Induction

INDUCTION. - In current language (apart from the technical meaning in physical science, e.g., magnetic i.) it means "conjecture," "supposition." In philosophical language, to which the term principally refers, i. (Gr. ἐπαγωγή, from ἐπ-άγω, I lead, I bring in) is, in general, the logical-gnoseological process by which human thought ascends from particular facts of experience to universal principles or values. In this comprehensive sense i. is also the formation of the concept from the concrete data of experience. In a more restricted sense, which has become common in logic, i. is the inference of universal propositions (referring to the cause, the law or the nature of a phenomenon) from particular cases or instances: such, for example, is the general conclusion about the medicinal properties of a substance from positive experience in some cases. In this latter sense i. is distinguished, with technical terminology due to the school of Wolff, into complete, if it proceeds from the consideration of all possible cases, and incomplete, if from a partial consideration of them. It should be noted, however, that by extension one speaks of i. also when the phenomenon under examination does not admit a multiplicity of cases, e.g., the rotational motion of the earth: i. is then the determination of the law or cause or essential nature of the phenomenon by means of the methods of scientific observation.

Complete i., as a general affirmation derived from the ascertainment of the predicate in all possible subjects (e.g., the conclusion that every species of biological organism is constituted of cells, from its verification in plants, in animals, in man) has evidently an absolute logical value, inasmuch as it entails a total equivalence between the premises and the conclusion. By reason of this character of logical necessity, complete i. (moreover very rarely possible in the observation of nature) presents no scientific interest nor particular problems: rather than i. it may be called the limiting case of i., in which, that is, i. itself assumes substantially the meaning of deduction (v.), understanding this in its most generic sense as necessary formal inference. On the contrary, incomplete i., which represents in itself a progress and an extension of knowledge, has essential importance in the scientific process and method and has given rise in the history of philosophical thought to relevant problems touching the vital points of logic and metaphysics. It expresses in fact the movement of thought from the particular to the universal, from the contingency of the fact to the absolute of the principle, and is therefore at once the process by which the premises of demonstration (v.) are put into act, and the disintegrated multiplicity of experience is organized into a scientific-philosophical system of concepts and principles. Through it there is accordingly effected the most essential emergence of human knowledge, and its scope extends as far as experience itself, wherever this presents itself in human knowing as the foundation of meaning and value.

I. was already in the essence of the Socratic method (cf. Aristotle, Met., XIII, 4, 1078 b 27 ff.; Xenophon, Memor., IV, 6, §§ 13-15), but as a simple conceptual universalization through a process of comparison between particular sensible representations and common opinions. I. was therefore the way toward the formation of the concept. This general sense is preserved in Aristotle, who constantly refers the origin of the universal to i.: ἐπαγωγή δὲ ἡ ἀπὸ τῶν καθ' ἕκαστον ἐπὶ τὰ καθόλου ἔφοδος (Top., I, 12, 105 a 13); ἡ μὲν δὲ ἐπαγωγή ἀρχή ἐστι καὶ τοῦ καθόλου (Eth. Nic., VI, 3, 1139 b 28). In a general sense i. therefore also includes abstraction (ἀφαίρεσις: ibid.). But the universal to which i. leads is not only the concept but also the proposition: in this sense i. is the argumentative process inverse to deduction and moves from particulars: ἔστι δ' ἡ μὲν ἀπόδειξις ἐκ τῶν καθόλου, ἡ δ' ἐπαγωγή ἐκ τῶν κατὰ μέρος (Anal. post., I, 13, 81 a 40; cf. Anal. pr., II, 23, 68 b 13). As such, i. has in human knowledge the essential task of leading to the formulation of the principles of the syllogism (Eth. Nic., loc. cit.); if therefore the syllogism, as a process that deduces the particular from the universal, is, in the order of being, prior to i. (in Aristotle the universal is prior in nature), in the order of human knowledge, on the other hand, i. precedes as dialectical and investigative argumentation: δῆλον δὴ ὅτι ἡμῖν τὰ πρῶτα ἐπαγωγῇ γνωρίζειν ἀναγκαῖον (Anal. post., II, 19, 100 b 3-4). It was a much discussed question whether Aristotle means to speak of complete or of incomplete i. In a well-known text of the Anal. pr. (II, 23) he seems to require for i. a total enumeration of singulars (ἡ γὰρ ἐπαγωγὴ διὰ πάντων: 69 b 28) and therefore not to recognize the value of argumentation in incomplete i. But both the examples that he adduces here and elsewhere (Top., I, 2), and the expressions with which he determines the value and specific meaning of i., seem to leave no doubt that he has clearly in mind incomplete i. (of which, moreover, he always made wide use in his researches in natural history). In fact "i. is an instrument of proof more manifest, more known to sense and common to the many" (τοῖς πολλοῖς κοινόν), but less urgent and less efficacious than the syllogism (Top., loc. cit.); it is not an easy operation, and requires knowledge of the characters of similarity of singular things (ibid., ch. 16); he who uses i. properly "does not demonstrate, but nevertheless shows something" (Anal. post., II, 5); from the memory of the same fact repeated many times (πολλάκις) experience is born (ἐμπειρία); and such experience, in which the universal is contained and becomes present ("the one beyond the many"), "is the principle of art and of science" (ibid., ch. 15 [19]; cf. Anal. post., I, 31). Similar and analogous expressions cannot be understood of complete i., which supposes the knowledge of all particulars and as such has a demonstrative value substantially equal to the syllogism (cf. Fr. A. Trendelenburg, Elementa logicae Aristoteleae, 7th ed., Berlin 1892, pp. 111-12).

If incomplete i. and its essential logical-gnoseological meaning were not unknown to Aristotle, he nevertheless did not arrive at a formulation of the conditions for its methodical use in science. And such, it may be said, substantially remains the doctrine of i. down to Bacon. The Aristotelian definition is taken up again in Boethius (De diff.

top., II: PL 64, 1183 a D). Albert the Great, following in Aristotle's footsteps, distinguishes induction from the syllogism (I Top., tract. 3, ch. 4), while nevertheless admitting the possibility of a material reduction of the former to the latter (II Pr. anal., tract. 7, ch. 4). Peter of Spain inserts into Aristotle's definition an element that later becomes common in logic to indicate the fundamental requirement of incomplete induction: « Inducrio est a singularibus sufficienter enumeratis ad universale progressio » (Sum. logic., 5; but already Aristotle in Anal. pr., I, 30, 46 a 20, had spoken of a "sufficient" observation of phenomena). St. Thomas, commenting on Aristotle, states: « oportet supponere quod accepta sint omnia quae continentur sub aliquo communi; aliquin nec inducens poterit ex singularibus acceptis concludere universale » (in Anal. post., II, lect. 4ᵃ, n. 4; cf. I, lect. 1ᵃ, 3, 8). The text would seem to attribute value only to complete induction; but in reality the term supponere allows one to understand that the complete enumeration is to be taken in a virtual sense, inasmuch as even a partial consideration, if it refers to an essential and therefore common characteristic of the individual, is equivalent, virtually, to a complete enumeration. In St. Thomas, incomplete induction, even in its broader sense of a process from the particular to the universal, is still indicated in the exposition and assimilation of Aristotle's doctrine of ἐγεινάς (experimentum: see the text cited above): « Experimentum enim est ex collatione pluriam singularium in memoria receptorum »; « ex multit memoriis unius rei accipit homo experimentum de aliquo » (In I Met., lect. 1ᵃ; ed. R. Cathala, nn. 15, 17; cf. in Post. anal., II, lect. 20). The legitimacy of a general conclusion in incomplete induction is explicitly affirmed by Scotus in a well-known passage: « ...licet experientia non habeatur de omnibus singularibus, sed de pluribus, nec quod semper sed quod pluries, tamen expertus infallibiliter novit quod ita est et quod semper et in omnibus, et hoc per istam propositionem quiescentem in anima: quidquid evenit ut in pluribus ab aliqua causa non libera, est effectus naturalis illius causae » (In I Sent., dist. 3, q. 4, n. 9).

The value of induction, not only gnoseological but also scientific (in the modern sense of the term), together with the fundamental lines of its method, is clearly announced in F. Bacon. For science « spes unica est in inductione vera » (Nov. org., I, aph. 14). But the induction really useful to science is not that which proceeds « per enumerationem simplicem » without taking account of contrary cases and therefore « gives precarious conclusions and exposes itself to the danger of contradictory instances » (ibid., aph. 105): scientific induction must proceed « per eliminazioni ed esclusioni (per exclusiones ac reiectiones) and thus arrive at affirmative conclusions after as many negative ones as will be necessary » (ibid.). Bacon reproaches the ancients for having too easily generalized from a few affirmative instances, afterwards forcing contrary cases into the conclusions already established (ibid., aph. 125). The true method of induction must unfold through three essential moments: a) tables of positive instances (tabulae praesentiae), which note the different cases in which the fact or phenomenon takes place; b) tables of negative instances (tabulae absentiae), which note the similar cases in which the fact does not occur; c) tables of quantitative variations (tabulae graduum), which describe the cases and circumstances in which the fact varies in intensity, both in the same subject and in different subjects (ibid., II, 10 ff.). Through this method (the experientia litterata, that is, written down, noted: ibid., I, 101-103), and, where possible, by means of the "prerogative instances" or privileged cases, in which the phenomenon occurs in entirely new and singular circumstances, or does not present itself in the usual circumstances, Baconian induction aimed at the determination of the "form," that is, the internal structure of things, from which the various effects proceed (ibid., II, aph. 1). Bacon, however, did not give a philosophical theory of induction, nor did he discuss the problem of by what path the general conclusion from an incomplete enumeration is justified. The essential lines of the method of scientific induction (beyond its shrewd application in experimental research) are also indicated by Galileo (cf. Opere, ed. naz., IV, Florence 1895 p. 27): he explicitly affirms that only induction incomplete induction has value in science: "induction, if it had to pass through all particulars, would be impossible or useless; impossible when the particulars were innumerable; and when they were numerable, considering them all would be useless, or, to put it better, concluding by induction would be null" (op. cit., VI, p. 701). From Galileo it may be said that induction, availing itself of experiment, becomes the classic instrument of physical research. The definitive theorization of its method is due to J. Stuart Mill. For Mill, induction "is that operation of the mind by which we infer that what we know to be true in one or more particular cases is also true in all cases that resemble that one in certain definable respects" (System of logic, III, ch. 2, p. 333, § 1): induction properly so called is neither that "complete" induction, nor a pure description in simple terms of a complex fact, as, for example, in the case of one who, coasting along a land and finding himself at last at the point of departure, "concludes" that it is an island. True induction is an extension, a generalization of experience (ibid., ch. 3, 354, § 1): such a generalization, which is founded on the principle of the constancy of the order of nature, or, what is the same, on the law of causality as constant succession, is not legitimized by the mere observation of many positive cases (such is induction "per enumerationem simplicem", which established the fact but did not proceed to an examination of the fact): one must further have the certainty that no contrary cases occur. To this end the four methods of experimental research are ordered, which complete Bacon's three "tables": a) method of agreements: if two or more cases of the phenomenon under examination have in common only one among many circumstances, that one is the cause of the phenomenon; b) method of differences: if a case in which the phenomenon occurs and a case in which it does not occur differ in a single circumstance, present in one case and absent in the other, that circumstance is the cause (or at least part of the cause) of the phenomenon; a variant of these two methods is the combined method of agreements and differences: if two or more cases in which the phenomenon occurs have in common a single circumstance, and two or more cases in which it does not occur have in common only the absence of the same circumstance, this is the cause (or part of the cause) of the phenomenon; c) method of residues: once there has been subtracted from a given phenomenon that part which is already known as the effect of a determinate antecedent, what remains of the phenomenon is the effect of the remaining antecedents; a typical example was the discovery of the planet Neptune (1846): certain irregularities that appeared in the motions of Uranus and were not referable to already known causes led Le Verrier to suppose the existence (and also, approximately, the position and mass) of a new planet, later confirmed by direct observation; d) method of concomitant variations: every phenomenon that undergoes variations each time another phenomenon varies in a determinate way is a cause or an effect of that phenomenon, or at least is connected with it by some causal fact: such, for example, is the dilation of bodies always associated with heat and its variations, or the retardation of motion that varies with the variation of friction. Stuart Mill's formulas (which, like Bacon's "rules", find in part a spontaneous application in common experience itself; and therefore it is a purely polemical attitude to attribute to the ancients only induction "per enumerationem simplicem") represent the most complete definition of the scientific method of induction (v. the detailed exposition, ibid., ch. 8, p. 448 ff.).

But, besides as a problem of scientific method, induction had presented itself, principally with the affirmation of empiricism, as an essentially philosophical problem, bound up with the general question of the value of human knowledge.

The philosophical problem of induction has a double aspect: gnoseological and logical, according as induction itself is considered in its general (and fundamental) function as a process of knowledge from the concrete-sensible datum to the conceptual universal, or instead as the inference of a general proposition from its observation in some particular cases. Under the first aspect the problem of induction is the same problem as that of the origin of the concept, that is, of the derivation in human knowledge of necessity and universality. It is a fundamental contention of empiricism that such necessity and universality cannot derive from experience. According to Hume there is no certainty in experience except of that which is given through it at a precise moment of time (Enquiry concerning human understanding, sec. 4, part 2, Ital. trans. by G. Prezzolini, Bari 1927, p. 39; cf. Treatise, I, part 3, sec. 2 ff.). The assertion is taken up by Kant: "Experience indeed teaches us that a thing is made in this or that way, but not that it cannot be otherwise" (Kritik d. r. Vernunft, intr., § 2; Ital. trans. by G. Gentile, Bari 1940, p. 39); the necessity of conceptual knowledge, which is an undisputed presupposition of criticism, is consequently referred by Kant to the a priori organization of thought. The derivation of the universal from experience (and therefore from induction: St. Thomas, In Post. anal., I, lect. 30) is instead, at least in a certain sense, essential to the Aristotelian-Thomist system: from the repetition of the sensory presentations of a given object, and from their permanence in memory (the experimentum), the singular datum comes to be determined in its ontological unity, and therefore, through the intuitive-abstractive power of thought, emerges in the intellective act as a universal value. This doctrine has a profound correspondence with the actual condition of human knowing. It is further clarified, however, by the virtual objectivity that, in Thomist thought itself, is attributed to the agent intellect (cf. Sum. Theol., 1a, q. 84, a. 5; C. Gent., I, III, ch. 47; De ver., q. 11, a. 1, 3). For it must be said that if the sensible datum is of itself capable of founding a general representation, it could not however posit itself as an absolute necessity of value unless there were immanent in the same intellect a relation to a higher objective necessity. It is thought that, as a transcendental relation to being, which is in itself positivity and necessity, contains the roots of universality and absolute necessity: that universality which may be called founding and which pours itself, according to the finite participations and determinations of being, into the concrete data of experience. Induction presents itself in this way as the emergence or ascent of the objects and contents of experience to the universality and necessity of being, according to the determinations that such universality assumes in their own ontological concreteness. Consequently — and in this direction the overcoming of both empiricism and rationalism is to be ascertained — the derivation of the universal is not purely empirical but at the same time transcendental, inasmuch as thought itself, as a relation to being, is already of itself in a metaphysical situation of virtual transcendence in comparison with the individual moments of experience.

The logical problem of induction was already clearly formulated by ancient skepticism in the dilemma: either induction claims to conclude from all particulars, and this is an impossible undertaking, the particulars being infinite, or only from some of them, and then it will be insecure "since it is possible that some of the particulars omitted in the induction may conflict with the universal" (Pyrrh. hypoth., II, 15). Jevons, Goblot, Bolzano, Hamilton, Windelband, besides the whole empiricist school, on the basis of this dilemma recognize in incomplete induction a purely probabilistic value. It seems first of all that it must be affirmed that induction not only is not to be reduced to the syllogism (a reduction attempted first by Wolff: Logica, part 1, section 4, chap. 6, and later accepted by not a few others), but neither should it be understood as a formal logical process analogous to the syllogism. Even as an "argumentation," induction is an emergence from the particular to the universal. Its logical antinomy lies precisely in the passage from some cases to a universal conclusion: no experience has been had, nor can it be had, of all individual existing bodies, and yet the general formula is accepted: all bodies are heavy. As a pure process of formal inference, induction would not be legitimate. From this difficulty arose the problem of the "foundation" of induction. Such a foundation cannot be of a logical order, but is necessarily of either an empirical or a metaphysical order, or is a synthesis of both. The grounding of induction in pure experience is precisely the hallmark of empiricism: according to Hume, the extension of experience is made on the basis of a psychological habit by which one expects in the future what one has experienced in the past; for Stuart Mill, the principle that sustains induction: "there is a constant causality in nature" is itself derived from experience and therefore is the fruit of previous inductions (op. cit., III, chap. 3, p. 355, § 1). Metaphysical grounding, on the other hand, is characteristic of rationalism and idealism: rationalism in fact transfers into nature the order and necessity of thought; and in idealism the very problem of induction is dissolved insofar as nature is posited as a function of spirit and of its becoming. While empirical grounding ends in the denial of the legitimacy of the inductive process, metaphysical derivation is a systematic position not supported by the phenomenology of human knowing. It must be held that induction cannot be founded on a purely rational principle, e.g.: "the same causes produce the same effects"; for it is indeed indispensable to know that there actually exist the same causes, or permanent causes. It is therefore correct that induction is based on the principle, derived from experience, that the order of nature (of beings) is constant (already explicit in St. Thomas: Sum. Theol., 1ᵃ, q. 19, a. 4, c.; q. 41, a. 2, c.). But the very breadth of experience that attests the constancy of the being and activity of the world is strengthened by the metaphysical consideration that recognizes in being (in every being) an essential ontological determination and therefore also an immanent principle of operative determination: "eo modo aliquid operatur quo est" (Sum. Theol., q. 75, a. 2). The principle of the uniformity of nature therefore admits a metaphysical justification derived from the principle of the determination of being. And this principle (which in the last instance relates to an absolute rational Value, the principle of all being) is the general presupposition of the possibility of knowledge, of action and of human history: the metaphysical indeterminacy of reality would dissolve every firmness in being and therefore every possibility of truth and of action.

This does not, however, deny the margin of probability that must be joined with the inductive process in the knowledge of nature. Not only in the order of human reality (history) does induction present a radical uncertainty because of the principle of freedom, in which action posits itself, at its source, as an undefined possibility, but also in nature induction becomes all the more dialectical the more it draws upon the most complex and profound phenomena of being. Where, as in the subatomic world, the vast interaction of cosmic causes can at any moment modify the effect, the function of induction contracts to a pure determination of statistical probability, more or less broad. Indeed, since the operative process of beings is in reality bound up with the whole dynamism of the universe, one may in general admit, with Boutroux (L'idea di legge naturale, Italian trans. by A. Pasa, Verona 1940, pp. 252-53), an inadequacy of physical laws to confine reality and time within rigid schemes. This matters neither for science nor for action, to which the inductive process confers a solidity of affirmations and explanations that finds in the grandiose successes of science itself its clearest and most permanent confirmation.