INDUCTION

The name "induction," derived from the Latin translation of Aristotle's epagoge, will be used here to cover all cases of nondemonstrative argument, in which the truth of the premises, while not entailing the truth of the conclusion, purports to be a good reason for belief in it. Such arguments may also be called "ampliative," as C. S. Peirce called them, because the conclusion may presuppose the existence of individuals whose existence is not presupposed by the premises.

Thus, the conclusion "All A are B " of an induction by simple enumeration may apply to A 's not already mentioned in the finite number of premises having the form "A_i is B." Similarly, in eduction (or arguments from particulars to particulars) the conclusion "Any A is B " is intended to apply to any A not yet observed as being a B.

It would be convenient to have some such term as adduction to refer to the sense of induction here adopted, which is broader than the classical conception of induction as generalization from particular instances. Most philosophical issues concerning induction in the classical sense arise in connection with the more general case of nondemonstrative argument.

In what follows it will be convenient to use Jean Nicod's expression "primary inductions" to refer to those nondemonstrative arguments "whose premises do not derive their certainty or probability from any induction." Problems of philosophical justification are most acute in connection with such primary inductions.

It may be added that "mathematical induction" is a misnomer because the useful types of reasoning so labeled are rigorously demonstrative. Given that the first integer has a certain property and also that if any integer n has that property then so does n + 1, the next, it follows demonstratively that all the integers have the property in question. Inductive arguments, as here conceived, do not constitute mathematical or logical proofs; by definition induction is not a species of deduction.

Types of Inductive Arguments

In addition to the types of arguments already mentioned, the following are most frequently discussed:

1. Elaborated induction (as it might be called) consists of more or less sophisticated variations of induction by simple enumeration, typically including supplementary information concerning the mode of selection of the individuals named in the premises and perhaps including reference to negative instances.
2. Proportional induction is inference from the frequency of occurrence of some character in a sample to the frequency of occurrence of the same character in the parent population—that is, from "m ₁/n ₁ A 's selected by a stated procedure P are B " to "m ₂/n ₂ A 's are B." Here the ratio stated in the conclusion may be other than the one stated in the premise; it is often advantageous to locate the final ratio within a certain designated interval.
3. Proportional eduction is argument from sample to sample. From the same premises as in proportional induction a conclusion is drawn concerning approximate frequency of occurrence in a further sample obtained by the same procedure or by another one.
4. Proportional deduction (commonly called "statistical syllogism") is inference from "m/n C 's are B " (where m/n is greater than 1/2) and "A is a C " to "A is a B."

In all the above cases modern writers usually insist upon inserting some more or less precise indication of probability or likelihood, either within the conclusion itself or as an index of reliability attached to the mark of inference ("therefore," "hence," or the like). Careful attention to the probability or likelihood attributed to a given inductive conclusion is a distinct merit of modern treatments of the subject.

The foregoing list cannot claim to be exhaustive, nor are its items to be regarded as mutually irreducible. There is no general agreement concerning the basic forms of inductive argument, although many writers regard simple enumeration as in some sense the most fundamental.

History of Inductive Methods

Interest in the philosophy and methodology of induction was excited by the extraordinary successes of natural science, which tended to discredit the rationalistic conception of knowledge about matters of fact. The classical writers on the subject, from Francis Bacon on, have lamented the powerlessness of deduction to do more than render explicit the logical consequences of generalizations derived from some external source. If recourse to intellectual intuition or to self-evidence is repudiated as a source of factual knowledge, nothing better seems to remain than reliance upon the empiricist principle that all knowledge concerning matters of fact ultimately derives from experience. However, experience, whether conceived as sporadic and undirected observation or as the systematic search for specific answers extorted by planned experiment, seems to supply knowledge only of particular truths. Empiricists are therefore faced with the problem of accounting for the crucial step from knowledge of experiential particulars to reasoned acceptance of empirical generalizations sufficiently powerful to serve as the major premises of subsequent logical and mathematical deduction.

The aspiration of early writers was, characteristically, to demonstrate the conclusions of acceptable inductive arguments as true; not until the end of the nineteenth century did a more modest conception of inductive argument and scientific method, directed toward acquiring probability rather than certainty, begin to prevail.

Problem of Induction

The celebrated problem of induction, which still lacks any generally accepted solution, includes under a single heading a variety of distinct, if related, problems. It is useful to distinguish the following:

1. The general problem of justification: Why, if at all, is it reasonable to accept the conclusions of certain inductive arguments as true—or at least probably true? Why, if at all, is it reasonable to employ certain rules of inductive inference?
2. The comparative problem: Why is one inductive conclusion preferable to another as better supported? Why is one rule of inductive inference preferable to another as more reliable or more deserving of rational trust?
3. The analytical problem: What is it that renders some inductive arguments rationally acceptable? What are the criteria for deciding that one rule of inductive inference is superior to another?

These problems may be briefly labeled "justification," "differential appraisal," and "analysis." Many writers on induction have also occupied themselves with the task of codification, the formulation of a coherent, consistent, and comprehensive set of canons for the proper conduct of inductive inference. Important as it is, this task is not distinctively philosophical, except insofar as it requires in advance answers to the questions listed above.

In practice the three problems here distinguished cannot be pursued separately; a comprehensive general defense of inductive procedures involves specification, inter alia, of legitimate forms of inductive argument, and selection between alternative inductive rules or methods must rely, explicitly or not, upon determination of what, if anything, makes an inductive argument "sound." The why of inductive argument cannot profitably be isolated from the how.

It is characteristic of much recent investigation of the subject to concentrate on the last two of the problems listed, often in the hope of formulating precise canons of inductive inference (an inductive logic). These comparative and analytical versions of the problem of induction are thought worth pursuing even by writers who reject the general problem of justification as insoluble.

hume's view of causation

For better or worse, all modern discussion of the philosophy of induction takes off from David Hume's celebrated analysis of causation, whose connection with the philosophical problems of induction (a word that Hume never used) arises from his view that all reasoning concerning matters of fact is founded on the relation between cause and effect. Although Hume may be held to have given undue prominence to causation (his skeptical conclusions do, in fact, challenge every kind of nondemonstrative argument, whether or not grounded in causal imputation), it is easy to overlook and to be misled by the special form in which he conceived the problem of justification.

Hume, unlike such later writers as J. S. Mill, was not satisfied to analyze the notion of cause and effect into the notions of spatial contiguity, temporal succession, and joint occurrence; he fatefully added to these the criterion of "necessary connexion." That objects of certain kinds have been conjoined or associated in past experience might be no more than an extended coincidence. Something more is needed before one event can properly be recognized as the cause of the other; we must be able to pass from post hoc to propter hoc. In predicting a putative effect of a given event we can ensure contiguity and succession by choosing to look only for a spatiotemporally proximate event, and memory (if that can be relied on) will furnish knowledge of constant conjunction in the past. Whether we are truly justified in predicting the occurrence of the putative effect will therefore turn entirely upon whether there is good reason to assert that it is necessarily connected with its neighbor. Hume, in effect, challenged his reader to find anything in the observation of a single case of supposed causal action (for instance, in the favorite example of a collision between two billiard balls) that answers to the required "necessary connexion" between two events. No observation, however attentive, will discover more than contiguity and an internal habit of expecting association. Nor will examination of a series of cases, all exactly alike, help at all: A sum of zeroes is still zero.

But what did Hume mean by "necessary connexion"? Although he did not tell us in so many words, his main proof that we can "never demonstrate the necessity of a cause" rests simply upon the conceivability, and hence the logical possibility, of an event's being bereft of its putative cause. He seems, therefore, to have implied that our notion of a cause and its effect requires the existence of the one to be entailed by the existence of the other. If so, it does not need much argument to show that we can have no impression (direct sensory experience) of such entailment. Hume concluded that necessity cannot reside in the external world but must arise, as an idea, from an internal impression of the mind, a "determination to carry our thoughts from one object to another."

Repeated observation of the association of events leads us to the habit of expecting the association to continue "by means of an operation of the soul … as unavoidable as to feel the passion of love, when we receive the benefits" (Enquiry concerning the Human Understanding, Sec. 5, Part 1). Our idea of necessary connection is nothing more than an internal response to the habit of expecting effects: "Upon the whole, necessity is something in the mind, not in objects." At this point skepticism is just around the corner; we are on the verge of such famous conclusions as that "all probable reasoning is nothing but a species of sensation" (Treatise, Book I, Part 3, Sec. 8).

The reference to habit or custom explains nothing, of course, and is at best only a concise reference to the truism, which according to Hume's view simply has to be accepted, that men do in fact expect events to be accompanied by effects. Without such habits of causal expectation men could hardly have survived—but this reflection, itself based on induction, cannot be a reason for belief in causation. For a philosopher so critical of such allegedly occult entities as power and energy, Hume was strangely carefree in his reliance upon habit or custom as a vera causa. In keeping with his own principles he ought to have turned as skeptical an eye on habit as on cause and ought to have concluded that our idea of habit is derived from nothing more than a habit of expecting that a man who acts in a certain way will continue to do so. But now the account looks circular. Have we any better reason to believe in the existence of habits—even if construed, in as reductionist a fashion as possible, as mere constant conjunctions—than we have to believe in causes? And would not everything that tended to show we have no sufficient basis in external experience for belief in the objective reality of causal connection also tend to show, by parity of reasoning, that we have no basis for believing in the existence of those habits that are invoked at least to explain, if not to justify, our ordinary causal beliefs?

It has seemed to nearly all of Hume's readers that his method must lead to a skepticism more sweeping than he himself was perhaps willing to recognize or to accept. If Hume had been correct about the origin of the idea of necessity, he would have been committed to a totally skeptical answer to the general problem of justification. Whether or not we can escape from the bondage of causal expectation, we are at any rate free to see that such a habit can provide no reason, in Hume's sense, for the belief in causal connection. And once we see this, wholesale skepticism concerning inductive inference seems inescapable.

Hume's skeptical conclusions cannot be dismissed on the ground that they originated in an oversimplified psychology of ideas and impressions, for his argument can, with little difficulty, be made independent of any psychological assumptions. Cause and effect are logically independent, not because repeated search fails to find any logical connection, as Hume's own account misleadingly suggests, but because it is a part of what we mean by cause and effect that the two shall be logically separable. It is tempting to say, then, that there is no reason why the separable consequent should follow its antecedent in any particular instance. We can very well imagine or conceive the cause's occurring without its usual consequent, and, in Hume's words, "nothing of which we can form a clear and distinct idea is absurd or impossible" (A Treatise of Human Nature, Book I, Part 1, Sec. 7).

neo-humean arguments

Even if Hume was wrong in including logical necessity in the idea of causal connection, a neo-Humean can correct his argument without weakening its skeptical force. It is reasonable to say that what distinguishes a causal connection from a merely accidental association is that empirical rather than logical necessity obtains between the two events. This, in turn, may be rephrased by saying that the observed conjunction is a case of lawful and not merely accidental association. But then Hume's challenge to discover such lawfulness in experience remains as formidable as ever; no matter how many instances of joint occurrence we encounter, we will never observe more than the de facto association and will never have ultimate, noninductive grounds for believing in a de jure connection.

Thus, Hume's problem can be put into modern dress, without restriction to causal inference, as follows: An inductive inference from an observed association of attributes (A_n—B_n ) can justify inference to another case (A_{n + 1}—B_{n + 1}) or inference to the corresponding generalization ("All A are B ") only if the association is somehow known to be lawlike, not merely accidental. Yet how can this be known in primary inductions that do not themselves rest upon the assumed truth of other laws? Certainly not by immediate experience, nor a priori, nor, without begging the question, by appeal to induction.

The sharpest form of this version of the problem (called by its author the "new riddle of induction") is that of Nelson Goodman. Suppose all emeralds examined before a certain time t have been green; use the label "grue" for the property of being green up to the time t and being blue thereafter. Then all the evidence supports equally well the competing laws "All emeralds are green" and "All emeralds are grue." Here an instance of the comparative problem is raised in a particularly pointed and instructive way.

Goodman's challenge awaits an answer. Some writers have hoped to defend the received or standard modes of inductive argument by invoking criteria of relative simplicity. But apart from the yet unsolved problem of clarifying what simplicity is to mean in this connection, there seems no good reason why nature should obligingly make correct inference simple; often enough the bestconfirmed law is less simple than others that would accord with the given evidence. Goodman's own suggestion to restrict defensible inductions to "entrenched" predicates (roughly speaking, those that have been frequently employed in previous inductive judgments) seems less than satisfying.

From the standpoint of the philosophy of induction the chief significance of Hume's memorable discussion (apart from its tonic effect in disturbing "dogmatic slumber") is that it brought into full daylight the problem of distinguishing between a merely accidental series of associations and the genuine laws that we seek by means of inductions.

deductive standard of justification

A demand that induction be justified arises, of course, from some supposed deficiency or imperfection. If all were obviously well with inductive argument, there would be no point in asking for any defense or justification. It is therefore of the first importance to be clear about the alleged weakness or precariousness of induction and the corresponding standard of justification to which appeal is covertly made. We need to know what is supposed to be the trouble with induction, for only when the disease is understood will the search for a remedy have much prospect of success.

The root of the trouble is plain enough in the writings of a hundred writers who have trodden in Hume's footsteps. All have been haunted by the supposedly superior certainty of demonstrative reasoning. If valid deduction from premises known to be true transmits certainty to the conclusion, even the best induction will seem inferior by comparison. (John Locke said that induction from experience "may provide us convenience, not science"—Essay concerning Human Understanding, Book IV, Ch. 12, Sec. 10.) The nagging conviction that induction somehow falls short of the ideals of rationality perfectly exemplified in valid deductive argument has made the problem of induction needlessly intractable.

If Hume, for instance, did not require that induction be shown as somehow satisfying the criteria of valid deduction, an answer to his question about how "children and peasants" learn from experience would be easy. The method employed, as he himself stated, is that of arguing from similarity of causes to similarity of effects. However, such an answer would obviously not have satisfied him, because this method will not guarantee the truth of the conclusion drawn; that is, it is not the kind of method that would be acceptable as justifying a valid deduction. Hume would have liked an inductive conclusion to follow from (be entailed by) premises known to be true, for anything less would not have seemed genuinely reasonable. Having shown, in effect, that no reason of this kind can be produced for primary inductions, he was forced to regard the question of justification as demonstrably insoluble. This conclusion has the notable inconvenience of leaving the comparative problem also insoluble (while the analytical task vanishes for lack of an object).

Hume's conclusion must be granted if his is the only sense of "reason" in point. If we never have a reason for an inductive conclusion unless we know the conclusion to follow strictly from premises known to be true, then we can have no reason for believing in primary inductive conclusions; it is as reasonable to expect that thistles will bear figs, or something equally absurd, as it is to expect anything else extending beyond past experience. (Whether we can in fact bring ourselves to believe anything so absurd is beside the point.) Only in recent times have serious efforts been made to escape from the spell of the deductive model, used by Hume and his innumerable followers, by inquiring whether there may not be other proper and relevant senses of "reasonable." It will be argued later that belief in induction is reasonable in principle and that belief in one kind of inductive conclusion is more reasonable than belief in another.

The lasting attraction of the deductive model is not hard to understand. The raison d'être of deductive argument seems enticingly plain: Valid deductions are truth-transmitting and truth-preserving—which, given an interest in obtaining novel truth, seems enough to show the point of deductive reasoning. (That this cannot be the whole story is obvious from the uses of deductive reasoning in exhibiting the consequences of propositions hypothetically entertained—not to mention reductio arguments and other uses.) By contrast the raison d'être of induction seems unclear and mysterious. It would be easy, although unsatisfying to the genuinely perplexed, to say that sound inductive arguments are "likelihood-transmitting," for likelihood is as unclear a concept as inductive correctness. Thus, it is natural to ask for and to expect a detailed answer to the question "Why should a reasonable man rely upon likelihood in default of truth?" Even if the power of sound induction to confer likelihood upon conclusions is regarded as sufficient to make inductive argument reasonable beyond further cavil, the question how such likelihood is conferred will remain. Attention thus shifts to the analytical task.

It may be added that an enduring source of disquiet concerning inductive argument is its disorderliness and formlessness by contrast with deductive argument. In deductive argument we flatter ourselves upon readily perceiving the underlying principles and their necessary connection with logical form. By contrast with such classic simplicity, and order the realm of inductive argument seems disconcertingly complex, confused, and debatable: An inductive argument accepted by one judge may be rejected, on good grounds, by another, equally competent judge; supposedly sound arguments from different sets of true premises may yield opposed conclusions; the very soundness of induction seems not to be clear-cut but to admit of gradations of relative strength and reliability. Given all this, it is not surprising that although many students have labored to introduce order into the field, others, abandoning any hope of so doing, have turned away from induction as a tissue of confusions.

Types of Solution

The answers given in the literature to Hume's problem can be briefly summarized as follows:

1. Hume's challenge cannot be met; consequently, induction is indefensible and ought to be expunged from any reasoning purporting to be rational.
2. In the light of Hume's criticisms, inductive arguments as normally presented need improvement, either (a ) by adding further premises or (b ) by changing the conclusions into statements of probability. In either case a conclusion's validity is expected to follow demonstratively from the premises, and inductive logic will be reconstructed as a branch of applied deductive logic.
3. Although inductive argument cannot be justified as satisfying deductive standards of correctness, it may be proved that inductive policies (rather than rules or principles) are, in a novel sense to be explained later, reasonable. Induction can be vindicated if not validated.
4. Hume's problem is generated by conceptual and linguistic confusions; it must therefore be dissolved, rather than solved, by exposing these confusions and their roots.

These approaches are not all mutually exclusive. Thus (3), the pragmatic approach, is usually combined with (1), repudiation of induction as an acceptable mode of reasoning. Apart from (4) all the approaches accept or make substantial concessions to Hume's major assumption—namely, that the only wholly acceptable mode of reasoning is deductive. This is true even of those who hold (3), the "practicalists," who might be supposed, at first glance, to be relaxing the criteria of rationality.

rejection of induction

The rejection of induction as a proper mode of scientific reasoning is sometimes found in the guise of advocacy of the so-called hypothetico-deductive method. According to such a view, the essence of genuinely scientific reasoning about matters of fact is the framing of hypotheses not established by given empirical data but merely suggested by them. Inference enters only in the control of hypotheses by the verification of their observable consequences: Negative instances strictly falsify a hypothesis, whereas positive instances permit its use, pending further experimental tests, as a plausible, if unproved, conjecture. Science, as well as all reasoning about matters of fact aspiring to the reliability of scientific method, needs only the kind of reasoning to be found in deductive logic and in mathematics. Some such position was already adumbrated in the writings of William Whewell. It has at least the merit of drawing attention to the role of hypotheses in scientific method, a welcome corrective to the excessive claims of early partisans of inductive logic.

The most influential, and possibly the most extreme, of contemporary writers following this line is Karl Popper, who often maintained that what is called induction is a myth, inasmuch as what passes under that title "is always invalid and therefore clearly not justifiable." In his own conception of scientific method such repudiation of induction is linked with the thesis that the purpose of scientific theorizing is falsification (demonstration of error) rather than verification or confirmation (provisional support of an approximation to the truth). Those who agree would rewrite putatively inductive inferences to make them appear explicitly as hypothetical explanations of given facts. (Thus, instead of inferring "All A are B " from premises of the form "A_n is B," the first statement is offered as a more or less plausible explanation of why all the A_n should have been found to be B.)

In spite of its enthusiastic advocacy, it is hard to see where this proposal accomplishes more than a superficial change in the form in which inductive arguments are written and a corresponding alteration in the metalanguage in which they are appraised. Any hypothetical explanation of given empirical data is intended to reach beyond them by having empirical consequences amenable to subsequent tests. If all explanations consonant with the known facts (always an infinite set) were treated as equally unjustified by the evidence, Hume's problem would certainly be set aside, but only at the cost of ignoring what provoked it—namely, the apparent existence of rationally acceptable nondemonstrative arguments. It can hardly be denied that there are nondemonstrative arguments lending reasonable support to their conclusions; otherwise it would be as reasonable to expect manna from heaven as rain from a cloud. Anti-inductivists have seldom been hardy enough to brand all inductive arguments as equally invalid, but as soon as they discriminate between alternative hypotheses as more or less corroborated, more or less in accord with available facts, they are faced, in a new terminology, with substantially the original problems of justification and differential appraisal.

inductive support for induction

To the layperson the most natural way of defending belief in induction is that it has worked in the past. Concealed in this reply, of course, is the assumption that what has already worked will continue to do so, an assumption that has seemed objectionably circular to nearly all philosophers of induction. A stubborn minority (including R. B. Braithwaite and Max Black), however, insists that the appearance of circularity arises only from overhasty application of criteria applicable to deduction. Even in the limiting case, where the rule governing the supporting argument from previous efficacy is the very rule that is to be defended, it can be plausibly argued that no formal circularity is present. Nor is there the more subtle circularity that would obtain if knowledge of the conclusion's truth were needed to justify use of the self-supporting argument. In spite of spirited objections, this line of reasoning has not yet, in the writer's opinion, been shown to be mistaken.

The point that inductive support of induction is not necessarily circular has some importance as illustrating the interesting self-applying and self-correcting features of inductive rules; in virtue of these features, scrutiny of the consequences of the adoption of such rules can, in favorable cases, be used to refine the proper scope of inductive rules and the appropriate judgments of their strength.

A more serious weakness of this kind of defense, if it deserves to be called that, is lack of clarity about what counts as success in using the rule, which is connected in turn with the insufficiently discussed question of the raison d'être of induction considered as an autonomous mode of reasoning.

But even if this controversial type of inductive support of inductive rules ultimately survives criticism, it will not dispose of the metaphysical problems of induction. Those satisfied with Hume's conception of the problem are at bottom objecting to any use of inductive concepts and of the language in which they are expressed unless there is deductive justification for such use. They will therefore reject any reliance upon induction by way of defense, however free from formal defect, as essentially irrelevant to the primary task of philosophical justification. It must be admitted that inductive support of induction, however congenial to the layman, does not go to the roots of the philosophical perplexity.

a priori defenses

A few twentieth-century writers (notably D. C. Williams and R. F. Harrod) maintained that certain inductive arguments, unimproved by the addition of supplementary premises or by modification of the form of the conclusion, can be proved to be valid. Williams argued, with surprising plausibility, that the probable truth of the conclusion of a statistical syllogism can be shown to be necessitated by the truth of the premises, solely by reference to accredited principles of the mathematical theory of chances. While admiring the ingenuity displayed in this approach, critics have generally agreed in finding it fallacious. That some modes of inductive argument are certified as sound or acceptable on broadly a priori (perhaps ultimately linguistic) grounds is, however, a contention of some versions of the linguistic approach.

deductive reconstruction

The effort to provide justification for induction through a reconstruction of inductive arguments so as to make them deductively valid has chiefly taken two forms.

SEARCH FOR SUPREME INDUCTIVE PRINCIPLES

If a given nondemonstrative argument, say from the amalgamated premise P to a conclusion K (where K, for the present, is regarded as a categorical statement of fact containing no reference to probability), is looked at through deductive spectacles, it is bound to seem invalid and so to be regarded as at best an enthymeme, needing extra premises to become respectable. It is easy, of course, to render the original argument deductively valid by supplying the additional premise "If P then K " (this premise will be called Q ). In order for induction to be defended in the classical way, however, the premises have to be true and known to be true. Since P was supposed not to entail K, the new premise, Q, will be a contingent statement of fact, knowledge of whose truth is presumably to be derived either by deduction from more general principles or by induction from empirical data. In either case, if the deductive standard of justification is to be respected, the process must continue until we obtain general factual principles, neither capable of further empirical support nor needing such support.

The line of thought is the following: Since K does not follow strictly from P, the fact that the truth of propositions resembling P in assignable ways is regularly associated with the truth of propositions resembling K is a contingent fact about the actual universe. Looked at in another way, if events occurred purely at random, it would be impossible to make successful inductions; conversely, if inductions of a certain sort do systematically produce true conclusions, there must be a contingent regularity in the universe that should be capable of expression in the form of supreme principles or postulates of induction. Only if such postulates are true can inductions be sound; they must therefore be the assumed but unexpressed premises of all sound inductive arguments.

Favored candidates for the role of such enabling postulates have been the principle that the future resembles the past (Hume), a general principle of causation to the effect that every event has a sufficient cause (Mill), a principle of spatiotemporal homogeneity, which makes locations and dates causally irrelevant (Mill again), and a principle of limited independent variety ensuring that the attributes of individuals cluster together in a finite number of groups (J. M. Keynes, C. D. Broad; Keynes's principle, however, was intended to ensure only the probability of inductive conclusions). Any of these, if true, records the presence in the universe of a certain global regularity or order that permits inductive procedures to produce the desired true conclusions. For example, if we somehow knew in advance that a given attribute C of an observed event must have some other attribute invariably associated with it, and if we further knew that the associated attribute must be included in a finite list of known attributes, say E ₁, E ₂, …, E_n, then there would be a good prospect that repeated observations of similar events would eliminate all but one of the possible associations, E ₁—E_i. Refinements aside, this is how Mill, for instance, conceived of inductive method; his celebrated "methods" (which have received attention out of all proportion to their merits) reduce, in the end, to deductive procedures for eliminating unfit candidates for the title of necessary or sufficient conditions. (Later attempts to develop eliminative induction follow substantially the same path.)

It is clear that the whole interest of this program rests upon the considerations that can be advanced in favor of the supreme premises. If the supreme premises can be known to be true, the remaining processes of inference become trivial (so that there is no need for an autonomous logic of induction); if not, the entire project floats in the void.

The task of formulating plausible principles of the sort envisaged by this program has proved harder than Mill supposed. However, it may be argued that the search for them is pointless and misguided. For one thing, they would accomplish too much: If known to be true, they would allow the conclusions of selected primary inductions to be demonstrated as true, which is too much to expect. It is generally agreed (and rightly so) that the conclusion of even the best inductive argument may without contradiction turn out to be false—if only through bad luck.

Still more serious is the problem of how, from the standpoint of this program, the desired supreme premises could ever be known to be true. Since appeal to induction is excluded at this point on the score of circularity, and since the principles themselves cannot be analytic if they are to serve their desired purpose, there seems no recourse at all. At this point those who search for supreme inductive principles find themselves with empty hands. Mill, for instance, was compelled to let his whole program rest upon the supposed reliability of simple enumeration (the method he regarded as the weakest), in whose defense he had nothing better to say than that it is "universally applicable" (which, on his principles, delightfully begs the question); Keynes, forsaking his empiricist principles for a half-hearted flirtation with Immanuel Kant, could do no better than to suggest that the ultimate principles rest upon "some direct synthetic knowledge" of the general regularity of the universe. Induction may indeed beg to be spared such defenders as these; better the robust skepticism of Hume or Popper than the lame evasions of Mill or Keynes. The conclusion seems inescapable that any attempt to show (as Bacon and many others have hoped) that there are general ontological guarantees for induction is doomed to failure from the outset.

RECOURSE TO PROBABILITY

A more promising way, at least at first sight, of hewing to the deductive line is to modify the conclusion of an inductive argument by including some explicit reference to probability. This approach, influential since Keynes's spirited exposition of it, still has many adherents. If there is no prospect of plugging the deductive gap between P and K by adding further premises known to be true, then perhaps the same end can be achieved by weakening the conclusion. If K does not follow from P, why not be satisfied with a more modest conclusion of the form "Probably, K " or perhaps "K has such and such a probability relative to P "?

The most impressive projects of this sort so far available have encountered severe technical difficulties. It is essential to Keynes's program, for instance, that the probability of a generalization relative to an unbroken series of confirmatory instances steadily approach unity. The conditions necessary for this to be possible in his program are at least that the generalization have an initial nonzero probability and that infinitely many of the confirmatory instances be independent, in the sense of having less than maximal probability of occurrence given the already accumulated evidence. The supreme ontological principles to which Keynes was ultimately driven to appeal (see the preceding section) hardly suffice to satisfy these conditions; subsequent criticism—for example, by Nicod and G. H. von Wright—has shown that even more rigorous conditions are needed. (Von Wright has argued that the desired asymptotic convergence will result only if in the long run every instance of the generalization is scrutinized—which would certainly render the theory somewhat less than useful in practical applications.) For all his importance as a founder of confirmation theory, the theory advocated by Keynes must be judged a failure.

CARNAP'S CONSTRUCTION

The merits of Rudolf Carnap's impressively sustained construction of inductive logic, following in the tradition of Laplace and Keynes but surpassing the work of both in elaboration and sophistication, are still in dispute. Taking probability to express a logical relation between propositions, Carnap has shown how, in certain simplified languages, it is possible to define the breadth or logical width of a given proposition. (Roughly speaking, the degree of confirmation given by a proposition x to a proposition y is the ratio of the width of x · y to the width of x.) The definition of logical width depends on the class of possible universes expressible in the language in question. In order to assign a definite measure of logical width it is necessary to adopt some method of weighting the various possible universes ("state descriptions," in Carnap's terminology) compatible with a given proposition.

One of the merits of Carnap's analysis is to have shown that there is an entire continuum of alternative weighting procedures and associated inductive methods, each of which is internally coherent. The arbitrariness thereby recognized in inductive procedure has worried even the most sympathetic of Carnap's readers; still more disturbing is the emergence of what might be called the paradox of the unconfirmable generalization—the impossibility of ensuring, by Carnap's principles, that an unbroken series of positive instances will raise the probability of a generalization above zero. (Carnap retorts that an instance confirmation—that is, the conclusion of an eduction—does acquire progressively increasing probability, but this is insufficient to satisfy those critics who still hope to find a place for authentic generalization within inductive method.) It is too soon to decide whether such problems as these are more than the teething pains of a new subject. The ingenious modifications of Carnap's program suggested by, among others, J. G. Kemeny and Jaakko Hintikka offer some hope for their elimination.

More serious is the fundamental difficulty that flows from Carnap's conception of confirmation statements as analytic. If it is a truth of logic (broadly speaking) that given the selected definition of confirmation, presented evidence confirms a given hypothesis to such-and-such a degree, then how could such an a priori truth justify any rational belief in the hypothesis? Or, again, if someone were to adopt a different definition of confirmation and thereby be led to a contrary belief, then how could he be shown to be in error?

Carnap's answer is based on the notion that the bridge between confirmation, as defined by him, and rational belief is to be found in some principle for the maximization of expected utility (due allowance being made, however—in his sophisticated rendering of that principle—for subjective estimates of probabilities and utilities). Yet it seems that because considerations of probability also enter into the calculation of probabilities and expected utilities, a logical circle is involved here. Since Carnap's discussions of this fundamental point are still comparatively rough and provisional, it would be premature to reach any final judgment on the success that he and those who agree with him are likely to achieve in coping with this basic difficulty. (It might be said that difficulty with the connection between probability judgments and practice is not peculiar to Carnap's work, since it arises in one form or another for all theorists of induction who take the trouble to work out in detail the consequences of their principles and assumptions.) It may be held, however, that Carnap's relatively cursory judgments about the justification of induction belong to the least satisfactory parts of his work on inductive logic.

How much the recourse to probability will accomplish depends, of course, upon how the reference to probability is construed. With empirical interpretations of probability, such as those favored by "frequentists," the probability conclusion still extends beyond the premises by covert reference to finite or infinite sets of events not covered by the given premises. The inductive leap remaining in the reconstructed argument will thus still leave the problem of induction unsolved. If, however, probability is construed in some logical way (as by Keynes or Carnap), the amended conclusion will say less than the premises and will therefore be untouched by subsequent empirical test; the deductive validity of the reconstructed argument will be saved only at the cost of rendering problematic its relevance to prediction and empirical control. In converting a purportedly inductive argument into a valid deductive one, the very point of the original argument—that is, to risk a prediction concerning the yet unknown—seems to be destroyed.

pragmatic defenses

Answers of the pragmatic type, originally offered by Peirce but independently elaborated with great resourcefulness by Hans Reichenbach, are among the most original modern contributions to the subject. To many they still offer the best hope of avoiding what seems to be the inevitable failure of the attempts so far discussed. The germ of the pragmatic strategy is the reflection that in ordinary life, situations sometimes arise where, in default of reliable knowledge of consequences, problematic choices can still be justified by a "nothing to lose" argument. Faced with a choice between an operation for cancer and a sure death, a patient may choose surgery, not because of any assurance of cure but on the rational ground that nothing is lost by taking the chance.

REICHENBACH'S "VINDICATION"

According to Reichenbach, the case is similar in what he takes to be the paradigmatic inductive situation. Given an antecedent interest in determining the probability of occurrence of a designated character (construed, by him, as the limit, in an infinitely long run of events, of the relative frequency of occurrence of that character), Reichenbach argues that the only rationally defensible policy is to use the already ascertained relative frequency of occurrence as a provisional estimate of the ultimate limiting value. A man who proceeds in this way can have no guarantee or assurance that his estimates, constantly revised as information about the series gradually accumulates, will bring him into the neighborhood of a limiting value of the frequency, for the provisional values of the relative frequencies may, in fact, diverge. In that case no predictive policy at all will work, and successful induction is impossible.

However, if this should not be the case and the series really does have a limiting value for the relative frequency in question, we can know in advance, and with certainty, that the policy is bound eventually to lead the reasoner to estimates that will remain as close to the limit as desired. There is therefore nothing to lose by adopting the inductive policy: If the series of events under scrutiny is sufficiently regular to make induction possible, the recommended policy is bound to yield the desired result ultimately (and we know before we start that it will do so), whereas if the series is irregular enough to defeat the standard inductive policy, nothing will avail, and we are no worse off than if a contrary decision had been made.

This type of justification is often called "vindication," as Herbert Feigl termed it. It is claimed that in a sense the type of vindication sketched above resolves Hume's problem by bypassing it. We know for certain that what Hume desired—namely, certification of the soundness of inductive argument by the standard of demonstrative reasoning—cannot be supplied. But it would be fainthearted to leave the matter there. By conceiving the practice of induction as the adoption of certain policies, applied in stoic-acceptance of the impossibility of assured success in obtaining reliable knowledge concerning matters of fact, we are able to see that such policies are, in a clear sense, preferable to any of their competitors. Standard induction is preferable to soothsaying because we know that it will work (will approach limiting values in the long run) if anything will.

To these plausible claims it has been objected that the analogy with genuinely practical decisions to act upon insufficient evidence is misconceived, for in the state of perfect ignorance postulated by defenders of the pragmatic approach no method at all can be regarded as superior to any other. Vindicationists have been relatively undisturbed by such general criticism; they have, however, felt obliged to seek remedies for a grave technical flaw that threatens to wreck their entire program. Given the assumption that the best to be achieved by an inductive policy is asymptotic convergence to a limiting relative frequency, it is obvious that no policy for inductive estimation in the short run is excluded as unreasonable. Thus, from the standpoint of pragmatic vindication an unbroken run of A 's found to be B would not make it unreasonable to predict the subsequent occurrence in the short run of A 's that are not B, provided only that the adopted estimates are chosen so as to converge eventually to the limit (if it exists). But since the long run is in fact never attained, even by immortal beings, it follows that the pragmatic defense yields no criteria for inductive decisions in short-run cases, to which inductive prediction is confined, and offers no differential reasons for preferring one inductive policy to another.

In spite of strenuous attempts (notably by Wesley Salmon) to improve Reichenbach's original conception by providing supplementary reasons for rejecting unwanted nonstandard policies, the prospects for vindicationism remain dubious. Even if some plausible way could be found of assigning, on vindicationist principles, a special status to the standard policy of induction, the approach would be vulnerable to the objection that it conceives inductive method in an eccentrically restricted fashion. The determination of limiting values of relative frequencies is at best a special problem of inductive method and by no means the most fundamental.

PEIRCE'S VIEWS

Peirce, whose views on induction have exerted a lasting influence on the subject since the posthumous appearance of his Collected Papers, had a more complex conception of scientific method than latter-day vindicationists. Induction, conceived by him as a process of testing statistical hypotheses by examining random samples, has to be understood in its relations to two other procedures, statistical deduction and abduction.

Statistical deduction consists of inference from the frequency of occurrence of an attribute in a population to the probable and approximate occurrence of that attribute in a sample randomly drawn from it. Given Peirce's definition of probability as limiting frequency and his conception of randomness, it follows demonstratively that most of the samples drawn will have nearly the same composition as the parent population; statistical deduction is thus "valid" in the sense that it generates conclusions that are true most of the time.

Abduction, the creative formulation of statistical hypotheses and the only mode of scientific inference introducing new ideas, is a kind of inversion of statistical deduction. It has almost no probative force, its value being rather that it provides new generalizations needing independent verification and having "some chance of being true."

When the three procedures are used in combination, induction is seen to be a self-correcting method that if indefinitely followed must in the long run lead the scientific community, although not the individual reasoner, indefinitely close to the truth. In such asymptotic convergence to the truth lies the peculiar validity of induction.

Peirce cannot be held to have succeeded in his effort to defend the rationality of inductive policies in terms of long-range efficacy in generating conclusions approximately and for the most part true. Since the intended justification of induction depends essentially upon the randomness of the samples used, it must be objected that there is normally no way of guaranteeing in advance the presence of such randomness. (To this objection Peirce had only the lame and unsupported rejoinder that inductive inference retains some probative force even in the absence of the desired randomness.) The following are among the most obvious weaknesses of Peirce's views about induction.

The self-corrective tendency of induction, which Peirce, in his last writings on the subject, came to view as the heart and essence of inductive method, remains obscure, in spite of his eulogies. That inductive estimates will need, on Peirce's principles, repeated adjustments as further evidence accumulates is clear enough, but that this process will show any convergence toward a limiting value cannot be guaranteed a priori. If the samples to be examined were random in Peirce's severe sense of that term, we could at least count upon an overall predominance of approximately correct estimates, but even then we should have no reason, in the absence of additional guarantees, to expect the better estimates to come near the end of the testing process. In any case, supposing realistic conditions for the testing of hypotheses (such as our necessary reliance on cases that we are in a position to examine), it seems clear that the conditions for the kind of sampling demanded by Peirce cannot be fulfilled.

Peirce's references to the long run seem on the whole incoherent. Much of the time he seems to have been thinking of what would prove to be the case in an actual but infinitely extended series of trials. Toward the end of his life, however, he appears to have recognized that his definitions of probability and of the validity of induction needed to be construed more broadly, by reference to the "would be" of events, conceived as real general characters or habits. How such general features of events can in fact be disclosed, even by very lengthy series of trials, Peirce never made plain. Yet the need for clarification is great for anybody attracted by his approach. The infinitely long run is a chimera, and to be told that a certain method, if consistently pursued, would in such a long run eventually lead as close as we pleased to the truth is to be told nothing that can be useful for the actual process of verification. All verification is necessarily performed in the finite run, however extended in length, and what would happen if per impossibile the "run" were infinite is not relevant to the relative appraisal of given hypotheses. We need a method for adjudicating between rival hypotheses, if not now then in the foreseeable future, and this Peirce's conception cannot provide. Because of his reliance upon the infinitely long run Peirce's pragmaticism, which initially seems so hardheaded in its emphasis upon success and practical consequences, ends by being as utopian as any of the metaphysical conceptions that he derided.

justification as a pseudo problem

In view of the quandaries that beset all known attempts to answer Hume's challenge, it is reasonable to consider whether the problem itself may not have been misconceived. Indeed, it appears upon examination that the task of logical justification of induction, as classically conceived, is framed so as to be a priori impossible of solution. If induction is by definition nondeductive and if the demand for justification is, at bottom, that induction be shown to satisfy conditions of correctness appropriate only to deduction, then the task is certainly hopeless. But to conclude, for this reason, that induction is basically invalid or that a belief based upon inductive grounds can never be reasonable is to transfer, in a manner all too enticing, criteria of evaluation from one domain to another domain, in which they are inappropriate. Sound inductive conclusions do not follow (in the deductive sense of "follow") from even the best and strongest set of premises (in the inductive sense of "strongest"); there is no good reason why they should. Those who still seek a classical defense of induction may be challenged to show why deductive standards of justification should be appropriate. Perhaps the retort will be that there is no clear sense in which assertion of a conclusion is justified except the sense in which it is known to follow strictly from premises known to be true, so the burden of argument rests upon anybody who claims the existence of some other sense.

LINGUISTIC APPROACH TO THE PROBLEM

The challenge to the claim that inductive arguments cannot be said to be justified might be met in the following way: Suppose a man has learned, partly from his own experience and partly from the testimony of others, that in a vast variety of circumstances, when stones are released they fall toward the ground. Let him consider the proposition K, that any stone chosen at random and released will do likewise. This is, in the writer's opinion, a paradigm case for saying that the man in question (any of us) has a good reason for asserting K and is therefore justified in asserting K rather than not-K. Similarly, this is a paradigm case for saying that the man in question is reasonable in asserting K and would be unreasonable in asserting not-K, on the evidence at hand. Anybody who claimed otherwise would not be extraordinarily and admirably scrupulous but would be abusing language by violating some of the implicit criteria for the uses of "good reason," "justified," and "reasonable," to which he, like the interlocutor with whom he succeeds in communicating, is in fact committed.

Any man—say, one from Mars—who used these words according to criteria that would really make it improper for him to apply them in the kind of situation envisaged would not, in the end, be understood by us. Worse still, he would be trying, if he were consistent, to change our actual concepts of reason and reasonableness so that it would be logically impossible to have reasons for assertions concerning the unknown or to be reasonable in expecting one matter of fact rather than another on the basis of empirical evidence. (He would be behaving like a man who insisted that only stallions deserved to be called horses.) Nor would such distortion achieve anything significant, for the man who proposed to make "empirical reason" as impossible of application as "being in two places at once" would find himself forced to reintroduce essentially the same concept under some such label as "generally accepted as a reason" or "what commonly passes for a reason." The distinction between what ordinary men and what scientists call "good reasons" and "bad reasons" is made for a good purpose, has practical consequences, and is indispensable in practice. Thus, the dispute between the advocate of the linguistic approach and his opponent seems to reduce to a verbal one, ripe for oblivion.

Given the intertwined complexity of the concepts entering into alternative formulations of the problem of induction and the seductive plausibility of the distortions to which such concepts are subject, no brief reply such as the above can be expected to clarify and to expose the conceptual confusions upon which traditional formulations of the problem rest. A full discussion would at least also have to consider the relevant senses of "knowledge" and "possibility" and related epistemological notions. The outline of the strategy is perhaps sufficiently plain; the line to be taken is that close and detailed examination of how the key words in the statement of the problem occur will show that criteria for the correct uses of such terms are violated in subtle and plausible ways. If this can be established, the celebrated problem of justifying induction will dissolve, and the confused supposition that induction needs philosophical justification or remains precarious in its absence will disappear.

The comparative problem and the analytical problem do not dissolve under this attack. Advocates of the linguistic approach can be fairly reproached for having been too often content to show to their own satisfaction that the general problem of justification is rooted in confusion, while neglecting the constructive tasks of rendering clearer the criteria for preferential appraisal of inductive arguments.

To those unsympathetic with the linguistic approach such an attack upon the traditional problem has sometimes seemed to be operating with dubious and insufficiently elaborated theories of meaning or use and to be altogether too glib in its attribution of semantical confusions. Moreover, a number of critics have thought that an appeal to ordinary language cannot be ultimately decisive from a philosophical standpoint. Even if it were established that it is a violation of ordinary language to describe the conclusion of some inductive arguments as supported by less than good reasons, the critics ask, what is there in the nature of things that requires us to continue talking in the ordinary way or to be bound by the encapsulated metaphysical prejudices of those originally responsible for establishing the rules of use to which appeal is now made? The linguistic philosopher necessarily uses such key words as reasonable in his polemic against the traditional approaches to the problem. But to use the crucial terms in a discussion of the nature of the inductive problem, it might be urged, is to beg the very question at issue. A lunatic or an eccentric philosopher might well use the expression "good reason" in a way that would be blatantly improper, yet he might be able to prove, by appeal to his own criteria, that he had "good reasons" to use the phrase in the way he did. But are we ourselves in any better position? Are we not obligated to break through the linguistic barrier and at least to show why the alleged criteria for good reasons to which appeal is made should continue to receive our allegiance?

There is no short way of dealing with this type of objection. It may be helpful, however, to sketch the general view upon which the present writer, as a defender of the linguistic approach, would rely.

DEFENSE OF THE LINGUISTIC APPROACH

All normal adult human beings follow the same broad and systematic patterns for drawing inferences concerning the unobserved and apply the same general principles for appraising such nondemonstrative inferences. For instance, all normal persons expect observed cases of association of attributes to be confirmed in further experience unless there are countervailing factors (the principle of simple enumeration), all count increase in the number of independent confirmatory instances of a law as strengthening (or at least not weakening) the probability of the law's truth, and all alike share the inductive beliefs that underpin causal notions. It is, therefore, not fanciful to conceive of all sane adult human beings as participating in a complex system of ways of learning from experience that might be called the inductive institution. Like other institutions (warfare, the law, and so on), it has a relatively fixed, though not immutable, structure, transmitted from one generation to the next and crystallized in the form of prohibitions and licenses, maxims of conduct, and informal precepts of performance. Like other institutions, the inductive institution requires that its participants have mastered a system of distinctive concepts (among them the concepts of good reason, sound argument, and relative likelihood) having both descriptive and normative aspects.

Such mastery is shown in capacity to use the corresponding language correctly—which, in turn, implies recognition of, though not invariable obedience to, associated rules for assertion, for evaluation, and for the appraisal of actions. Understanding what people mean by reasons for empirical conclusions requires acceptance of certain types of situations as paradigmatic of empirical evidence; to call given facts sound reasons for some conclusion is to imply the acceptability of certain criteria for judging one reason to be better than another; asserting that some belief about the hitherto unobserved is reasonable commits the speaker to holding that other things being equal, action based on such belief should be approved.

The philosophical problem of justifying induction can arise only for somebody who is a member of the inductive institution and is therefore already bound by its constitutive rules. A spectator can understand bridge without being a player, but all of us are necessarily players of the "inductive game" before we achieve the reflective self-consciousness characteristic of philosophical criticism.

The constitutive rules of the inductive institution (whose precise delineation remains a still unfinished task for philosophers of induction) are highly abstract, schematic, and limited in their practical usefulness. Indeed, the general principles of inductive inference are about as relevant to practice as the abstract principles of justice are to decisions on concrete legal issues. In particular situations concerning the soundness of empirical hypotheses the reasoner is compelled to fall back upon his specific knowledge of relevant facts and theories. In this way the conduct of concrete inductive inference resembles the exercise of a craft or skill more than it does the automatic application of a decision procedure. Yet the constitutive rules provide important general constraints that cannot be violated without generating nonsense. To be in command of inductive language, whether as a master of advanced techniques of statistical inference or as a layperson constantly and more or less skillfully anticipating future experience, is necessarily to be subject to the implicit norms of belief and conduct imposed by the institution.

The inductive concepts that we acquire by example and formal education and modify through our own experiences are not exempt even from drastic revision. The norms may be usefully thought of as formal crystallizations into linguistic rules of general modes of response to the universe that our ancestors have, on the whole, found advantageous to survival, but the earlier experience of the race never has absolute authority. Piecemeal reform of the inductive institution can be observed in the history of modern science.

What is clearly impossible, however, is the sort of wholesale revolution that would be involved in wiping the inductive slate clean and trying to revert to the condition of some hypothetical Adam setting out to learn from experience without previous indoctrination in relevant rules of inductive procedure. This would be tantamount to attempting to destroy the language we now use to talk about the world and about ourselves and thereby to destroy the concepts embodied in that language. The idea of ceasing to be an inductive reasoner is a monstrosity. The task is not impossibly difficult; rather, its very formulation fails to make sense. Yet it remains important to insist that the inductive institution, precisely because its raison d'être is learning from experience, is intrinsically self-critical. Induction, like the Sabbath, was made for humankind, not vice versa. Thus, constantly renewed experience of the successes and failures of the specific inductive procedures permitted within the general framework of the inductive institution provides a sound basis for gradual reform of the institution itself, without objectionable circularity.

Yet even if no feature of the institution is exempt, in principle, from criticism and reconstruction, the entire institution cannot be called into question all at once without destroying the very meaning of the words in which the philosophical problems of induction are stated. Wholesale philosophical skepticism about matters of fact is senseless and must be shown to be so. If this is the "linguocentric predicament," we must make the best of it.

The view here outlined must be carefully distinguished from what is commonly called conventionalism. The argument is not that the constitutive inductive rules hold by convention but rather that the sweeping question "Why should we accept any inductive rules?" can be shown to make no sense.

Our sketch may be usefully compared with Hume's view of induction as a habit or custom. Both views agree in regarding inductive practices as being, on the whole, social and contingent facts obtaining at given periods in human history. It is, after all, a contingent fact that there have existed animals sufficiently rational to be able to speak and hence to have inductive concepts. The present conception differs significantly from Hume's, however, in regarding the inductive institution as partly constituted by normative inductive rules to which the philosopher, like every reasoning individual, finds himself already committed. Thus, the encompassing social fact of the existence of the inductive institution includes within itself the means for appraisal and criticism of inductive procedures; we cannot regard inductive inference as something merely "given," as a natural fact, like the Milky Way, that it would be absurd to criticize. To understand induction is necessarily to accept its authority. However (to repeat), questions about the general or ultimate justification of induction as such, questions of the form "Why should any induction be trusted?" must be recognized as senseless. If we persist in trying to raise them, we come, as Wittgenstein expressed it, to the "limits of language," and we can see that we have done so by perceiving that what we had hoped were important and fundamental questions are no better than nonsense masquerading as sense. The foregoing will undoubtedly strike critics of the linguistic approach as too facile, for the tangle of philosophical problems that have been dubbed "the problem of induction" constitute, in their depth, their importance, their elusiveness, and their capacity to bewilder and confuse, a very paradigm of philosophical perplexity.

The preceding survey indicates that no wholly satisfactory philosophy of induction is yet available. The work still to be done may be summarized as follows: For those who recognize the crucial role of probability in inductive inference, to develop a consistent, systematic, and relevant reconstruction of the concept of probability; for those who reject induction as an outmoded myth, to elaborate a detailed and comprehensive account of scientific practice that will be reasonably close to the best actual procedures used in reasoning about matters of fact; for those who pin their hopes on the construction of an inductive logic, to remove the constraints imposed by the study of artificially simplified languages and to show in detail how analytical statements of probability can be relevant to the practice of inductive prediction; for vindicationists, to solve the comparative problem of selecting competing hypotheses and to show how eventual convergence in the long run can bear upon short-run judgment; for those who regard induction as a pseudo problem, to articulate the theory of language presupposed and to demonstrate in convincing detail the origins and the character of the stubborn confusions that have infested the subject.

See also Laws of Nature; Probability and Chance.

Bibliography

comprehensive studies

The best introduction to the whole subject of induction is still William Kneale, Probability and Induction (Oxford: Clarendon Press, 1949). See especially Part II, "The Traditional Problem of Induction." A shorter and more up-to-date discussion is Stephen F. Barker, Induction and Hypothesis (Ithaca, NY: Cornell University Press, 1957), which is especially good on the role of simplicity in inductive inference. Georg Henrik von Wright, The Logical Problem of Induction (Oxford, 1941; 2nd ed., 1957), is invaluable for its ample discussion of the history of the subject and also contains penetrating criticism. John Patrick Day, Inductive Probability (London: Routledge and Paul, 1961), uses somewhat opaque symbolism but is very comprehensive. Induction: Some Current Issues, edited by Henry E. Kyburg Jr. and Ernest Nagel (Middletown, CT: Wesleyan University Press, 1963), is a conference report containing edited versions of discussions by Black, Braithwaite, Nagel, Salmon, and others.

terminology

For a good brief treatment of terminology, see Kneale, op. cit., pp. 24–48, which includes discussion of Aristotle's uses of epagoge. Kneale is good also in his treatment of intuitive induction and mathematical induction.

S. F. Barker, "Must Every Inference Be Either Deductive or Inductive?," in Philosophy in America, edited by Max Black (Ithaca, NY: Cornell University Press, 1965), pp. 58–73, answers in the negative; it is an illuminating effort to distinguish criteria for nondemonstrative inference.

history of inductive methods

André Lalande, Les théories de l'induction et de l'expérimentation (Paris: Boivin, 1929), contains brief discussions of Isaac Newton, John Herschel, Claude Bernard, and other scientists, as well as of Bacon and Mill. There is no satisfactory general history of inductive methods.

Most students of Bacon confine themselves to Book I of the Novum Organum (available in many editions). The best philosophical commentary on Bacon remains R. L. Ellis's general preface to The Works of Francis Bacon, edited by James Spedding, R. L. Ellis, and D. D. Heath (London: Longman, 1857–1854). For a lucid and sympathetic appreciation of Bacon's general program—but with comparatively little discussion of Bacon's views on induction—see the commemorative address by C. D. Broad, The Philosophy of Francis Bacon (Cambridge, U.K.: Cambridge University Press, 1926).

For Mill, see A System of Logic, 8th ed. (London, 1872), especially Book III, Ch. 3, "On the Ground of Induction," and Ch. 21, "Of the Evidence of the Law of Universal Causation." Ernest Nagel, ed., John Stuart Mill's Philosophy of Scientific Method (New York: Hafner, 1950), is a useful selection from the Logic, including part of Mill's Examination of Sir William Hamilton's Philosophy. The editor's introduction is helpful.

A serious student of Mill will wish to consult his unjustly neglected critic, William Whewell, whose The Philosophy of the Inductive Sciences (London: J.W. Parker, 1847)—especially Part II, titled "Novum Organum Renovatum"—is still instructive. A useful summary of Whewell's views is C. J. Ducasse's "William Whewell's Philosophy of Scientific Discovery," pp. 183–217 in Theories of Scientific Method: The Renaissance through the Nineteenth Century, edited by Edward H. Madden (Seattle: University of Washington Press, 1960).

For Peirce's views, see the Collected Papers of Charles Sanders Peirce, edited by Charles Hartshorne, Paul Weiss, and Arthur Burks (8 vols., Cambridge, MA: Harvard University Press, 1931–1958), especially "Ampliative Reasoning," Vol. II, pp. 272–607. Unfortunately, Peirce's writings on induction are scattered throughout the volumes, out of chronological order. A concise critical examination is Thomas A. Goudge, "Peirce's Treatment of Induction," in Philosophy of Science 7 (1940): 56–68. Part VI of the same author's The Thought of C. S. Peirce (Toronto: University of Toronto Press, 1950) has a fuller treatment, for which the rest of the book provides good background.

For Keynes, see A Treatise on Probability (London: Macmillan, 1921). Chs. 18–22 are the most relevant; they include, inter alia, Keynes's original ideas concerning analogy. Jean Nicod, "The Logical Problem of Induction," Part II of Foundations of Geometry and Induction (London: K. Paul, Trench, Trubner, 1930), contains lucid and trenchant criticisms of Keynes's position. For more recent appraisal, see von Wright, op. cit., pp. 127–132.

Rudolf Carnap, Logical Foundations of Probability (Chicago: University of Chicago Press, 1950), is an elaborate treatise that, although still unfinished and in part superseded by Carnap's later studies, remains the most important original source for his views.

The Philosophy of Rudolf Carnap, edited by P. A. Schilpp (La Salle, IL: Open Court, 1963), is now indispensable to the serious student because of its important essays by Carnap's defenders and critics and the detailed comments and replies by Carnap. John G. Kemeny, "Carnap's Theory of Probability and Induction," pp. 711–738, is an outstandingly successful effort to convey the gist of Carnap's position sympathetically; Ernest Nagel, "Carnap's Theory of Induction," pp. 785–825 (vigorously attacked in Carnap's rejoinder, pp. 989–995), expresses at length the misgivings of those who see in Carnap's view a retreat from empiricism; Hilary Putnam, "'Degree of Confirmation' and Inductive Logic," pp. 761–783, is a highly ingenious attempt to demonstrate that a logic based on confirmation must violate accepted canons of scientific method.

Carnap's "The Aim of Inductive Logic," pp. 303–318 in Logic, Methodology, and Philosophy of Science, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski (Stanford, CA: Stanford University Press, 1962), is an important pioneering discussion of the application of inductive logic. Carnap concludes, "Induction, if properly reformulated, can be shown to be valid by rational criteria."

problem of justification

There is no substitute for reading Hume. His own summary, An Abstract of a Treatise of Human Nature (London, 1740; reprinted, with introduction by J. M. Keynes and Piero Sraffa, Cambridge, U.K.: Cambridge University Press, 1938), should not be overlooked. Pp. 11–20 express the essence of Hume's position. The serious student must, of course, read A Treatise of Human Nature, edited by L. A. Selby-Bigge (Oxford: Clarendon Press, 1896). See especially Book I, Part 3. Sec. 6 contains the famous skeptical attack on the objectivity of causal connection.

See also Hume's Enquiry Concerning the Human Understanding, edited by L. A. Selby-Bigge, 2nd ed. (Oxford: Clarendon Press, 1902), especially Sec. 4, "Sceptical Doubts concerning the Operations of the Understanding."

Bertrand Russell, "On Induction," Ch. 6 of The Problems of Philosophy (London: Williams and Norgate, 1912), is a lucid and concise version of what is basically Hume's skeptical position.

The best statement of the "new riddle of induction" is in Nelson Goodman, Fact, Fiction and Forecast (Cambridge, MA: Harvard University Press, 1955), Chs. 3 and 4. Goodman's position, with some unpublished improvements, is sympathetically discussed in Israel Scheffler, The Anatomy of Inquiry (New York: Knopf, 1963), pp. 291–326. The same book is valuable for its lucid appraisal of various theories of inductive confirmation.

rejection of induction

See Karl Popper, The Logic of Scientific Discovery (London: Hutchinson, 1959), especially Ch. 1. John Oulton Wisdom, Foundations of Inference in Natural Science (London: Methuen, 1952), is written from a Popperian standpoint; Parts II–IV, pp. 85–232, deal with induction. For criticism of Popper's general position, see John Arthur Passmore, "Popper's Account of Scientific Method," in Philosophy 35 (1960): 326–331.

a priori defenses

Donald C. Williams, The Ground of Induction (Cambridge, MA: Harvard University Press, 1947), claims to have answered Hume on a priori grounds. For a dissenting verdict, see the extended review by Max Black in Journal of Symbolic Logic 12 (1947): 141–144.

Roy F. Harrod, Foundations of Inductive Logic (London: Macmillan, 1956), is still another ingenious attempt to show that induction must be successful. For criticism, see Popper's review in British Journal for the Philosophy of Science 9 (1958–1959): 221–224; Harrod's reply appears in 10 (1959–1960): 309–312.

inductive support for induction

Consult Richard B. Braithwaite, Scientific Explanation (Cambridge, U.K.: Cambridge University Press, 1953), Ch. 8, "The Justification of Induction," pp. 255–292.

Abner Shimony's severely critical "Braithwaite on Scientific Method," in Review of Metaphysics 7 (1953–1954): 644–660, has not been rebutted. On this topic see also H. E. Kyburg Jr., "R. B. Braithwaite on Probability and Induction," in British Journal for the Philosophy of Science 9 (1958–1959): 203–220.

A defense of the soundness of self-supporting inductions can be found in Max Black, "Inductive Support of Inductive Rules," pp. 191–208 in Problems of Analysis (Ithaca, NY: Cornell University Press, 1954), and "Self-Supporting Inductive Arguments," pp. 209–218 in Models and Metaphors (Ithaca, NY: Cornell University Press, 1962). For criticism of Black's views, see Peter Achinstein, "The Circularity of a Self-Supporting Argument," in Analysis 22 (1961–1962): 138–141. See also Black's reply in the same journal, 23 (1962–1963): 43–44, and Achinstein's rejoinder, ibid., 123–127.

deductive reconstruction

See Bertrand Russell, Human Knowledge: Its Scope and Limits (New York: Simon and Schuster, 1948). Russell states and defends postulates of scientific inference in the concluding part, pp. 421–507. Arthur W. Burks, "On the Presuppositions of Induction," in Review of Metaphysics 8 (1955): 574–611, is a persuasive exposition by the ablest contemporary defender of deductive reconstruction. See also Burks's article in The Philosophy of Rudolf Carnap, edited by Schilpp (above), and his Chance, Cause, Reason: An Inquiry into the Nature of Scientific Evidence (Chicago: University of Chicago Press, 1977). Deductive reconstruction is criticized in most of the articles listed below in the section on induction as pseudo problem.

C. D. Broad, "On the Relation between Induction and Probability," in Mind 27 (1918): 389–404, and 29 (1920): 11–45, is an early and unjustly neglected discussion whose second part is a suggestive development of Keynes's ideas.

pragmatic defenses

Herbert Feigl's views are contained in two of his articles, "De Principiis Non Est Disputandum …?," in Philosophical Analysis, edited by Max Black (Ithaca, NY: Cornell University Press, 1950), and "Validation and Vindication," in Readings in Ethical Theory, edited by Wilfrid Sellars and John Hospers (New York: Appleton-Century-Crofts, 1952), pp. 667–680. The first paper argues the case for vindication as a special mode of justification, and the second enlarges upon the idea.

Hans Reichenbach, The Theory of Probability (Berkeley: University of California Press, 1949), is the locus classicus of recent pragmatic defenses; see especially "The Justification of Induction," pp. 469–482. The same author's Experience and Prediction (Chicago, 1938) contains a more popular presentation.

Isabel P. Creed, "The Justification of the Habit of Induction," in Journal of Philosophy 37 (1940): 85–97, and Everett J. Nelson, "Professor Reichenbach on Induction," in Journal of Philosophy 33 (1936): 577–580, are two thorough appraisals and criticisms of Reichenbach's position.

Max Black, "'Pragmatic' Justifications of Induction," in his Problems of Analysis (Ithaca, NY: Cornell University Press, 1954), pp. 157–190, is a full-scale attack upon "pragmatic vindication."

Wesley C. Salmon, "The Short Run," in Philosophy of Science 22 (1955): 214–221, is an admirable, if ultimately unsuccessful, attempt to avoid reliance upon the infinitely long run. In "Should We Attempt to Justify Induction?," Philosophical Studies 8 (1957): 33–48, Salmon elaborately defends the pragmatic approach; Black's "Can Induction Be Vindicated?," in his Models and Metaphors (Ithaca, NY: Cornell University Press, 1962), pp. 194–208, is a detailed reply to Salmon.

Salmon's "Vindication of Induction," in Current Issues in the Philosophy of Science, edited by H. Feigl and G. E. Maxwell (New York: Holt, Rinehart and Winston, 1961), pp. 245–256, is the latest version of Reichenbach's approach by its ablest contemporary advocate. See also Salmon's contributions to Nagel and Kyburg, eds., Induction, above.

induction as a pseudo problem

For an introduction to the growing number of studies written from this standpoint, see Alice Ambrose, "The Problem of Justifying Inductive Inference," in Journal of Philosophy 44 (1947): 253–272; Max Black, "The Justification of Induction," in his Language and Philosophy (Ithaca, NY: Cornell University Press, 1949), pp. 59–88; Frederick L. Will, "Will the Future Be Like the Past?," in Mind 56 (1947): 332–347; and Paul Edwards, "Russell's Doubts about Induction," in Mind 58 (1949): 141–163. Perhaps the most accessible brief account is Peter F. Strawson, "The 'Justification' of Induction," in Introduction to Logical Theory (London: Methuen, 1952), pp. 248–263. Brief criticism of the "linguistic approach is to be found in many of the works listed above. No satisfactorily broad statement of the position of the "linguists" or their critics is yet available.

bibliographies and surveys

Lengthy bibliographies are supplied in the above-mentioned books by Keynes (to 1921), Carnap (to 1951), and von Wright (to 1955).

Black's "Induction and Probability," in Philosophy in the Mid-century, edited by Raymond Klibansky (Florence: Nuova Italia, 1958), pp. 154–163, is a critical survey of the work of the decade prior to its publication.

H. E. Kyburg Jr., "Recent Work in Inductive Logic," in American Philosophical Quarterly 1 (1964): 249–287, is a highly useful analysis of the main trends, with criticism and an extensive bibliography.

other recommended titles

Hacking, Ian. Logic of Statistical Inference. Cambridge, U.K.: Cambridge University Press, 1965.

Horwich, P. Probability and Evidence. Cambridge, U.K.: Cambridge University Press, 1982.

Jeffrey, R. C. The Logic of Decision. 2nd ed. Chicago: University of Chicago Press, 1983.

Jeffrey, R. C. Probability and the Art of Judgment. New York: Cambridge University Press, 1992.

Jeffrey, R. C. "The Valuation and Acceptance of Scientific Hypotheses." Philosophy of Science 23 (1956): 237–246.

Kaplan, M. Decision Theory as Philosophy. New York: Cambridge University Press, 1996.

Kyburg, H. The Logical Foundations of Statistical Inference. Dordrecht: Reidel, 1974.

Levi, I. The Fixation of Belief and Its Undoing. Cambridge, U.K.: Cambridge University Press, 1991.

Lewis, David. "A Subjectivist's Guide to Objective Chance." In Studies in Inductive Logic and Probability. Vol. 2, edited by Richard C. Jeffrey. Berkeley: University of California Press, 1980.

Salmon, Wesley. The Foundations of Scientific Inference. Pittsburgh: University of Pittsburgh Press, 1966.

Salmon, Wesley. "Inductive Inference." In Philosophy of Science: The Delaware Seminar, edited by B. Baumrin. New York: Interscience, 1963.

Skyrms, B. Choice and Chance. 4th ed. Encino, CA: Wadsworth, 2000.

Skyrms, B. Pragmatics and Empiricism. New Haven, CT: Yale University Press, 1984.

Sober, Elliot. "Bayesianism—Its Scope and Limits." In Bayes' Theorem, edited by Richard Swinburne. Oxford: Oxford University Press, 2002.

Swinburne, Richard, ed. The Justification of Induction. Oxford: Oxford University Press, 1974.

Van Fraassen, Bas. Laws and Symmetry. Oxford: Clarendon Press, 1989.

Max Black (1967)

Bibliography updated by Benjamin Fiedor (2005)

INDUCTION

A method or activity by which one proceeds from observation to generalization. Although some regard it as the counterpart of deduction, it should more properly be seen as the counterpart of demonstration, since its use leads to the acceptance of principles from which conclusions can be demonstrated. This article sketches the historical genesis of the notion and various views concerning the ground on which it is based.

Historical Genesis. Inductive method seems to have had its origin in the philosophizing of socrates. "For two things may be fairly ascribed to Socrates: inductive arguments and universal definition, both of which are concerned with the starting point of science" (Aristotle, Metaphysics 1078b 27). In Western philosophy, this Socratic attribution is later confirmed by cicero: "This form of argument which attains the desired proof by citing several parallels is called induction, in Greek ἐπαγωγή; Socrates frequently used this in his dialogue" (Topica 10.42; Loeb Classical Library 413).

For Aristotle, the sciences are distinguished by their methodological differences; thus the manner of demonstrating and the type of certitude in the speculative sciences are other than in the technical sciences (De Partibus Animalium 639b 30). Again, in the natural sciences a uniform method does not apply, since one must always seek whether demonstration or classification or some other procedure is best suited (De Anima 402a 11). Moreover, Aristotle forbids the transfer of a method from one science to another, for example, in proving geometrical truths by the methods of arithmetic (Analytica Posteriora 75a 38–75b 7). He further points out that the sciences are differentiated by their degree of exactness and that this degree depends on the object and the method chosen (Ethica Nicomachea 1094b 12, 23, 1098a 26; Metaphysics 982a 25, 1078a 9).

Inductive Procedure. Aristotle also distinguishes formally between two opposed procedures, the process from principles and the process to principles; in so doing, he recalls that Plato sought to determine which of the two is more advantageous (Ethica Nicomachea 1095a 30). In the Republic (510–511), Plato maintains that geometry and similar sciences start from accepted hypotheses and argue from these to conclusions, whereas dialectic first searches out the principles and then proceeds to conclusions. Just as Plato prefers the dialectical method, so too does Aristotle recommend the process leading from experience to principles, i.e., the process of generalization or abstraction by which man goes from what is more evident to him to what is more evident by nature (Physica 184a 17–21).

At a much later period in the history of philosophy, Francis bacon would wish to proceed from sensations to axioms (Novum Organum 1.19), and René descartes would insist on beginning with objects that are "the easiest to know" (Discourse 2). Yet the naturalness of this procedure had long before been noted by Plato and Aristotle. All animals have sensations that, for some, are retained through memory; in man, memories form the basis of experience, and experience provides the basis of science (Analytica Posteriora 99b 15–100b 17; Metaphysics 980a 22–982a 1). Aristotle, however, insists on the experiential starting point more than does Plato. In the Aristotelian view, sciences, whether inductive or deductive, cannot overlook sensations (Analytica Posteriora 81a 37–81b 9); physicists, astronomers, and zoologists must reckon with phenomena (De Partibus Animalium 639a 2–642b 4; Cael. 306a 16). Aristotle is quick to blame those who, lacking sufficient experience, would explain nature (De Generatione et Corruptione 316a 5); even in morals, he says, one must begin with phenomena (Ethica Nicomachea 1145b 2). In some places he even indicates a preference for experience over reasoning. Speaking of the multiplication of bees, for example, he affirms that observation is more reliable than speculation (Generatione Animalium 760b 30); elsewhere he regards as

empty a general or logical explanation of the sterility of mules (ibid., 747b 28; De Generatione et Corruptione 316a 10; De Anima 403a 29). For Aristotle, observation must be continuous and complete; he explains that if one, being on the Moon, would observe once that Earth faces the Sun and that an eclipse is taking place, he would not know its cause, but he would come to know it after numerous observations (Analytica Posteriora 87b 39–88a4). On this score, he takes Democritus to task for his poor explanation of dentition based on limited observation of some animals alone (De Generatione Animalium 788b 9–19). Surely, the originality of Descartes is not to be found in this rule: "make everywhere divisions so complete and revisions so general [as to be] certain to omit nothing" (Discourse 2).

Nature of Induction. It is this method or procedure—from observation to generalization—that Aristotle usually calls induction. H. Maier has suggested that perhaps he is the first one to have used the word in this technical sense (Die Syllogistik des Aristoteles [3 v. Tübingen 1896–1900] 2.1:374). Aristotle specifies that the basis of induction is the resemblance or similarity among particular objects (Topica 108b 7; De Rhetorica 1356b 12). In relating induction to the syllogistic form, he also prescribes a completeness that includes all particular cases (Analytica Priora 68b 8–29). This prescription was challenged by Bacon (Novum Organum 1:105) and by all those who have failed to notice that the Aristotelian "particular" refers to the species and not to the individual. Other logicians have objected to Aristotle's limiting of induction to the observation of facts and his assigning of causal explanation to demonstration (Analytica Posteriora 92a 34). In fact, it seems that induction is justified by the very existence and recognition of causal relations (see causality).

A variant of Aristotelian induction is the paradigm, or the exemplary model. As in induction, its basis is similarity or resemblance. But contrary to induction, it does not lead to the universal; rather, it concludes to the particular from the particular. The following is an example: If a war between two neighboring countries ends up as a liability for the aggressor, another war of the same type will likewise be fateful for the new assailant (Analytica Priora 68b 38–69a 19; De Rhetorica 1355b 26–1358a 35, 1393a 22–1394a 18). Example itself was considered by Aristotle as an instrument proper to rhetoric but improper to scientific logic (Analytica Posteriora 71a 1–10; De Rhetorica 1356b 5). Actually, it involves an analogical judgment (άναλογίζεσθαι) identical with that of induction.

Ground of Induction. The precise formulation of the ground of inductive reasoning has long been seen in the regula philosophandi of Sir Isaac Newton: "Effectuum generalium ejusdem generis eaedem sunt causae" (General effects of the same kind have the same causes). This ground, for Newton, is nothing more than an application of the principle of causality (see causality, princi ple of).

The attack on causality launched by David hume is also an attack on induction or, at least, an attempt to seek its foundation elsewhere. Indeed the author of the Treatise of Human Nature and of the Philosophical Essays maintains that induction is based on habit or on a personal disposition that has nothing to do with truth or with the nature of things. "Having observed the constant relation between two things, for example heat with flame or solidity with weight, we are determined only by force of habit to conclude from the existence of one to the existence of the other. Otherwise, it is impossible to explain why we conclude from a thousand cases that which we could not conclude from a single case" (Philosophical Essays 5).

Yet another ground is proposed by Thomas reid and the scottish school of common sense. "In the phenomena of nature, what is to be will probably be like to what has been in similar circumstances" (Essays on the Intellectual Powers of the Human Mind, 6, ch. 5, 12); this is to say that "nature is governed by invariable laws." For P. P. Royer-Collard (1763–1843), these laws are of two types, stable and general. Royer-Collard's first principle of induction is that "the universe is governed by stable laws," so that once known at one moment, they are known at all times. His second principle is that "the universe is governed by general laws," so that once known for a single case, they are alike for all cases. The later developments of Reid and of Royer-Collard seem reducible to a simpler Newtonian formulation: nature is governed by laws. The character of a law, or the sign that reveals its existence, is that it applies equally well to all cases covered by the law. Stated somewhat differently, the same cause, in the same circumstances, will produce the same effect. (see uniformity.)

Jean Nicod (1893–1924), basing his analysis on John Maynard Keynes's Treatise on Probability (1921), has come to the following conclusions: (1) induction by simple enumeration is a type of basic proof that cannot be dispensed with without resorting to sophistry; (2) this type of reasoning retains its value even without a prior postulate of determinism; (3) inductive proof can increase the probability of a hypothesis even if newly observed facts are merely repetitions without variation of facts already known; (4) if causes are eliminated as a ground of induction, the most that can be attained by this type of argument is a moderate degree of probability; and (5) it has yet to be demonstrated that inductive reasoning can raise the degree of probability of a law to that of absolute certitude (Le Problème logique de l'induction, Paris 1961).

Recent Theories. Recent attempts at solving the problem of induction have moved in two main directions. First, many thinkers basically accept Hume's criticism and take it as starting point in their investigation of the topic; they try to offer some kind of "vindication" or "validation" as a substitute for a definitive justification. Within this philosophical literature, induction has been examined mostly from a logical perspective; sometimes presumably an "exclusively" logical one. The post-Humean tendency has been to reduce the issue of induction to the process whereby a general proposition is obtained from several particular instances. There has also been a complementary effort to explain that process as the fulfillment of a psychological need of human beings. To date, these viewpoints sprung from a common source have been seen as different and very much unrelated.

Other authors, moving in the second direction, examine and reject the presuppositions on which Hume's famous attack is based. They prefer to unravel the confusions that cloud the issue. Noting that many of the approaches that accept Hume's criticism seem to presuppose some inductive procedures at any rate, they try to use the reduction ad absurdum against Hume's criticism. Hume's dilemma is thus interpreted as an indirect proof that the presuppositions on which it is based are at fault. In general, philosophers working along these lines find nothing wrong with a conclusion that in some way "says more than its premises." This approach has received some valuable support from nonpositivistic philosophers of science. William Kneale and Henry B. Veatch, among others, have criticized the attempt to reduce all necessity to logical necessity, a doctrine clearly formulated in Wittgenstein's Tractatus (6.37).

Other philosophical traditions have also tried to clarify the riddle of induction from their own viewpoints. Within a Thomistic framework, Lonergan has explained inductive conclusions as based on the principle that "similars are similarly understood" (Insight: A Study of Human Understanding 288). The real problem of induction is the problem of criteria of relevant similarity. There cannot be a difference in understanding the data unless there is a difference in the data themselves. In a further elaboration of Lonergan's theories, Philip McShane's Randomness, Statistics and Emergence provides an analysis of statistical science as a type of general knowledge of random aggregates.

Another group of philosophers has dealt with induction from a phenomenological perspective. Phenomenologists reject the empiricist principle upon which many of the investigations on induction depend: when such a principle is abandoned, it becomes questionable whether there still is a "problem of induction." The phenomenological notion of eidetic intuition is frequently mentioned in connection with induction, as the factor that guarantees the validity and certainty of inductive generalizations. In this sense, eidetic intuition is a necessary though not sufficient condition for the validity of inductive conclusions. Its peculiar role is to provide an insight into the contents of inductive generalizations, thereby providing the universality and necessity that cannot be found in simple repetition (cf. Joseph Kockelmans, The World in Science and Philosophy 123).

During the mid-20th century much thought was given to the origin and evolution of science. These investigations indirectly affected the discussions on induction, mostly in the sense of establishing the limits of its extent without, however, denying its existence as a valid way of knowledge. Thomas Kuhn's notion of paradigm (The Structure of Scientific Revolutions, Chicago 1962) and Norwood R. Hanson's analysis of seeing as "seeing that x " (Patterns of Discovery ) emphasize the importance of formulating imaginative hypotheses in order to account for the facts that, for the most part, do not appear to be in a "pure" form, just ready for public inspection. The evolution and growth of science is not explained as a process of accumulation, but as dependent on the formulation, development, and eventual substitution of paradigms. Within the perspective of these approaches, induction may play a role in the origin of theories and may give some indications about their possible explanatory and predictive power. In this way, even though induction does exist, its role is by no means as decisive and primary as Baconian inductivism would want it to be.

As a result of long discussions on the topic, inductive inferences enjoyed an increasingly secure though less extended place in the world of science.

See Also: methodology (philosophy).

Bibliography: m. j. adler, ed., The Great Ideas: A Syntopicon of Great Books of the Western World, 2 v. (Chicago 1952); v. 2, 3 of Great Books of the Western World 1:805–815. r. eisler, Wörterbuch der philosophischen Begriffe, 3 v. (4th ed. Berlin 1927–30) 1:741–745. r. houde, "The Logic of Induction," The Logic of Science, ed. v. e. smith (New York 1963) 17–34. p. conway, "Induction in Aristotle and St. Thomas," Thomist 22 (1959) 336–365. o. hamelin, Le Système d'Aristote (Paris 1920; 2d ed. 1931). For induction in the philosophy of science, see e. h. madden, ed., The Structure of Scientific Thought (Boston 1960). For induction in comparative logic and Indian philosophy, see p. massonoursel, La Philosophie comparée (Paris 1923). t. shcherbatsky, Buddhist Logic, 2 v. (pa. New York 1962). p. edwards, "Russell's Doubt about Induction," Mind 58 (1949) 141–163. i. p. creed, "The Justification of the Habit of Induction," Journal of Philosophy 37 (1940) 85–97. f. l. will, "Is There a Problem of Induction?" ibid. 39 (1942) 505–513. w. h. v. reade, The Problem of Inference (Oxford 1938). a. j. ayer, Language, Truth and Logic (New York 1952). s. barker, Induction and Hypothesis (Ithaca, NY 1957). m. black, Problems of Analysis (London 1954); "The Justification of Induction," Language and Philosophy (Ithaca, NY 1949). c. d. broad, Induction, Probability and Causation (Dordrecht, the Netherlands; New York 1968). m. bunge, Causality (Cleveland, New York 1963). r. carnap, Foundation of Logic and Mathematics (Chicago 1939). n. r. hanson, Patterns of Discovery (Cambridge, Eng. 1965). r. harre, Theories and Things (London, New York 1961); An Introduction to the Logic of the Sciences (New York 1967); The Principles of Scientific Thinking (Chicago 1970). j. katz, The Problem of Induction and Its Solution (Chicago 1962). j. kockelmans, The World in Science and Philosophy (Milwaukee 1969). j. kockelmans and t. kisiel, Phenomenology and the Natural Sciences (Evanston, IL 1970). w. kneale, Probability and Induction (Oxford 1949). h. e. kyburg and e. nagel, eds., Induction: Some Current Issues (Middletown, CT 1963). c. i. lewis, An Analysis of Knowledge and Valuation (La Salle, IL 1946). b. lonergan, Insight: A Study of Human Understanding (New York 1957). e. h. madden, ed., The Structure of Scientific Thought (Boston 1960). p. mcshane, Randomness, Statistics, and Emergence (Notre Dame 1970). j. piaget, The Psychology of Intelligence (New York 1950); The Construction of Reality in the Child (New York 1954); Genetic Epistemology (New York 1970). k. r. popper, The Logic of Scientific Discovery (New York 1951); Conjectures and Refutations (London 1963). h. reichenbach, Experience and Prediction (Chicago 1938). d. williams, The Ground of Induction (Cambridge, MA 1947).

[r. houde/

l. camacho]

Induction

In mathematics, induction is a technique for proving certain types of mathematical statements. The induction principle can be illustrated by arranging a series of dominoes in a line. Suppose two facts are known about this line of dominoes.

1. The first domino is knocked over.
2. If one domino is knocked over, then the next domino is always knocked over.

What can be concluded from these statements? If the first domino is knocked over, then the second domino is knocked over, which knocks over the third, fourth, fifth, and so on, until eventually all of the dominoes fall.

Induction is a simple but powerful idea when applied to mathematical statements about positive integers. For example, consider the following statement: n ² ≥ n for all positive integers, n. To prove that this statement is true using induction, it is necessary to prove two parts: first, that the statement is true for n = 1; and second, that if the statement is true for a positive integer n = k, then it must be true for n = k + 1. Demonstrating both of these parts proves that the mathematical statement has to be true for all positive integers.

Suppose using the induction principle it has been shown that n ² ≥ n. It is then instructive to see how the statement is true for all positive integers, n. The first part says that n ² ≥ n is true for n = 1, which is, in effect, knocking over the first domino. According to the second part, n ² ≥ n is also true for n = k + 1 when it is true for n = k, so it is true for 1 + 1 = 2. This proves that the next domino is always knocked over. Now apply the second part again and take k = 2. Continuing this process proves that n ² ≥ n is true for all positive integers.

Using the induction principle, it can also be shown that 2n is always an even number for all positive integers, n. Substitute 1, 2, 3, and 4 for n, and the results are 2, 4, 6, and 8, which are all even numbers. But how can it be certain that, without fail, every positive integer n will result in an even number for 2n ? It looks obvious, but often what looks obvious is not necessarily a valid proof. The induction principle, however, provides a valid proof.

The mathematical statement we want to prove is that 2n is an even number when n is a positive integer. To test the first part, we know that for n = 1, 2n is 2 × 1, or 2. The first even number is 2. So the statement is true for n = 1. To test the second part, suppose that 2n is an even number for some positive integer n = k. Therefore, 2k is even. Remember, adding 2 to any even number always produces an even number. So 2k + 2 is also an even number, but 2k + 2 = 2(k + 1). Hence, 2(k + 1) is an even number. Assuming that the statement is true for n = k leads to the fact that the statement is true for n = k + 1. Therefore, the induction principle proves that 2n is an even number for all positive integers, n.

see also Proof.

Rafiq Ladhani

Bibliography

Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.

Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.

in·duc·tion / inˈdəkshən/ • n. 1. the action or process of inducting someone to a position or organization: the league's induction into the Baseball Hall of Fame. ∎  [usu. as adj.] a formal introduction to a new job or position: an induction course. ∎  enlistment into military service.2. the process or action of bringing about or giving rise to something: isolation, starvation, and other forms of stress induction. ∎  Med. the process of bringing on childbirth or abortion by artificial means, typically by the use of drugs.3. Logic the inference of a general law from particular instances. Often contrasted with deduction. ∎  (induction of) the production of (facts) to prove a general statement. ∎  (also mathematical induction) Math. a means of proving a theorem by showing that if it is true of any particular case, it is true of the next case in a series, and then showing that it is indeed true in one particular case.4. Physics the production of an electric or magnetic state by the proximity (without contact) of an electrified or magnetized body. See also magnetic induction. ∎  the production of an electric current in a conductor by varying the magnetic field applied to the conductor.5. the stage of the working cycle of an internal combustion engine in which the fuel mixture is drawn into the cylinders.

induction
1. A method of logical inference in which a general but not necessarily true conclusion is drawn from a set of particular instances. In machine learning, for example, the term induction is used to describe an approach to machine learning in which generalized structures or statements are inferred from particular examples.

2. A process for proving mathematical statements involving members of an ordered set (possibly infinite). There are various formulations of the principle of induction. For example, by the principle of finite induction, to prove a statement P(i) is true for all integers i ≥ i₀, it suffices to prove that

(a) P(i₀) is true;

(b) for all k ≥ i₀, the assumption that P(k) is true (the induction hypothesis) implies the truth of P(k+1).

(a) is called the basis of the proof, (b) is the induction step.

Generalizations are possible. Other forms of induction permit the induction step to assume the truth of P(k) and also that of P(k–1), P(k–2), …, P(k–i)

for suitable i. Statements of several variables can also be considered. See also structural induction.