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Induction is the conscious mental process by which we pass from the perception of particular phenomena (things and events) to the knowledge of general truths. The sense perception is expressed logically in the singular or particular judgment (symbolically: "This S is P", "Some S's are P", "If S is M it may be P"); the general truth, in the universal judgment ("All S is P", "S as such is P", "If S is M it is P").
Although induction is equally applicable in all departments of generalization from experience, in the historical and anthropological no less than in the physical sciences, still it is in its application to the discovery of the causes and laws of physical phenomena, animate and inanimate, that it lends itself most readily to logical analysis. Hence it is that logical textbooks ordinarily speak of "physical" induction. The process is often described as a ratiocinative or inferential process, and from this standpoint is contrasted with deductive reasoning. But if by logical inference we are to understand the conscious passage of the mind from one or more judgments as premises to another new judgment involved in them as conclusion, then this is certainly not the essence of the inductive process, although there are indeed ratiocinative steps involved in the latter, subsidiary to its essential function which is the discovery and proof of some universal truth or causal law of phenomena. Induction is really a logical method involving many stages and processes besides the central step of generalization itself; and it is opposed to deduction only in the sense that it approaches reality from the side of the concrete and individual, while deduction does so from that of the abstract and universal.
The first of these steps is the observation of some fact or facts of sense-experience, usually a repeated coexistence in space or sequence in time of certain things or events. This naturally prompts us to seek its explanation, i.e. its causes, the total combination of proximate agencies to which it is due, the law according to which these causes secure its regular recurrence, on the assumption that the causes operative in the physical universe are such that acting in similar circumstances they will always produce similar results. Logic prescribes practical directions to guide us in observing, in finding out accurately what accompanies or follows what, in eliminating all the merely accidental concomitant circumstances of a phenomenon, so as to retain for analysis only those that are likely to be causally, as distract from casually, connected with the event under investigation.
Next comes the stage at which the tentative, empirical generalization is made; the suggestion occurs that the observed connection (between S and P) may be universal in space and time, may be a natural causal connection the ground of which lies in a suspected agency or group of agencies operative in the total sense-experience that gives us the elements under investigation (S and P). This is the formation of a scientific hypothesis. All discovery of laws of physical nature is by way of hypotheses; and discovery precedes proof; we must suspect and guess the causal law that explains the phenomenon before we can verify or establish the law. A hypothesis is conceived as an abstract judgment: "If S is M it is P", which we—relying on the uniformity of nature—forthwith formally generalize: "Whenever and wherever S is M it is P", a generalization which has next to be tested to see whether it is also materially accurate. A hypothesis is therefore a provisional supposition as to the cause of a phenomenon, made with the object of ascertaining the real cause of the latter. Logic cannot, of course, suggest to us what particular supposition we ought to make in a given case. This is for the investigator himself. This is where the scientific imagination, originality, and genius come into play. But logic does indicate in a general way the sources from which hypotheses are usually drawn and, more especially, it lays down conditions to which a hypothesis must conform if it is to be of any scientific value. The most fertile source of hypotheses is the observation of analogies, i.e. resemblances between the phenomenon under investigation and other phenomena whose causes are already partially or fully known. When the state of our knowledge does not enable us to make any likely guess about the cause of the phenomenon, we must be content with a working hypothesis which will be perhaps merely a description of the events observed. A hypothesis that purports to be explanatory must be consistent with itself throughout, free from evident and irremediable conflict with known facts and laws, and capable of verification. This latter condition will be fulfilled only when the hypothesis is based on some analogy with known causes. Were the supposed cause totally unique and sui generis, we could form no conjecture as to how it would work in any given or conceivable set of circumstances, and we could therefore never detect whether it was really there or not. A hypothesis may be legitimate and useful in science even though it may turn out to be inaccurate; few hypotheses are altogether accurate at first. It may even have to be rejected altogether as disproved after a time and yet have served to lead to other discoveries or have put investigators on the right track. Or, as is more usually the case, it may have to be moulded, modified, limited, or extended in the course of verifying it by further observation and experiment.
It is to help the investigator in this work of analyzing the facts of sense-experience so as to discover and prove causal connections or natural laws by the formation and verification of hypotheses, that modern logicians have dealt so exhaustively with the "canons of inductive inquiry", or "experimental methods" first outlined by Herschel in his "Preliminary Discourse on the Study of Natural Philosophy" and first popularized by John Stuart Mill in his "System of Logic". These canons—of agreement, difference, concomitant variations, residues, positive and negative agreement, combined agreement and difference—all merely formulate various ways of applying to the analysis of phenomena the principle of eliminating what is casual or accidental so as to leave behind what is causal or essential; they are all based upon the principle that whatever can be eliminated from a set of things or events without thereby eliminating the phenomenon under investigation, is not causally connected with the latter, and whatever cannot be so eliminated without also eliminating the phenomenon is causally connected with it. Stating a hypothesis in the symbols, "If S is M it is P", we have in M the supposed real or objective cause of P, and also the mental or logical ground for predicating P of S. We test or verify such a hypothesis by endeavouring to establish, through a series of positive experiments or observations, that whenever and wherever M occurs so does P; that M necessitates P; and, secondly, through a series of negative experiments or observations, that wherever and whenever M is absent so is P, that M is indispensable to P, that it is the only possible cause of P. If these tests can be applied successfully the hypothesis is fully verified. The supposed cause of the phenomenon is certainly the real one if it can be shown to be indispensable, in the sense that the phenomenon cannot occur in its absence, and necessitating, in the sense that the phenomenon must occur when it is present and operative. This sort of verification (often only very imperfectly and sometimes not at all attainable) is what the scientist aims at. It establishes the two propositions "If S is M it is P", and "If S is not M it Is not P"—the latter being equivalent to the reciprocal of the former (to "If S is P it is M"). Whenever we attain to this ideal (of the reciprocal hypothetical) we can infer from consequent to antecedent, from effect to cause, just as reliably as vice versa. But over what range of phenomena are we to carry on our negative observations and experiments in order to make sure that our hypothesis offers the only possible explanation of the phenomenon, that M is the only cause in the universe capable of producing P—that, for instance, the necessity which beset the early Christians of securing a place of refuge for themselves and of burial for their dead could alone account for the formation of the Roman catacombs as we find them? This is obviously a matter for the prudence of the investigator, and, incidentally, it indicates one limitation of the certitude we can reach by induction. What is known as a crucial instance or experiment will, if it occurs enable us summarily to dismiss one of two conflicting hypotheses as erroneous, thus establishing the other, provided this other is the only conceivable one in the circumstances—that is to say, the only one reasonably suggested by the facts; for there is scarcely any hypothesis to which some fanciful alternative might not be imagined; and here again prudence must guide the investigator in forming his conviction. Is he, for instance, to suspend his assent to the physical hypothesis of a universal ether because the alternative of actio in distans is at any rate not evidently an intrinsic impossibility?
When a hypothesis cannot be rigorously verified by establishing the reciprocal universal judgment, it may nevertheless steadily grow in probability in proportion to the number and importance of other cognate phenomena which it is found capable of accounting for, in addition to the one it was invented to explain. A hypothesis is rendered highly probable if it foretells or explains cognate phenomena; this is called by Whewell consilience of inductions (Novum Organum Renovatum, pp. 86, 95, 96). This process of verification runs somewhat on these lines: "If M be a really operative cause then in such and such circumstances it ought to produce or account for the effect X, and in such others for Y and so on; but (by observation or experiment we proceed to find that) in these circumstances these effects are produced or explained by it; therefore probably they are due to M." They are probably attributable only, because the argument does not formally yield a certain conclusion; but the more we extend our hypothesis, and the larger the groups of phenomena it is found competent to explain, the firmer does our conviction naturally grow, until it reaches practical or moral certitude that we have hit on the true law of the phenomena examined. Thus, for instance, was Newton's gravitation hypothesis gradually extended by him so as to explain the motions of the moon and the tides, the motions of the satellites around the planets and of these around the sun, until finally it came to be regarded as applicable throughout the whole material universe. The aim of the inductive process is to explain isolated facts by bringing them under some law, i.e. by discovering all the causes to the co-operation of which they are due and laying down those general propositions called laws of nature which embody and express the constant mode of operation of those causes. It is thus that we transform the observed sequences of sense-experience into understood or intellectually explained consequences of cause and effect. Scientific explanation also aims at reducing these separate and narrower laws themselves to higher and wider laws by showing them to be partial applications of the latter, thus obeying the innate tendency of the human mind to synthesize and unify, as far as may be, the manifold and chaotic data of sense experience.
The inductive generalization by which, after examining a limited number of instances of some connection or mode of happening of phenomena, we assert that this connection, being natural, will always recur in the same way, is a mental passage from particular to general, from what is within experience to what is beyond experience. Its legitimacy needs justification. It rests on the assumption of a few important metaphysical principles. One of these is the principle of causality: "Whatever happens has a cause." Since by the cause of a thing or event we mean whatever contributes positively to its being or happening, the principle of causality is clearly a self-evident, necessary, analytic principle. And it is obviously presupposed in all inductive inquiry: We should not seek for the causes of phenomena did we believe it possible that they could be or happen without causes. A somewhat wider objective principle than this is the principle of sufficient reason: "Nothing real can be as it is without a sufficient reason why it is so"; and, applied to the subjective, mental, or logical order the principle states: "No judgment can be true without a sufficient reason for its truth". This principle, too, is presupposed in induction; we should not seek for general truths as an explanation or reason for the individual judgments that embody our sense-experience did we not believe it possible to find in the former a rational explanation of the latter. But there is yet another principle, more directly assumed, involved in the inductive generalization, viz. the principle of the uniformity of nature: "Natural or non-free causes, i.e. the causes operative in the physical universe apart from the free will of man when they act in similar circumstances always and everywhere produce similar results"; "Physical causes act uniformly."
Since human free will is excluded from the scope of this Principle, it follows that the phenomena which issue directly from the free activity of man do not furnish data for strict induction. It would, however, be a mistake to conclude that the influence of free will renders all science of human and social phenomena impossible. Such is not the case. For even those phenomena have a very large measure of uniformity depending largely, as they do, on a whole group of influences and agencies other than free will: on racial and national character, social habits and surroundings, education, climate, etc. They are, therefore, manifestations of stable causes and laws, though not of mechanical or physical laws, and form a suitable, though difficult domain, for inductive inquiry—difficult, because the operative influences are hidden under a mass of chaotic data which must be prepared by statistics and averages based on painstaking and long-continued observations and comparisons.
In the domain of physical induction proper we have to do only with natural or non-free causes. Above these, therefore, the question next arises: by what right do we assume the universal truth of the principle of uniformity as just stated, or what kind or degree of certitude does it guarantee to our inductive generalizations? Obviously it can give us no higher degree of certitude about the latter than we have about the principle itself. And this latter certitude will be determined by the grounds and origin of our belief in the principle. How, then, do we come to formulate consciously for ourselves, and give our assent to, the general proposition that the causes operative in the physical universe around us are of such a kind that they are determined each to one line of action, that they will not act capriciously, but regularly, uniformly, always in the same way in similar circumstances. The answer is that by our continued experience of the order and regularity and uniformity of the ordinary course of nature we gradually come to believe that physical causes have by their nature a fixed, determined line of action, and to expect that unless something unforeseen and extraordinary interfere with them, they act beyond our experience as they do within it. Mill is right in saying that the principle is a gradual generalization from experience, and, furthermore, that it need not be consciously grasped in all its fullness anterior to any particular act of inductive generalization. But this as not enough; for, whether we take it partially or fully in a given case the question still remains: What is our ultimate rational justification for extending it at all beyond the limits of our actual personal experience? The answers given to this question by logicians, as indeed their entire expositions of the inductive process, are as divergent and conflicting as their general philosophical views regarding the ultimate nature of the universe and of all reality. The fact to be explained and justified is that we believe the world outside our personal experience to be of a piece with the world within our experience. But the Empirical or Positivist philosophy, represented by Hume and Mill, makes all rational justification of this belief impossible; for if there is no world outside experience; it reduces all reality in ultimate analysis to the present actual sensations of the individual's consciousness; and the alleging of mere custom, mere actual experience of uniformity, as a reason for belief in unexperienced uniformity, it regards not as a rational expectation based on a reasoned view about the nature of reality but simply a blind leap in the dark. The explanation of the current Monistic Idealism, which would identify the laws of physical phenomena with the laws of logical thought and reduce all reality to one system of intellectually necessary thought-relations, is no less unsatisfactory, for it confounds the phenomena of existing, contingent being with the metaphysical relations between abstract, possible essences—relations which have their ultimate basis only in the nature of the Necessary Being, God Himself. The answer of scholastic philosophy is that the ultimate rational justification for our belief in the uniformity of nature is our reasoned conviction that nature is the work of an All-Wise Creator and Conserver, Who has endowed physical agencies with regular constant modes of activity with which He will not interfere unless by way of miracle for motives of the higher or moral order. The certitude of our belief in the principle and its applications is thus hypothetical, physical, not absolute, not metaphysical: "If God continues to conserve and concur with created physical agencies, if He does not miraculously interfere with them, if no other unknown cause intervene, then those agencies will continue to act uniformly."
Physical induction sometimes inquires into the constitutive ("formal" and "material") causes of phenomena (as, for instance, in chemical and physical researches into the constitution of matter), sometimes into their purpose (or "final" causes, as in the biological sciences); but mainly into their proximate efficient causes, i.e. the total group of proximate agencies sufficient and indispensable for the production of any given phenomenon. To these primarily is inductive research restricted, for the agencies operative in the physical universe are so intimately interwoven and interdependent that, were we to trace the chains of causality outward and backward from any effect indefinitely, we should see that in a sense all the agencies in the universe are in some remote way operative in the production of any single effect. Much controversy has been needlessly imported into Logic regarding the concept of cause. The rejection of "efficiency" or "positive influence" from this concept and the substitution of "invariable and unconditional sequence" is a feature of Empiricism. But it can have no influence on inductive generalization about the conduct of phenomena in space and time. For reliable generalization about the latter the only objective condition needed is uniformity or regularity of occurrence. The scope of induction will, however, be unduly and unjustifiably narrowed if by physical cause we are always to understand with Mill something which is itself a phenomenon, perceptible by the senses, and if we are to eschew all inquiry into causes which are not themselves sense-phenomena but active qualities rooted in the natures of things and discernible only by intellectual reasoning. No doubt it is to inductive research for mere phenomenal antecedents—for material masses and energies—and to their exact mathematical measurement in terms of mechanical work that the applied sciences-owe their greatest triumphs. But though the only concern of the engineer is to know how to secure useful coexistences and sequences of material masses and motions, yet the man of thought, be he physical scientist or philosopher, will rightly resent being prohibited by Positivism from prosecuting a further investigation into the rational why and wherefore of these occurrences, into the natures and properties which reason alone can discover through those phenomena. Men will ever and rightly insist on inquiring inductively after veræ causæ, which, though they produce effects perceptible by the senses, are not themselves phenomena. However, when we push back our inquiry into the more remote conditions, causes, origin, and constitution of wider and wider fields of phenomena, analogies from known proximate causes—which aided us in our more specialized researches—begin to fail us; and so our wider theoretical conceptions—about atoms, electrons, ether, etc.—must ever remain more or less probable hypotheses, never fully verified. When, finally, we inquire into the absolutely ultimate origin, nature, and destiny of the universe, where analogies fail us altogether, we must abandon induction proper, which seeks to compare and classify the causes it discovers, and have recourse to the a posteriori argument, which simply infers, from the existence of an effect, that there must exist a cause capable of producing it, but gives us no further information about the nature of this cause than that it must have higher perfection, excellence, being, than the effect produced by it. Such, for instance, are the arguments by which we prove the existence of God.
Scientific induction, as just set forth, was not unknown to Aristotle and the medieval scholastics. It is not, however, the process referred to by Aristotle as epagogé (Anal. Prior, II, 23) and usually described as the "inductive syllogism", or "enumerative induction". This is simply the process of inferring that what can be predicated of each member of a class separately can be predicated about the whole class. It is of no scientific value; for, when the enumeration of instances is perfect, or complete the conclusion is not a scientific universal, a general law, but a mere collective universal; and when the enumeration of individuals is imperfect, or incomplete, the collective conclusion is hazardous, more or less probable, but not certain. Aristotle was, however, well aware of the possibility of reaching a certain conclusion after an incomplete enumeration of instances, by abandoning mere enumeration and undertaking an analysis of the nature of the instances as in modern induction. He refers to this process repeatedly under the name of empeiría in the "Posterior Analytics" (c. xix; xxxi; i, §4; cf. Rhet., II: parádeigma), though he did not investigate the conditions under which such analysis would produce certitude. The prevalent belief that the medieval scholastics treated only "enumerative induction" is erroneous. They were also familiar with scientific induction, using the terms experimentum, experientia, to translate Aristotle's empeiría. Albertus Magnus (In An. Post. I, tr. I, c. ii, iii), Duns Scotus (I Sent., dist. iii, q. iv, n. 9), and St. Thomas Aquinas (In An. Post. II, lect. xx) examined it, without, however, attempting to treat of the conditions of its application, for the very good reason that the apparatus for scientific research did not exist in their day. But the achievements of Roger Bacon, a Franciscan monk of the thirteenth century, in this direction, are perhaps sounder than those of his better known namesake, Francis Bacon, of the sixteenth and seventeenth. With the progress of the physical sciences in the eighteenth and nineteenth centuries the attention of logicians was concentrated almost exclusively on the application of the inductive method to the discovery and proof of the laws of nature; and at the present time its philosophical foundations are giving rise to considerable discussion.