INDUCTION. — In ordinary language (apart from the technical meaning in physical science, e.g., magnetic induction) it means “conjecture,” “supposition.” In philosophical language, to which the term chiefly refers, induction (Gr. ἐπαγωγή, from ἐπ-άγω, I lead, bring forward) is, in general, the logical-epistemological process whereby human thought rises from particular facts of experience to universal principles or values. In this comprehensive sense, induction also means the formation of the concept from the concrete data of experience. In a more restricted sense, which has become common in logic, induction is the inference of universal propositions (referring to the cause, law, or nature of a phenomenon) from particular cases or instances: such, for example, is the general conclusion concerning the medicinal properties of a substance drawn from positive experience in certain cases. In this latter sense induction is distinguished, in technical terminology due to the school of Wolff, as complete if it proceeds from consideration of all possible cases, and incomplete if it proceeds from only partial consideration of them. It should nevertheless be noted that, by extension, one also speaks of induction when the phenomenon under examination does not admit of a multiplicity of cases, e.g., the earth’s rotation: induction then consists in determining the law, cause, or essential nature of the phenomenon by means of the methods of scientific observation.
Complete induction, as a general affirmation derived from establishing the predicate in all possible subjects (e.g., the conclusion that every species of biological organism is constituted of cells, drawn from its verification in plants, animals, and human beings), evidently has absolute logical value, insofar as it entails complete equivalence between the premises and the conclusion. Because of this character of logical necessity, complete induction (moreover, only very rarely possible in the observation of nature) presents neither scientific interest nor particular problems: rather than induction, it may be called the limiting case of induction, in which the induction itself deduction (v.), understanding the latter in its most general sense as necessary formal inference. By contrast, incomplete induction, which in itself represents an advance and an extension of knowledge, is of essential importance in the scientific process and method and has given rise, in the history of philosophical thought, to significant 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 fact to the absoluteness of principle, and is therefore at once the process demonstration (v.) are put into effect, and that by which the disintegrated multiplicity of experience is organized into a scientific-philosophical system of concepts and principles. Through it, therefore, the most essential emergence of human knowledge takes place, and its scope extends as far as experience itself, wherever the latter presents itself in human knowing as the foundation of meaning and value.
Induction was already present 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. Induction was therefore the path toward the formation of the concept. This general sense is preserved in Aristotle, who constantly refers the origin of the universal to induction: ἐπαγωγή δὲ ἡ ἀπὸ τῶν καθ' ἕκαστον ἐπὶ τὰ καθόλου ἔφοδος (Top., I, 12, 105 a 13); ἡ μὲν δὲ ἐπαγωγή ἀρχὴ ἐστι καὶ τοῦ καθόλου (Eth. Nic., VI, 3 1139 b 28). In the general sense, induction therefore also includes abstraction (ἀφαίρεσις: ibid.). But the universal to which induction leads is not only the concept but also the proposition: in this sense induction is the inverse argumentative process of deduction and proceeds 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, induction has the essential task in human knowledge of leading to the formulation of the principles of the syllogism (Eth. Nic., loc. cit.); thus, although the syllogism, as a process that deduces the particular from the universal, is, in the order of being, prior to induction (in Aristotle the universal is priu τῇ ῥόσει), in the order of human knowledge it is instead preceded by induction, as dialectical argumentation and inquiry: δῆλον δὴ ὅτι ἡμῖν τὰ πρῶτα ἐπαγωγή γνωρίζειν ἀναγκαῖον (Anal. post., II, 19, 100 b 3-4). It was a much-discussed question whether Aristotle meant complete induction or incomplete induction. In a well-known passage of the Anal. pr. (II, 23), he seems to require for induction a total enumeration of the singulars (ἡ γὰρ ἐπαγωγή διὰ πάντων: 69 b 28), and therefore not to recognize incomplete induction as having argumentative value. But both the examples he gives here and elsewhere (Top., I, 2), and the expressions with which he determines the specific value and meaning of induction, seem to leave no doubt that he has incomplete induction clearly in mind (of which, moreover, he always made extensive use in his investigations of natural history). In fact, “induction is an instrument of proof more evident, more familiar to the senses, and common to the many” (τοῖς πολλοῖς κοινόν), but less compelling and less effective than the syllogism (Top., loc. cit.); it is not an easy operation, and requires knowledge of the characteristics of resemblance among singular things (ibid., chap. 16); the person who uses induction properly “does not demonstrate, but nevertheless shows something” (Anal. post., II, 5); from the memory of the same fact repeated many times (πολλίκες) experience (ἐμπερίβο) arises; and this experience, in which the universal (“the one beyond the many”) is contained and made present, “is the principle of art and science” (ibid., chap. 15 [19]; cf. Anal. post., I, 31). Such and similar expressions cannot be understood of complete induction, which presupposes knowledge of all particulars and, as such, has demonstrative value substantially equal to that of the syllogism (cf. Fr. A. Trendelenburg, Elemente logicae Aristoteleae, 7th ed., Berlin 1892, pp. 111-12).
Although incomplete induction and its essential logical-epistemological significance were not unknown to Aristotle, he nevertheless did not arrive at a formulation of the conditions for its methodical use in science. And this, it may be said, substantially remained the doctrine of induction until Bacon. The Aristotelian definition is taken up by Boethius (De diff.
top.}, II: PL 64, 1183 a D). Alberto Magno, following Aristotle, distinguishes induction from the syllogism (I Top., tract. 3, cap. 4), while admitting the possibility of reducing it materially to the latter (II Pr. anal., tract. 7, cap. 4). Pietro Ispano inserts into Aristotle’s definition an element that subsequently became common in logic to indicate the fundamental requirement of incomplete induction: « Inducrio est a singularibus sufficienter enumeratis ad universale progressio » (Sum. logic., 5; but Aristotle himself, in Anal. pr., I, 30, 46 a 20, had already spoken of a “sufficient” observation of phenomena). S. Tommaso, 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 suggests that complete enumeration is to be understood in a virtual sense, insofar as even a partial consideration, if it refers to an essential and therefore common characteristic of the individual, is virtually equivalent to a complete enumeration. In S. Tommaso, incomplete induction, even in its broader sense as a process from the particular to the universal, is still indicated in the exposition and assimilation of the Aristotelian doctrine of ἐγεινάς (experimentum: V. 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 Scoto 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 merely epistemological but also scientific (in the modern sense of the term), together with the fundamental lines of its method, is clearly announced in F. Bacone. For science, « spes unica est in inductione vera » (Nov. org., I, af. 14). But the induction that is genuinely useful to science is not that which proceeds « per enumerationem simplicem » without taking contrary cases into account and therefore « gives precarious conclusions and exposes itself to the danger of contradictory instances » (ibid., af. 105): scientific induction must proceed « per eliminations ed esclusioni (per exclusiones ac reiectiones) and thus arrive at affirmative conclusions after as many negative ones as shall be necessary » (ibid.). Bacone reproaches the ancients for having too readily generalized from a few affirmative instances, subsequently forcing contrary cases into conclusions already established (ibid., af. 125). The true method of induction must unfold through three essential stages: a) tables of positive instances (tabulae praesentiae), which record the various cases in which the fact or phenomenon occurs; b) tables of negative instances (tabulae absentiae), which record 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 and recorded experience: ibid., I, 101-103), and, where possible, by means of the “prerogative instances,” or privileged cases, in which the phenomenon occurs under entirely new and singular circumstances, or does not appear under its usual circumstances, Baconian induction aimed at determining the “form,” that is, the internal structure of things, from which the various effects proceed (ibid., II, af. 1). Bacone did not, however, provide a philosophical theory of induction, nor did he discuss the problem of how the general conclusion is justified on the basis of an incomplete enumeration. The essential lines of the method of scientific induction (in addition to its shrewd application in experimental research) are also indicated by Galileo (cf. Opere, ed. naz., IV, Firenze 1895 p. 27): he explicitly states that only incomplete-
incomplete induction is of value in science: “induction, if it were to pass through all particulars, would be impossible or useless; impossible when the particulars were innumerable; and when they were innumerable, considering them all would be useless, or, better said, it would make concluding by induction null.” (op. cit., VI, p. 701). With Galileo, induction may be said, by availing itself of experiment, to become the classical 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 which resemble the former in certain definable respects” (System of logic., III, chap. 2, p. 333, § 1): induction properly speaking is neither “complete” induction nor the pure description, in simple terms, of a complex fact, as, for example, in the case of someone who, having sailed along a coastline and finally finding himself at the point of departure, “concludes” that it is an island. True induction is an extension, a generalization of experience (ibid., chap. 3, 354, § 1): such generalization, which is founded on the principle of the constancy of the order of nature, or, which is the same thing, 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 in addition have certainty that no contrary cases occur. To this end the four methods of experimental research are ordered, completing Bacon’s three “tables”: a) the method of agreement: if two or more cases of the phenomenon under examination have only one circumstance in common among many, that circumstance is the cause of the phenomenon; b) the method of difference: if a case in which the phenomenon occurs and a case in which it does not occur differ in only one 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 agreement and difference: if two or more cases in which the phenomenon occurs have only one circumstance in common, and two or more cases in which it does not occur have in common only the absence of that same circumstance, this circumstance is the cause (or part of the cause) of the phenomenon; c) the method of residues: after subtracting 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 appearing in the motions of Uranus and not referable to the causes already known led Le Verrier to suppose the existence (and also, approximately, the position and mass) of a new planet, subsequently confirmed by direct observation; d) the method of concomitant variations: every phenomenon that undergoes variations whenever another phenomenon varies in a determinate manner is either a cause or an effect of that phenomenon, or at least is connected with it by some causal fact: such, for example, is the expansion of bodies, always associated with heat and its variations, or the retardation of motion, which varies with variations in friction. The formulas of Stuart Mill (which, like Bacon’s “rules,” already find partial spontaneous application in common experience itself; and therefore it is purely polemical 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., chap. 8, p. 448 ff).
But, in addition to presenting itself as a problem of scientific method, induction had appeared, principally with the rise 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 twofold aspect: epistemological 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 certain particular cases. In the former
aspect, the problem of induction is the same as the problem of the origin of the concept, that is, of the derivation of necessity and universality in human knowledge. It is a fundamental tenet of empiricism that such necessity and universality cannot derive from experience. According to Hume, there is no certainty in experience except concerning what is given through it at a precise moment in time (Enquiring concerning human underst., sect. 4ᵃ, part 2ᵃ, Italian trans. by G. Prezzolini, Bari 1927, p. 39; cf. Treatise, I, part 3ᵃ, sect. 2 ff.). The assertion is taken up by Kant: “Experience teaches us indeed that a thing is made in this or that way, but not that it cannot be otherwise” (Kritik d. r. Vernunft, introd., § 2; Italian trans. by G. Gentile, Bari 1940, p. 39); the necessity of conceptual knowledge, which is an unquestioned 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: s. Tomm., In Post. anal., I, lect. 30) is instead, at least in a certain sense, essential to the Aristotelian-Thomistic system: from the repetition of sensory presentations of a given object, and from their persistence in memory (the experimentum), the singular datum becomes determined in its ontological unity, and hence, through the intuitive-abstracting power of thought, emerges in the intellectual act as a universal value. This doctrine corresponds profoundly to the actual condition of human knowing. It is nevertheless further clarified by the virtual objectivity attributed, in Thomistic thought itself, to the agent intellect. (cf. Sum. Theol., 1ᵃ, q. 84, a. 5; C. Gent., I, III, chap. 47; De ver., q. 11, a. 1,3). Indeed, one must say that, although the sensible datum is in itself capable of grounding a general representation, it could not nevertheless be posited as absolute necessity of value unless the intellect itself were immanent in a relation to a higher objective necessity. It is thought which, as a transcendental relation to being—which in itself is positivity and necessity—contains the roots of universality and absolute necessity: that universality which may be called foundational and which is poured, according to the finite participations and determinations of being, into the concrete data of experience. Induction thus presents itself 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 must be established—the derivation of the universal is not purely empirical but also transcendental, insofar as thought itself, as a relation to being, is already in itself in a metaphysical situation of virtual transcendence in comparison with the individual moments of experience.
The logical problem of induction had already been clearly formulated by ancient skepticism in the dilemma: either induction claims to conclude from all particulars, and this is an impossible undertaking, since the particulars are infinite, or only from some of them, and then it will be unreliable “since it is possible that one of the particulars omitted in the induction may conflict with the universal” (Pyrrh. hypoth., II, 15). Jevons, Goblot, Bolzano, Hamilton, Windelband, as well as the entire empiricist school, on the basis of this dilemma recognize in incomplete induction a purely probable value. It seems first of all necessary to affirm that induction not only must not be reduced to the syllogism (a reduction first attempted by Wolff: Logica, part 1ᵃ,
sez. 4ᵃ, cap. 6, e poi ammessa da non pochi altri), ma nemmeno va intesa come processo logico formale analogo al sillogismo. Anche come «argomentazione» l'i. è emergenza dal particolare all'universale. La sua antinomia logica sta appunto nel passaggio da alcuni casi a una conclusione universale: non si è fatta, né si può fare esperienza su tutti i singoli corpi esistenti, e tuttavia si accetta la formula generale: tutti i corpi sono pesanti. Come puro processo di inferenza formale, l'i. non sarebbe legittima. Da questa difficoltà è sorto il problema del «fondamento» dell'i. Tale fondamento non può essere di ordine logico, ma è, necessariamente, di ordine o empirico, o metafisico, o è sintesi di entrambi. La fondazione dell'i. nella pura esperienza è propria appunto dell'empirismo: secondo Hume l'estensione dell'esperienza è fatta in base a una abitudine psicologica per cui ci si aspetta in futuro ciò che si è sperimentato nel passato; per Stuart Mill il principio che regge l'i.: «si dà una causalità costante nella natura» è esso stesso derivato dall'esperienza e quindi è frutto di precedenti i. (op. cit., III, cap. 3, p. 355, § 1). La fondazione metafisica è propria invece del razionalismo e dell'idealismo: il razionalismo infatti trasferisce nella natura l'ordine e la necessità del pensiero; e nell'idealismo il problema stesso dell'i. è dissolto in quanto la natura è posta come funzione dello spirito e del suo divenire. Mentre la fondazione empirica termina alla negazione della legittimità del processo induttivo, la derivazione metafisica è una posizione sistematica non avvalorata dalla fenomenologia del conoscere umano. Occorre ritenere che l'i. non può essere fondata su un puro principio razionale, p. es.: «le stesse cause producono gli stessi effetti»; è infatti indispensabile sapere che esistono effettivamente delle stesse cause, o delle cause permanenti. È quindi esatto che l'i. si basa sul principio, derivato dall'esperienza, che l'ordine della natura (degli esseri) è costante (già esplicito in s. Tommaso: Sum. Theol., 1ᵃ, q. 19, a. 4, c.; q. 41, a. 2, c.). Ma la stessa ampiezza dell'esperienza che attesta la costanza dell'essere e dell'attività del mondo si avvalora dalla considerazione metafisica che riconosce all'essere (ad ogni essere) una determinazione ontologica essenziale e quindi anche un principio immanente di determinazione operativa: «eo modo aliquid operatur quo est» (Sum. Theol., q. 75, a. 2). Il principio di uniformità della natura ammette pertanto una giustificazione metafisica derivata dal principio della determinazione dell'essere. E tale principio (che in ultima istanza si rapporta a un Valore razionale assoluto, principio di tutto l'essere) è presupposto generale della possibilità della conoscenza, dell'azione e della storia umana: l'indeterminazione metafisica della realtà disolverebbe ogni fermezza nell'essere e quindi ogni possibilità di verità e di azione.
Con questo non è, peraltro, negato il margine di probabilità che va congiunto con il processo induttivo nella conoscenza della natura. Non solo nell'ordine della realtà umana (la storia) l'i. presenta una radicale incertezza per il principio della libertà in cui l'azione si pone, alla sua sorgente, come possibilità non definita, ma anche nella natura l'i. si fa tanto più dialettica quanto maggiormente attinge ai fenomeni più complessi e profondi dell'essere. Dove, come nel mondo infratomico, la vasta interazione delle cause cosmiche può ad ogni istante modificare l'effetto, la funzione dell'i. si contrae a una pura determinazione di probabilità statistica più o meno
ampia. E anzi, poiché il processo operativo degli esseri è in realtà solidale con tutto il dinamismo dell'universo, può in generale ammettersi con il Boutroux (L'idea di legge naturale, trad. II. di A. Pasa, Verona 1940, pp. 252-53) una inadeguazione delle leggi fisiche a stringere in rigidi schemi la realtà e il tempo. Ciò non ha importanza né per la scienza né per l'azione, cui il processo induttivo conferisce una solidità di affermazioni ed esplicazioni che trova nei grandiosi successi della scienza stessa la più chiara e permanente conferma.