment of induction dissolves the traditional question of justification in favor of the question of consequences. Whether this dissolution answers Hume's challenge or merely changes the subject is a matter on which subsequent philosophers have disagreed. What it makes clear, however, is that the problem of induction, like the problem of knowledge generally, cannot be separated from questions about the purposes for which knowledge is sought.