9 =:= = 8 ■ {— — —1 THILLY The Process of Inductive Inference i^^H M Volume II JNumbee 3 THE UNIVERSITY OF MISSOURI STUDIES EDITED BY FRANK THILLY Professor of Philosophy THE PROCESS OF INDUCTIVE INFERENCE BY THE EDITOR PUBLISHED BY THE UNIVERSITY OF MISSOURI April, 1904 PRICE, :i5 CENTS 3 THE PROCESS OF INDUCTIVE INFERENCE J Volume II Kumbek 3 THE UNIVERSITY OF MISSOURI STUDIES EDITED BY FRANK THILLY ;/ Professor of Philosophy THE PROCESS OF INDUCTIVE INFERENCE BY THE EDITOR PUBLISHED BY THE UNIVERSITY OF MISSOURI April, 1904 PRICE, 35 CENTS U^ V Copyright, 1904, by THE UNIVERSITY OF MISSOURI PRESS OF E. W. STEPHENS COLUMBIA, MO. TABLE OF CONTENTS Part I. Historical Survey Page §1. Aristotle i §2. Francis Bacon 3 §3. John Stuart Mill 6 §4. W. Stanley Jevons 13 §5. Christoph Sigwart 16 §6. Hermann Lotze 21 Part II. The Theory of Induction §1. Introduction 27 §2. The Nature of Induction 29 §3. The Validity of the Process 36 THE PROCESS OF INDUCTIVE INFERENCE PART I Historical Survey § I. Aristotle distinguishes between reasonings which proceed from first principles and reasonings which lead up to first principles.! The first kind of reasoning is the syllogism, the second induction (^iTzayajy-q') , Induction is a passage from particulars to universals, "as if the pilot skilled in his art is the best, so also is the charioteer, generally the skillful is the most excellent about each thing." - ^Nicomachean Ethics, Bk. I, chap ii. hXi.o, Posterior Analytics, Bk. I, chap, xviii, English translation in Bohn's library: "It is clear also that if any sense be deficient, a certain science must be also deficient, which we can not possess, since we learn either by induction or by dem- onstration. Now demonstration is from universals, but induction from particulars." See also, Bk. I, chap, i; Bk. II, chap. xix. ^The Topics, Bk. I, chap, xii, English translation Bohn's library: "These things then being determined, we must distinguish how many species of dialectic arguments there are. Now one is induction, but the other syllogism, and what indeed syllogism is, has been declared before, but induction is a progression from singulars to universals, as if the pilot skilled in his art is the best, so also is the charioteer, and generally the skillful is the most excellent about each thing." 149] I 2 UNIVERSITY OF MISSOURI STUDIES [^50 But what right have we to pass from particulars to uni- versals? Aristotle seems to justify the procedure in two ways. The particular, he helieves, arouses in the mind the universal. We perceive the particular, c. g., Callias, but perception includes the universal, e. g., man.^ That is, the perception of the particu- lar arouses in our minds the idea of the universal, of the all. In the words of Thomas Taylor, a translator and commentator of Aristotle : ''Induction is so far subservient to the acquisi- tions of science, as it evocates into energy in the soul, those uni- versal from which demonstration consists. For the universal, which is the proper object of science, is not derived from par- ticulars, since these are infinite, and every induction of them must be limited to a finite number. Hence the perception of the all and the every is only excited, and not produced, by induc- tion." ^ In other words the category of totality seems to be a function of the mind that is aroused or excited by the perception of particular things. The mind makes the leap from the part to the whole. In another place Aristotle tries to prove the inductive propo- sition deductively, that is, to justify the inductive leap to logic. He bases himself on the thought that we can enumerate all the species of the genus which occurs in our conclusion. Thus we conclude that whatever is devoid of bile is longlived, because every man, horse, mule, etc., is longlived, and what is devoid of bile is man, horse, mule, etc.^ That is, only so-called perfect induction is capable of proof.^ The thought implied by Aristotle is that what is true of the ^Posterior Analytics, Bk. II, chap xix. ^Quoted in the English translation of the Orffanoti, Bohn's library. ''Prior Analytics, Bk. II, chap, xxiii, 6See also, Whately, Logic; Apelt, Theorie der Induction; Jevons,. Lessons in Logic; Principles of Science ; Veitch, Institutes of Logic. 151] THE PROCESS OF INDUCTIVE INFERENCE 3 species is true of the genus, that the particular is an expression of the universal, that there is uniformity in the world. This is really the fundamental idea of his whole system of philosophy. The particular is an expression of the universal, of the form or idea, hence what is true of the particular must be true of the universal.''' § 2. Bacon, too, regards induction as a passage from the particular to the general. But he suggests a more careful method of procedure. "For the induction which proceeds by simple enumeration is childish," he says; "its conclusions are precarious, and exposed to peril from a contradictory instance; and it generally decides on too small a number of facts, and on those only which are at hand. But the induction which is to be available for the discovery and demonstration of sciences and arts, must analyze nature by proper rejections and exclusions; and then after a sufficient number of negatives, come to a con- clusion on the affirmative instances." ^ He also condemns hasty induction, and finds fault with the ancients for having employed it. "From a few examples and particulars (with the addition of common notions and perhaps of some portion of the received opinions which have been most popular), they flew at once to the most general conclusions, or first principles of science; taking the truth of these as fixed and immovable, they proceeded by means of intermediate propositions to educe and prove from them the inferior conclusions : and out of these they framed the art." ^ "There are and can be only two ways of searching into and discovering truth. The one flies from the senses and par- ^See Hegel's Logih. See also, Posterior Analytics, Bk. I, chap. xxxi. ^Novum Organum, Bk. I, cv. Quotations taken from the Sped- ding, Ellis, and Heath edition of Bacon's Works. ^Nov. Org., Bk. I, cxxv. 4 UNIVERSITY OF MISSOURI STUDIES [152 liculars to the most general axioms, and from these principles, tlie truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." ^^ Bacon's whole aim is to devise a method that will yield us general propositions of greater certainty than those obtained by the method of simple enumeration. The problem is to find the form or essence of things, that on which the phenomenon under consideration depends. The form of anything is inherent in each individual instance in which the thing itself is inherent.^ ^ "The investigation of Forms proceeds thus; a nature being given, we must first of all have a muster or presentation before the understanding of all known instances which agree in the same nature, though in substance the most unlike." ^^ Such a ^^Nov. Org., Bk. I, xix. "^^Nov. Org., Bk. II, xx; Bk. II, ii: "For though in nature nothing exists besides individual bodies, performing pure individual acts according to a fixed law, yet in philosophy this very law, and the investigation, discovery, and explanation of it, is the foundation as well of knowledge as of operation. And it is this law, with its clauses, that I mean when I speak of Fortns; a name which I the rather adopt be- cause it has grown into use and become familiar." Bk. II, iv: "For the Form of a nature is such, that given the Form the nature infal- libly follows. Therefore it is always present when the nature is present, and universally implies it, and is constantly inherent in it," etc. Bk. II, xvii: "For when I speak of Forms, I mean nothing more than those laws and determinations of absolute actuality, which govern and constitute any simple nature, as heat, light, weight, in every kind of matter and subject that is susceptible of them." See also, Bk. II, iii, v, xiii, xv, xvi. ^Nov. Org., Bk. II, xi. ^53] ^^^ PROCESS OF INDUCTIVE INFERENCE 5 list is called the table of essence or presence. "Secondly, we must make a presentation to the understanding of instances in which the given nature is wanting; because the Form, as stated above, ought no less to be absent when the given nature is absent, than present when it is present." ^^ This is the table of deviation or of absence in proximity. "Thirdly, we must make a presentation to the understanding of instances in which the nature under inquiry is found in different degrees, more or less." ^^ This is the table of degrees or of comparison. When all this has been done induction itself is brought into action. "The first work therefore of true induction (as far as regards the discovery of Forms) is the rejection or exclusion of the several natures which are not found in some instance where the given nature is present, or are found in some instance where the given nature is absent, or are found to increase in some instance when the given nature decreases, or to decrease when the given nature increases. Then indeed after the rejection and exclusion has been duly made, there will remain at the bottom, all light opinions vanishing into smoke, a Form affirmative, solid and true and well defined." i» It is apparently the business of the investigator to collect a number of diiYerent cases in which the phenomenon to be studied is present. Then he must collect cases in which it is absent, and also cases in which it varies. That which is always present when the phenomenon is present, and absent when the phenomenon is absent, and which varies with the phenomenon, is the sought-for form. It is clear. Bacon unconsciously bases himself upon the prin- ciple that there is a necessary connection between things, and ^Wov. Org-., Bk. II, xii. ^*JVov. Org., Bk. II, xiii. '^^Nov. Org., Bk. II, xvi. 6 UNIVERSITV OP^ MISSOURI STUDIES [154 that it is the business of true induction to find this connection. He reallv assumes that there is uniformity in nature, that things are so connected that when one appears the other will appear also. This principle he does not attempt to justify; indeed, as we have already said, he is not conscious of it at all. Since, however, he rejects the theory of innate ideas, and accepts empiricism, it is safe to say that he would have explained this principle as a product of experience after the fashion of Mill. § 3. John Stuart Mill ^^ defines induction as "that opera- tion of the mind, by which we infer that which we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain assignable respects. In other words, induction is the process by which we conclude that what is true of certain individuals of a class is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times." 1" So-called perfect induction is not induction at all. Nor are those cases in mathematics induction, in which the conclusion is a mere summing up of what was asserted in the various propositions from which it is drawn. Nor is it induc- tion to piece detached fragments together (the so-called colliga- tion of facts). ^^ Mill finds the ground of induction- in the principle of the uniformity of nature.^^ "The proposition that the course of nature is uniform, is the fundamental principle, or general axiom, of induction. But it would be a great error to offer this large generalization as any explanation of the inductive process. It is itself an instance of ^M Sysiem of Logic, Bk. III. ^''Log-ic, Bk. Ill, chap, ii, §1. See also, chap, i, §2. ^^Logic, Bk. Ill, chap, ii, §§3,4. ^^Logic, Bk. Ill, chap. iii. 155] THE PROCESS OF INDUCTIVE INFERENCE 7 induction, and induction by no means of the most obvious kind. Far from being the first induction we make, it is one of the last, or at all events one of those which are latest in attaining strict philosophical accuracy." This great generalization, however, is itself founded on prior generalizations. "The obscurer laws of nature were dis- covered by means of it, but the more obvious ones must have been understood and assented to as general truths before it was ever heard of. We should never have thought of affirming that all phenomena take place according to general laws, if we had not first arrived, in the case of a great multitude of phenomena, at some knowledge of the laws themselves ; which could be done no otherwise than by induction." ^o This principle is our warrant for all the other inductions in the sense in which the general propositions which we place at the head of our reasonings ever really contribute to their validity. The major premise of a syl- logism does not prove the conclusion, but is a necessary condition of its being proved. Every induction may be thrown into the form of a syllogism, by supplying a major premise. "If this be actually done, the principle which we are now considering, that of the uniformity of the course of nature, will appear as the ul- timate major premise of all inductions, and will, therefore stand to all inductions in the relation in which * * * the major propo- sition of a syllogism always stands to the conclusion; not contributing at all to prove it, but being a necessary con- dition of its being proved; since no conclusion is proved, for which there can not be found a true major premise." "The real proof that what is true of John, Peter, etc., is true of all mankind, can only be, that a different suppos- ition would be inconsistent with the uniformity which we know to exist in the course of nature. Whether there would be this inconsistency or not, may be a matter of long and "^Logic, Bk. Ill, chap, iii, §1. S UNIVERSITY OF MISSOURI STUDIES [156 delicate inquiry; but unless there would, we have no sufficient ground for the major of the inductive syllogism. It hence ap- pears, that if we throw the whole course of any inductive argu- ment into a series of syllogisms, we shall arrive by more or fewer steps at the ultimate syllogism, which will have for its major premise the principle, or axiom, of the uniformity of the course of nature." 21 The validity of all the Inductive Methods (Agreement, Dif- ference, Joint Method, etc.) depends on the assumption that every event, or the beginning of every phenomenon, must have some cause ; some antecedent, on the existence of which it is in- varibly and unconditionally consequent. 22 This assumption is itself an instance of induction. We arrive at this universal law, by generalization from many laws of inferior generality. As, however, all rigorous processes of induction presuppose the gen- eral uniformity, our knowledge of the particular uniformities from which it was first inferred was not, of course, derived from rigorous induction, but by the loose and uncertain mode of in- duction per enumerationem simplicem; and the law of universal causation, being collected from results so obtained, can not it- self rest on any better foundation. Induction per enumera- tionem simplicem is, however, not only not necessarily an illicit logical process, but is in reality the only kind of induction pos- "^^Logic, Bk. Ill, chap, iii, §1. ^Logic, Bk. Ill, chap, xxi, §i. See also, chap, v, §2: "The Law of Causation, the recognition of which is the main pillar of inductive sci- ence, is but the familiar truth, that invariability of succession is found by observation to obtain between every fact in nature and some other fact which has preceded it." Section 6: But, "invariable sequence is not causation, unless the sequence, besides being invariable, is uncon- ditional." "We may define, therefore, the cause of a phenomenon, to be the antecedent, or the concurrence of antecedents, on which it is in- variably and iinconditionally consequent." 157] THE PROCESS OF INDUCTIVE INFERENCE 9 sible.23 The precariousness of this process is in inverse ratio to the largeness of the generalization. It is for this reason that the most universal class of truths, the law of causation, for in- stance, and the principles of number and of geometry, are duly and satisfactorily proved by that method alone, nor are they susceptible of any other proof.^^ The assertion, that our induc- tive processes assume the law of causation, while the law of caus- ation is itself a case of induction, is a paradox, only on the old theory of reasoning, which supposes the universal truth, or ma- jor premise, in a ratiocination, to be the real proof of the partic- ular truths which are ostensibly inferred from it. According to Mill's doctrine, however, "the major premise is not the proof of the conclusion, but is itself proved, along with the conclusion from the same evidence. 'All men are mortal' is not the proof that Lord Palmerston is mortal ; but our past experience of mor- tality authorizes us to infer both the general truth and the partic- ular fact, and the one with exactly the same degree of assurance as the other. The mortality of Lord Palmerston is not an in- ference from the mortality of all men, but from the experience which proves the mortality of all men ; and is a correct inference from experience, if that general truth is so too. This relation ^Log-ic, Bk. Ill, chap, xxi, §2: "Is there not then an inconsistency in contrasting the looseness of one method with the rigidity of another, when that other is indebted to the looser method for its own foundation ? The inconsistency, however, is only apparent. Assuredly, if induction by simple enumeration were an invalid process, no process founded on it could be valid; just as no reliance could be placed on telescopes, if we could not trust our eyes. But though a valid process, it is a fallible one, and fallible in very different degrees; if, therefore, we can substi- tute for the more fallible forms of the process, an operation founded on the same process in a less fallible form, we shall have effected a very material imorovement. And this is what scientific induction does." "^Logic, Bk. Ill, chap, xxi, §§2,4. lO UNIVERSITY OF MISSOURI STUDIES [158 between our general beliefs and their particular applications holds equally true in the more comprehensive case which we are now discussing. Any new fact of causation inferred by induc- tion is rightly inferred, if no other objection can be made to the inference than can be made to the general truth that every event has a cause. The utmost certainty which can be given to a con- clusion arrived at in the way of inference, stops at this point. When we have ascertained that the particular conclusion must stand or fall with the general uniformity of the laws of nature — that it is liable to no doubt except the doubt whether every event has a cause — we have done all that can be done for it." In mat- ters of evidence as in all other human things, we neither require, nor can attain, the absolute. "Whatever has been found true in innumerable instances, and never found to be false after due ex- amination in many, we are safe in acting on as universal provis- ionally, until an undoubted exception appears; provided the na- ture of the case be such that a real exception could scarcely have escaped notice. When every phenomenon that we ever knew sufficiently well to be able to answer the question, had a cause on which it was invariably consequent, it was more rational to sup- pose that our inability to assign the causes of other phenomena arose from our ignorance, than that there were phenomena which w-ere uncaused, and which happened to be exactly those which we had hitherto had no sufficient opportunity of studying." ^^ "^Logic, Bk. Ill, chap, xxi, §4. Mill goes on to say: "It must, at the same time, be remarked, that the reasons for this reliance do not hold in circumstances unknown to us, and beyond the possible range of our experience. In distant parts of the stellar regions, where the phe- nomena may be entirely unlike those with which we are acquainted, it would be folly to affirm confidently that this general law prevails, any more than those special ones which we have found to hold universally on our own planet. The uniformity in the succession of events, otherwise called the law of causation, must be received not as a law of the universe, 159] THE PROCESS OF INDUCTIVE INFERENCE II Mill too, we see, grounds induction on the proposition that things are connected in such a way that when one appears the other will appear also. This proposition itself is a product of experience, the result of loose induction.^^ But we are satisfied with this loose induction on account of the great number of par- ticular cases observed by us in which we experience causality. Moreover, the particular case is not proved by the general prop- osition at all ; the general proposition is proved by the particular cases. Hence, everything is ultimately reduced by Mill to induc- tion; induction is the only possible process of inference; deduc- tion is not inference at all, but a mere deciphering of our notes. "All inference is from particulars to particulars; general propositions are merely registers of such inferences already made and short formulae for making more. The major premise of a syllogism, consequently, is a formula of this description; and but of that portion of it only which is within the range of our means of sure observation, with a reasonable degree of extension to adjacent cases. To extend it further is to make a supposition without evidence, and to which in the absence of any ground from experience for estimat- ing its degree of probability, it would be idle to attempt to assign any." ^Logic, Bk. Ill, chap, xxi, §1 : Some metaphysicians affirm "that the universality of causation is a truth which we can not help believing; that the belief in it is an instinct, one of the laws of our believing fac- ulty." But "belief is not truth, and does not dispense with the neces- sity of truth." "Now a mere disposition to believe, even if supposed instinctive, is no guarantee for the truth of the thing believed. If, in- deed, the belief ever amounted to an irresistible necessity, there would then be no use in appealing from it, because there would be no possibil- ity of uttering it. But even then the truth of the belief would not fol- low; it would only follow that mankind were under a permanent neces- sity of believing what might possiblj' not be true. * * * 3yt j^ fact there is no such permanent necessity. There is not one of these sup- posed instinctive beliefs which is really inevitable. * * * It is not true, as a matter of fact, that mankind have always believed that all the succession of events were uniform and according to fixed laws. * * * 12 UNIVERSITY OF MISSOURI STUDIES [l6o tlio conclusion is not an inference drawn from the formula, but an inference drawn according to the formula; the real logical an- tecedent or premise being the particular facts from which the general proposition was collected by induction. Those facts, and the individual instances which supplied them, may have been forgotten; but a record remains, not indeed descriptive of the facts themselves, but showing how those cases may be distin- guished, respecting which, the facts, when known, were consid- ered to warrant a given inference. According to the indications of this record we draw our conclusion: which is, to all intents and purposes, a conclusion from the forgotten facts. For this it is essential that we should read the records correctly : and the rules of the syllogism are a set of precautions to insure our doing so." "The proposition. All men are mortal (for instance) shows that we have had experience from which we thought it followed Even now a full half of the philosophical world, including the very same metaphysicians who contend most for the instinctive character of the belief in uniformity, consider one important class of phenomena, voli- tions, to be an exception to the uniformity, and not governed by a fixed law." Chap, xxi, §3, Note: "Dr. McCosh maintains that the uniform- ity of the course of nature is a different thing from the law of causation, and while he allows that the former is only proved by a long continu- ance of experience, and that it is not inconceivable nor necessarily in- credible that there may be worlds in which it does not prevail, he con- siders the law of causation to be known intuitively. There is, however, no other uniformity in the events of nature than that which arises from the law of causation; so long, therefore, as there remained any doubt that the course of nature was uniform throughout, at least when not modified by the intervention of a new (supernatural) cause, a doubt was necessarily implied, not indeed of the reality of causation, but of its universality. If the uniformity of the course of nature has any excep- tions — if any events succeed one another without fixed laws — to the ex- tent the law of causation fails; there are events which do not depend on causes." l6l] THE PROCESS OF INDUCTIVE INFERENCE I3 that the attributes connoted by the term man, are a mark of mor- tality. But when we conclude that the Duke of Wellington is mortal, we do not infer this from the memorandum, but from the former experience. All that we infer from the memorandum is our own previous belief, (or that of those who transmitted to us the proposition), concerning the inferences which that former experience would warrant." 2" § 4. According to Jevons ^s it cannot be said that the in- ductive process is of greater importance than the deductive pro- cess, because the latter process is absolutely essential to the former. Each is the complement and counterpart of the other. Induction is in fact the inverse operation of deduction, and cannot be conceived to exist without the corresponding operation. In deduction we deduce from certain conditions, laws, or identities governing the combinations of qualities, the nature of the com- binations agreeing with those conditions. Our work is to unfold what is contained in any statements, and the process is one of synthesis. In induction all is inverted. The truths to be ascer- tained are more general than the data from which they are drawn. The process by which they are reached is analytical, and consists in separating the complex combinations in which natural phenomena are presented to us, and determining the re- lations of separate qualities. Given events obeying certain unknown laws, we have to dis- cover the laws obeyed. Instead of the comparatively easy task of finding what effects will follow from a given law, the effects are now given and the law is required. We have to interpret the will by which the conditions of creation were laid down.^* "^Logic, Bk. II, chap, iii, §4. "^Elementary Lessons ift Logic; Principles of Science. ^Principles of Scie?ice, chap. vii. I . UNIVEKSITV OK MISSOURI STUDIES [162 Being ill possession of certain particular facts or events ex- pressed in propositions, we imagine some more general propo- sition expressing the existence of a law or cause ; and, deducing the particular results of that supposed general proposition, we observe whether they agree with the facts in question. Hypoth- esis is thus always employed consciously or unconsciously. Thus there are but three steps in the process of induction : ( i ) Framing some hypothesis as to the character of the general law. (2) Deducing consequences from that law. (3) Observing whetlier the consequences agree with the particular facts under consideration.^" ^Principles of Science, chap. xii. Here Jevons seems to identify in duction with scientific method in general, which is really a combination of induction and deduction. But there are passages in his Principles of Science, and particularly in his elementary work, which do not appear to be consistent with this interpretation. In chapter xi of the first-named book, for example, he says: "We must carefully avoid confusing to- gether inductive investigations which terminate in the establishment of general laws, and those which seem to lead to the knowledge of particu- lar events. That process only can be called induction which gives gen- eral laws, and it is by the subsequent employment of deduction that we anticipate particular events. If the observation of a number of cases shows that alloys of metals fuse at lower temperatures than their con- stituent metals, I may with more or less probability draw a general in- ference to that effect, and may thence deductively ascertain the proba- bility that the next alloy examined will fuse at a lower temperature than its constituents." "I hold that in all cases of inductive inference we must invent hypotheses, until we fall upon some hypothesis which yields deductive results in accordance with experience. Such accordance ren- ders the chosen hypothesis more or less probable, and we may then deduce, with some degree of likelihood, the nature of our future ex- perience, on the assumption that no arbitrary change takes place in the conditions of nature." See also, Lessons in Logic, Lesson xxx, pp. 258f, quoted in the second part of this paper, page 35. [63 J THE PROCESS OF INDUCTIVE INFERENCE 1 5 But the results of imperfect induction are never more than probable. "It is a question of profound difificulty on what grounds we are warranted in inferring the future from the pres- ent, or the nature of undiscovered objects from those which we have examined with our senses. We pass from Perfect to Im- perfect Induction when once we allow our conclusion to apply, at all events apparently, beyond the data on which it was founded. In making such a step we seem to gain a net addition to our knowledge; for we learn the nature of what was un- known. We reap where we have never sown. We appear to possess the divine power of creating knowledge, and reaching with out mental arms far beyond the sphere of our own observa- tion. Of imperfect induction itself, I venture to assert that it never makes any real addition to our knowledge, in the meaning of the expression sometimes accepted. As in other cases of in- ference, it merely unfolds the information contained in past ob- servations; it merely renders explicit what was impHcit in pre- vious experience. It transmutes, but certainly does not create knowledge." The results of imperfect induction, however well authenticated and verified, are never more than probable. "We never can be sure that the future will be as the present. We hang ever upon the will of the Creator : and it is only as far as He has created two things alike, or maintains the framework of the world unchanged from moment to moment, that our most careful inferences can be fulfilled. All predictions, all inferences which reach beyond their data, are purely hypothetical, and pro- ceed on the assumption that new events will conform to the con- ditions detected in our observation of past events. No experience of finite duration can give an exhaustive knowledge of the forces which are in ooeration. There is thus a double uncertainty ; even supposing the Universe as a whole to proceed unchanged, we do not really know the Universe as a whole. We know only a point I 6 UNIVERSITY OF MISSOURI STUDIES [164 in its infinite extent, and a moment in its infinite duration. We cannot be sure, then, that our observations have not escaped some fact, which will cause the future to be apparently different from the past ; nor can we be sure that the future really will be the outcome of the past. We proceed, then, in all our inferences to the unexamined objects and times on the assumptions which are always uncertain : ( i ) that our past observation gives us a complete knowledge of what exists; (2) that the conditions of things which did exist will continue to be the conditions which will exist." 31 Jevons identifies induction with specific method in general. But he also bases it on the principle that there are uniform con- nections in nature, and regards this principle as a product of ex- perience. §5. According to Sigwart, ^^ ^i^g logical justification of the inductive process, that is, the attainment of universal prop- ^^Prtttciples of Science, chap. vii. Compare chap, xi: "We can only argue from the past to the future, on the general principle set forth in this work, that what is true of a thing will be true of the like. So far then as one object or event differs from another, all inference is impos- sible, particulars as particulars can no more make an inference than grains of sand can make a rope. We must always rise to something which is general or same in the cases, and assuming that sameness to be extended to new cases we learn their nature." "There is no reason in the nature of things, so far as known to us, why yellow color, ductility, high specific gravity, and incorrodibility, should always be associated together, and in other cases, if not in this, men's expectations have been deceived. Our inferences, therefore, always retain more or less of a hypothetical character, and are so far open to doubt. Only in so far as our induction approximates to the character of perfect induction, does it approximate to certainty." See also, on this point, Hamilton, Dis- cussions; VtSXch., Institutes of Logic; Benno Erdmann, Z^o^//^. ^Logic, vol. II, chap. v. The quotations are taken from Helen Dendy's English translation. 165] THE PROCESS OF INDUCTIVE INFERENCE 1*J ositions from particular judgments of perception, rests upon the fact that it is an inevitable postulate of our effort after knowl- edge that the given is necessary, and can be known as proceeding from its grounds according to universal laws.^s Every inductive inference contains a universal principle. If it is to be an infer- ence and not merely an association of merely subjective validity, the transition from the empirically universal judgment all known A's are B's to the unconditionally universal all that is A is B, can only be made by means of a universal major premise; and that only upon condition of this being true are we justified in in- ferring from the particular known A's to the still unknown A's. But then the universal major premise cannot be obtained simply by means of a summation of facts, for this by itself can yield no more than it says, that is, a certain number of cases A was B and as a pure matter of fact contains no reason for passing be- yond these A's to other A's. The universal major premise must have some other origin than in previously observed facts, and our right to make use of it must have some other ground than the narration of cases which have been observed so far.^^ We shall never find in the given a sufficient ground to yield us the conviction that because so many perceived A's have without exception the attribute B, or because event B has followed so many times upon circum- stance A, therefore it must necessarily be so. ''The innumerable exceptions by which many attempted generalizations of this kind are met do as a matter of fact, refute, even to superfluity, the assumption that a universal law can be derived with infallible certainty from similar cases, however great their number. For a long time it is for most Europeans ^Logic, vol. II, chap, v, §93. ^Logic, vol. II, chap, v, §93, 8. (-2) iS UNIVERSITY OF MISSOURI STUDIES [l66 a good induction that all men are white; it is a good induction that all men have five fingers; for thousands of years it was a srood induction that all metals are heavier than water." ^s The universal proposition by which we are guided in our mental elaboration of the particular propositions given by per- ception is that the given is necessary; and since necessity sig- nifies for us the same as the invariable and universal connection of a ground with a consequence, we get as the postulate of our search for knowledge that every particular perception is an in- stance of a general rule, a conclusion which follows from subor- dination to a universal major premise. This presupposition has reference both to the co-existence of the permanent attributes which we find in a particular object, and to the connection be- tween the changes in the same or different objects; the concepts of things in which we synthesize certain perceptible attributes at first as subjective images, have objective significance just in so far as the connection is necessary, and there is a general rule according to which these attributes exist together in the partic- ular case; the particular event is necessary when it happens ac- cording to a rule which prescribes that under certain conditions a certain change will take place. Hence it follows that we are forced by the nature of knowledge to apprehend all particular objects and facts with which observation presents us as instances in which a universal rule expresses itself ; the problem of induc- tion is to find this universal rule, and to formulate it in such a way that the given shall everywhere correspond to it. "In other words, we are dealing with a process of reduction, which constructs the premises from which the particular percep- tion follows with syllogistic necessity, whether it expresses the co-existence of attributes in a thing, or a change, or the succes- sion of different changes ; and the problem is to determine these "^Logtc, vol. II, chap, v, §93, 11. 167] THE PROCESS OF INDUCTIVE INFERENCE I9 major premises in such a way that they may be in harmony with all the perceptions known to us." ^^ From this it follows in the first place that the propositions gained by induction are neyer proved in the strict sense, but from a logical point of view are only hypotheses; further that the fundamental principles even of induction are based upon the rules of the syllogism, by which it is determined whether the premises assumed lead necessarily to the conclusion. If a per- ception does not agree with what is at first the assumed hypothe- sis of a universal proposition, then, assuming the process of in- ference to be correct, one of the premises is necessarily false; but the most comprehensive agreement of the hypothesis with the facts can never show it to be necessarily true, it can at most make it probable. When, however, any one hypothesis breaks down because the inferred universal proposition is opposed by exceptions, our universal presupposition that the given is neces- sary is not on that account destroyed; it is only the special as- sumption with reference to the necessary connection between a special ground and a special consequence which fails.^''' Induction is a process of finding universal major proposi- tions when the conclusions are given, — a reduction and therefore the reverse process of the syllogistic deduction from given major propositions. All induction consists in framing hypothetically universal propositions and comparing their consequences with the given. ^* Sigwart, like Jevons. identifies induction with scientific method in general; it consists of forming hypotheses, deducing their consequences, and verifying them. The first step in the process, the passage from the particular to the universal, Sig- ^Logic, vol. II, chap, v, §93, 14. ^'Logic, vol. II, chap, v, §93, 15. ^LogiC) vol. II, chap, v, §93, 17. Note 2. 20 UNIVERSITY OF MISSOURI STUDIES [l68 wart bases on the principle that the given is necessary : there is :i necessary connection between things, they are related as ground and consequent so that when one appears the other must appear also. "Every particular perception is an instance of a general rule.'' This principle is not. however, as with Mill and Jevons, derived from experience, but is a postulate of our think- ing.3® 3»Somewhat similar to Sigwart's conception is that of Ueberweg. According to him, the incomplete induction would, according to the rules of the syllogism, justify only a particular conclusion. The valid- ity of the generalization of the conclusion rests partly upon the general assumption of a lawful causal nexus in the objects of our knowledge, partly upon the special assumption that in the given case some lawful causal nexus exists between the subject and predicate of the conclusion. The import of induction as a means of extending our knowledge rests upon the same relation to the real uniformity (according to the princi- ple of ground) on which the possibility of the syllogism as a form of knowledge is founded. The existence of the causal nexus precedes our inductions. Our knowledge of the causal nexus in a particular case presupposes a collection of facts, and our knowledge of the causal nexus in general form follows upon many special inductions. This knowledge is the condition (not of these inductions, in which case we should have a circulus vitiosus, but only) of the logical justification of these induc- tions. We at first generalize only according to psychical laws of asso- ciation ; our generalizations have logical justification in so far as they invariably coincide with the objective causal nexus. System der Lo^ik, §§I27ff. The same idea is very clearly brought out by Creighton in his Intro- ductory Logic, §88. In induction, he tells us, we begin with particular phe- nomena, and try to discover from them the law or principle which unites them. Certain facts are observed to happen together, and the problem is to find the ground or explanation of this connection. In- ductive inference is thus a process of reading the general law out of the particular facts. It is an insight into the nature of the whole or system, based upon the careful examination of the parts. Inference alwavs im- plies an effort on part of the mind to see how phenomena are neces- 169] THE PROCESS OF INDUCTIVE INFERENCE 21 § 6. Lotze ^^ declares that "in attaining to universal propositions of the form all S are M, induction has reached its first goal, and it is possible to rest content with the result, espec- ially when we are dealing with a question of practical life; for in such questions we can go without a reason, so long as we are certain that as a matter of fact M is really true of all instances of S, say of all men; we do not care so much to know whv it •holds of them, and why only of them and not perhaps of animals as well. The theoretic impulse however is not satisfied with merely joining M to its proximate subject; it would fain seek out within the limits of S the narrower group of attributes, which contains the ground of this coniunction. and which conditions the same attribute, wherever it may occur, perhaps even outside &arilj connected according to some general principle. And in carrying out this purpose, the mind must begin with the knowledge which it al- ready possesses. When the general law of connection is known, and the object is to discover the nature of some particular fact, the method ot procedure is deductive. But when the problem by which we are con- fronted is to read out of the facts of sense perception the general law of their connection, the method of inference which must be employed is that of induction. But from whatever point we set out, and whatever may be the immediate object of the inference, the result is always the same — an insight into the necessary connection of tacts according to some general principle. The essential point is to detect the general law or principle, and for this purpose one case may conceivably be as good as a hundred. Inductive inference, then, is not a process of pass- ing from a certain number of cases to a general conclusion which always remains probable because it has no proper justification. But its real nature consists in the discover}', through the aid of examples, of a uni- versal law ot connection. See also, the able works oi Hihhen. /nduciive Logic, and Welton, A Manual of Logic, volume II. ""Los^ic, English translation, edited by Bernard Bosanquet; Outlines of Losric, translated by G. T. Ladd. These translations have been used bv me in the text. 2 2 UNIVERSITY OF MISSOURI STUDIES [ I 70 S. Then the induction is pushed further; we use a series of .universal propositions of the form: SM, RM, TM, ... as our now premises and try to deduce from them an universal conclu- sion of the form all 2 are M. In this new conclusion we under- stand and denote by 2 the true subject or the conception of the genus, or, to put it in another way, that complex of attributes, on which the predicate M in all cases depends and from which it results. Thus in our first induction we shall reach the proposition SM ; in all animals an exchange of gas takes place in respiration ; in a second induction in which S is successively replaced by birds, fishes, and amphibia, we shall reach the conclusion 2M, all animals require an exchange of gases. This new conclusion at once throws light on the earlier one, by showing that what we had hitherto only observed as an isolated fact is really necessi- tated by the universal nature of animal life; a third induction sets alongside of 2M a new premise to the same effect, viz. all plants display though in another way the phenomenon of a change of gas; its conclusion SM, all organic beings whatever find themselves in like case, shows us the phenomenon in question bound up with a still more universal subject, and lastly by com- paring the behavior of bodies which resemble organic bodies in structure towards the surrounding atmosphere we might be led to the thought that under the conditions prevalent on the earth's surface, such an exchange of material is absolutely neces- sary to the development of those interdependent processes of change, which make up organic life. In all this it is to be no- ticed that the further we advance these inductions the less do we care to obtain as our result a categorical judgment of the form S is P; we are no longer seeking the highest general conception, to which a given phenomenon attaches a predi- cate; what we are in search of is a hypothetical judgment, 171] THE PROCESS OF INDUCTIVE INFERENCE 23 which will acquaint us with the most general condition C, upon which the phenomenon always depends and of which it is the consequence wherever it occurs." *^ "Stated in its complete logical form a law is alwa3^s a uni- versal hypothetical judgment which states that whenever C is or holds good, 2 is or holds good, and that whenever C under- goes a definite change into Q} through a variation of itself, dC, E also becomes E^ through a definite variation of itself dE which depends on dC. A law is hypothetical, because it is never meant to be a mere enumeration of what happens ; its sole function is to determine what should or must happen when cer- tain conditions are given." ^^ "In theoretical investigations of reality, we mean by a law the expression of the peculiar inward relation which exists between two facts and constitutes the ground at once of their coniunction; and in every single case there is but one law." ■^^ The logical idea upon which induction rests is by no means merely probable, but certain and irrefragable. It consists in the conviction, based upon the principle of identity, that every de- terminate phenomenon M can depend only upon one determinate condition, and accordingly that, where under apparently different circumstances or in different subiects P, S, T, U the same M occurs, there must inevitably be in them some common element ^^Logic, vol. II, pp. 35 £f. See also, OatUties of Logic, pp. 72 ff,, 121 ff . Outlines, p. 123: "That is to say, it is only rarely of much use to us to show that a P is united with a general generic concept S, and belongs to all species of S. As a rule, we desire still further to know on vihdit ground P belongs to S. This, expressed in general form, leads to the problem of searching for the conditions on which the occurrence of an event depends in all the otherwise diverse instances of its repeti- tion." ^Logic, vol. II, p. 68. ^'^Logic, vol. II, p. 71. H UNIVERSITY OF MISSOURI STUDIES [172 :i. which is the true identical condition of M or the true subject of -M.-"-* "It would be quite unjustifiable to object, that as a matter of experience the same consequence M is often produced by differ- ent equivalent conditions, and the same predicate M may occur in extremely different subjects. If there are two equivalent con- ditions for a result M, it is not - in virtue of that which makes them different, P or S, but of that which is the ground of their equivalence, that they are really conditions of the same result: so long as we cannot separate this common characteristic in the two, we have not yet found the true 2 of the conclusion, and have not therefore carried out the induction in the way in which it demands to be carried out. Again, if the same M is found as predicate in a number of extremely different subjects, and sub- jects (as is usually the case in practice) the several sums of «Zo^ic, vol. II, English translation, p. 23: "The law of identity guarantees that if the same S were once more perceived in a second ex- perience it would be impossible that the same predicate P should be absent or should be replaced by some other predicate Q." Outlines of Log-ic, p. 120: "It follows from the law of identity, that a truth which is valid once can not fail to be valid a second time; accordingly that tvery individual experience is once for all valid — that is to say, the same predicate is again valid at all times for all cases of the recurrence of the same subject. The difficult thing is simply to determine in praxi whether a second instance does actually repeat precisely the subject ob- served in the first case. For this the probabilities are different in differ- ent domains of research. For example, it is enough for the chemist, if he once knows that he has some element before him in a pure state, to observe its reaction towards some other element a single time, in order to establish it forever. The zoologist, on the contrary, will hold some peculiarity of a new animal, only one example of which has been dis- covered, to be 'normal,' that is, to be valid in general (since disease and malformation are possible in such a case), only when the analogies of other classes of animals justify him in this assumption." 173] THE PROCESS OF INDUCTIVE INFERENCE 25 whose marks are only partially known, we may of course make a great mistake if we combine what is common to the known marks of all of Ihem, and then assume it to be 2, the true subject of the mark in question M. I do not deny that in the practice of induction we are often placed in such unfavorable circum- stances; but all these difficulties in carrying out the inductive principle do not alter its universal logical validity, when it asserts that wherever different conditions have the same result M, or different subjects the same predicate M, there must be discover- able one and only one determinate 2, forming the single invari- able condition or the single true subject, to which the predicate or the result M is to be universally and necessarily ascribed in a conclusion of the form, 'every 2 is M.' " ^^ That is, induction is founded on the principle that every phenomenon has its definite condition, and this condition is al- ways the same. There is uniformity in nature, whatever has happened will happen again and will happen as it happened be- fore.^^ Things are identical with themselves: whatever is re- ^Logic, vol. I, pp. i36f. ^Logic, vol. II, p. 25: "Once suppose that a single observed case is valid only for itself and not for its repetitions in like case • » • once suppose that with like obiect and like conditions a different result may be true, and there is an end to all possibility of developing univer- sal truths from experience; there is an end not merelv to the discoverv ot laws but to the use of the word 'law' with any intelligible meaning. The art of induction, which is to bring us to universal laws, rests whollv on the acumen shown in developing pure and self-connected propositions of the form Z is U out of the impure and confused material of our per- ceptions, which come to us in the form S is P " ,5 UNIVERSITY OF MISSOURI STUDIES [174 mains what it is.-*' Induction is therefore ultimately based by Lotze on the principle of identity.*'^. -•"Compare this idea with the statement in Outlines, pp. l\i: "This simple logical meaning ot the proposition [of identity A=A] must without fail be distinguished from other theorems, partly true, partly doubtful, which, although they spring from the application of the uni- versal logical proposition of identity, still do so only from its applica- tion to a definite real content, and are not on a par with the proposition itself. For example, that every 'thing' is like itself, or that it is un- changeably like itself, is a metaphysical proposition which arises from an application of the logical proposition of identity to the concept of the 'existent.' The logical proposition itself says nothing at all of 'things.' It is also valid of events that happen, of conditions that take place, of the real as truly as of the unreal. And of all of them it merely says, that to be is to be, the changeable is changeable, the contradictory is contradictory, the impossible is impossible." I do not see how this thought agrees with the notion that the idea on which induction is based is certain and irrefragable. ■•"According to Kromann, Unsre Naturerkenntniss, all logic is based upon the law of identity. Hence induction, to be a logical process at all, must obey this law, or must be perfect, i. e., as much must be con- tained in the premises as is contained in the conclusion. It would have the following form: This oxygen has a specific gravity of 16: all oxygen is like this with respect to its specific gravity; hence all oxygen has a specific gravity of 16. Or it may read as follows: This oxygen has a specific gravity of 16; all oxygen is probably like this with respect, etc.; hence all oxygen has probably a specific gravity of 16. The in- duction is based on the identity of the examined case with all other cases of the same kind. The principle of identity: the world is ident- ical with itself, every thing is what it is, is not the result of experience, but a postulate, a primary hypothesis with which we approach the in- vestigation of reality. It is a necessary postulate of the will to live: the acceptance of it makes life possible. See also Bosanquet, Logic, Essen- tials of Logic, pp. 153, 162, 165. PART II The Theory of Induction ^ § I. An examination of the different theories of induction shows us that there are two questions at issue: (i) What is the nature of the process called induction? And (2), What is the validity of the process ? The first question is answered as follows : Induction is defined in a general way as a process of inferring from the particular to the universal. That is, whenever we derive a general statement from a particular statement or facts, we have induction. Most writers would be willing to accept this as a rough definition of the process. Some distinguish between scientific induction and unscientific induction, but look upon both forms as coming under the definition.2 Others, however, reject the unscientific form, or simple enumeration, and accept only that phase of induction which derives from particular facts the law of their necessary connection. According to them, induction seeks to discover not the casual, but the causal connections.^ Of these, some identify iRead before the pint meeting of the American Psychological Association and the Western Philosophical Association, Chicago, Janu- ary I, 1902, and published in the Philosophical Review of July, 1903. 2Bacon, Mill, Veitch, Lotze, Wundt. ^Sigwart, Ueberweg, Bosanquet, Hibben, Welton, Creighton; Shute, Discourse on Truth; Hamelin, Sur V induction. 175] 27 28 UNIVERSITY OF MISSOURI STUDIES [176 induction with scientific method in general, including under it the forming of hypotheses, deducing their consequences, and verify- ing tliem."* The second question also receives various answers. Ac- cording to some thinkers, only so-called perfect induction is cer- tain : imperfect induction is merely probable.^ Nearly all seem to agree, however, that induction is grounded on the principle of the uniformity of nature. This principle is interpreted differ- ently bv different thinkers, sometimes merely called by another name. Some speak of it as the principle of identity. What is once true will always be true ; whatever is, will remain so : the world is identical with itself.^ Some express the same idea by saving that the particular is the expression of the universal.'^ Some call the principle the principle of necessary connection : the given is necessary .^ Some identify it with the law of causation : every event must have some cause.^ Moreover, this principle of uniformity is conceived by some ■•Sigwart, Jevons, Hamelin. ^Apelt, Whately, Jevons. ^Lotze, Kromann, Bosanquet. 'Aristotle, Hegel. ^Sigwart, Ueberweg, Hibben, Welton, Creighton. — Venn, Empirical and /uducfive Lo£-ic, defines it thus: "Perhaps indeed as near an ap- proach as we can get to any definition is reached by saying that wherever any two or more attributes are repeatedly found to be connected to- gether, closely or remotely, in time or in space, there we have a uni- formity. And the general expression, the uniformity of nature, is in- tended to cover all such partial connections, and to imply that their ex- istence may be detected or reasonably inferred throughout all phenom- ena whatever" (p. 93). ''Mill, Jevons, Veitch, Benno Erdmann. 177] THE PROCESS OF INDUCTIVE INFERENCE 2g as a postulate of our thinking,^^ by others as the product of ex- perience.^^ § 2. Let us now attempt to answer the first question: What is the nature of induction ? Induction is a process of in- ference. We must be careful to distinguish between inference and association of ideas. The perception of fire may arouse in the child's conscimisness the thought of a burn, simply because these two things have been experienced together before. A knock at the door may arouse in my consciousness the image of a man making certain movements. But in neither case is there necessarily inference. In order to infer, I must consciously relate one judgment with another. I must ground it on some other judgment, or draw it from some other judgment. I must say, Because this is so, that is so; or, this is so, therefore that is so. In the words of Ladd: "The thinking subject reaches genuine logical inference whenever two judgments are related in such manner that one is made the 'reason' or 'ground' of the other, with a consciousness of the relation thus estab- lished between them." ^^ There are two kinds of reasoning, de- duction and induction. Both are processes of inference, and therefore essentially the same, that is, both consciously relate judgments with other judgments. In both cases a certain judg- ment is accepted on the ground of another; this is so, we say, because that is so; or, this is so, therefore that is so. The dif- i^Sigwart, Lotze, Kromann, Bosanquet, Hibben, Welton, Creigh- ton. — Venn, Empirical Logic : "I am very decidedly of opinion that the difficulty does not admit of any logical solution. It must be assumed as a postulate, so far as logic is concerned, that the belief in the Uniform- ity of Nature exists, and the problem of accounting for it must be rele- gated to Psychology" (pp. 131 f). "Mill, Jevons, Benno Erdmann. ^Psychology, Descriptive aiid Explanatory, pp. 463 f. go UNIVERSITY OF MISSOURI STUDIES [178 ference between the processes consists in this ; in induction we ground our judgment on particular instances, that is, pass from particulars to a universal proposition concerning them; while in deduction we ground our judgment on a universal proposi- tion, that is, we start from a universal proposition and draw from it other propositions according to the principle of identity. "In induction, then, we conclude that all A is B, because we have observed that ai and a2 (all essentially alike and capable of be- ing grouped under A) are B. In deduction we know, or assume as known, that A is B, and conclude that a3 (which we have never met with before) is B." ^^ When I infer that all swans are white, because the swans I have seen were white, I am reas- oning inductively. In induction we leap from a particular case or cases to all ; we infer that because a certain thing is true of a certain case or cases, it is true for all cases resembling the others. And here it is well to remember several important points. I. So far as the principle is concerned, it makes no difference whether the induction is true or false. It is just as much an in- ductive inference to conclude that all crows are black because some are, as to conclude that all men are mortal because some are. Hasty induction is induction, as much so as careful and scientific induction. The characteristic mark of induction con- sists in making the so-called "inductive leap," in jumping from one or more instances to a general conclusion.^* 2. Nor is it correct to limit induction to the discovery of causal relations. Whenever we infer a universal statement from a particular case or cases, leap from the particular to the uni- versal, we have induction. We do not strive to know merely the ^^Ladd, Psychology, p. 478. ^^"An imperfect, hasty, or unwarranted induction is still an induc- tion, only a bad one." Veitch, Z.£>^/c, p. 461. See also Mill, Logic Bk. Ill, chap, iv, §3 note. 179] THE PROCESS OF INDUCTIVE INFERENCE 3 1 causes of things ; we are interested in other relations also, for in- stance, in the co-existence of certain qualities, whether they are causally related or not.^^ Our purpose is to discover regularity, uniformity everywhere. Of course, if we identify causality with uniformity, as some writers do, if we call all those relations causal in which there is uniformity of sequence or co-existence, then induction means to discover causality. But if we do not define causality that way, if we do not conceive all uniform se- quences and co-existences as causally related, then we cannot define induction as the quest for causal relations ; for, as was al- ready said, we are interested in all kinds of regularity or orderli- ness. It is true that, wherever we find such regularity, we are tempted to read causality into it ; but that is another story, 3. And this leads us to another point. It is held by many writers that induction seeks to discover the inner, necessary rela- tions existing between things. In a certain sense, this is true. The thinker is always eager to find out what qualities are con- nected necessarily, that is, he wants to feel not only that certain qualities go together, but that they must somehow go together. He is not satisfied with the statement that all swans are white, because he does not understand the inner relation existing be- tween swan nature and whiteness, he does not see why swans should be white, he does not see any necessary relation here. He seeks to discover connections between things which will sat- isfy him. "Take, for instance, the simple effect of hot water cracking glass. This is usually learnt empirically. Most people have a confused idea that hot water has a natural and inevitable tendency to break glass, and that thin glass, being more fragile than other glass, will be more easily broken by hot water. Physical science, however, gives a very clear reason for the effect isSee Veitch, Logic, p. 461; Venn, Logic, p. 93; Sigwart, Logik. o 2 UNIVERSITY OF MISSOURI STUDIES [l8o In sliowing that it is only one case of the general tendency of heat to expand substances. The crack is caused by the success- ful effort of the heated glass to expand in spite of the colder glass witli which it is connected." ^^ That is, the scientist aims to bring his proposition under a proposition which is more general in its scope, one which expresses a more constant connection be- tween objects than the other, and therefore impresses us as nec- essary. He seeks for a simple formula under which he can em- brace a great many cases that seem to have nothing at all in com- mon. "Suppose some one observes that (a) the addition of fuel, (b) the action of blowing, and (c) cold weather increase the flame of the fire. He may at first be satisfied with the assump- tion that every one of these three phenomena is a cause of the in- crease of the flame. But when he discovers a great number of phenomena which are followed by an increase of flame, he finds it hard to think of them all. But if he can find that every time the flame is increased, something was added to the fire which, according to analysis, contains oxygen, he will reduce the mani- fold experiences to the simple formula : All things which contain oxygen and are added to fire increase the flame. He will prob- ably go farther and say: Oxygen is the cause of the increase of the flame." ^7 The truth is, the thinker aims to understand his facts, that is, to assimilate them to the known, to bring them into relation with what he already knows. You tell him that heat cracks the glass because heat is motion, expansive motion ; he understands that because he has seen many examples of motion breaking things. "We did not reject the assertion that there are black swans," says Mill, "while we should refuse credence to any ^®Je\ons, Lessons in Logic, p. 257. i^Uphues, Grtmdlegung der Logik; Nach Shute's Discourse on Truth bearbeitet, p. 182. l8l] THE PROCESS OF INDUCTIVE INFERENCE 33 testimony which asserted that there were men wearing their heads underneath their shoulders. The first assertion was more credible than the latter. But why more credible? So long as neither phenomenon had actually been witnessed, what reason was there for finding the one harder to be believed than the other ? Apparently because there is less constancy in the colors of ani- mals than in the general structure of their anatomy. But how- do we know this? Doubtless, from experience. Experience testifies that among the uniformities which it exhibits or seems to exhibit, some are more to be relied upon than others." ^^ But it must not be forgotten here that it is induction to conclude from our observations that heat cracks glass, that blowing makes the fire burn, that chlorine bleaches, even if we do not understand the reasons or see the so-called necessary connections. "We learn empirically that a certain strong yellow color at sunset, or an unusual clearness in the air, portends rain ; that a quick pulse indicates fever; that horned animals are always ruminants; that quinine affects beneficially the nervous system and the health of the body generally; that strychnine has a terrible effect of the opposite nature; all these are known to be true by repeated ob- servation, but we can give no other reason for their being true, that is, we cannot bring them into harmony with any other scien- tific facts ; nor could we at all have deduced them or anticipated them on the ground of previous knowledge." ^^ Induction is induction, whether we can bring it into harmony with other scien- tific facts or not. It must further be remembered that deduction frequently enters into those cases in which we reach so-called necessary connections. I discover by induction that heat cracks glass. I refer this empirical law to a larger induction, that heat expands substances. I say heat must crack glass under certain ^^Lo£'ic, Bk. Ill, ch. iv. See also ch. iii. ^^Jevons, LessoMs, p. 256. (3) 24 UNIVEKSITV OF MISSOURI STUDIES [iSz circunistances, because heat expands substances. If heat expands substances, it must expand glass ; and if the colder parts of the o-lass connected with the heated parts do not expand fast enough, the glass will break. This is really deduction. I subsume the case under a general rule. I think I understand it better when I see that it is really an instance of a general occurrence with which I am very familiar. 4. This brings us to another point. Several thinkers de- fine induction as forming hypotheses, drawing their conse- quences, and verifying them. This, it seems to me, is a false definition. If we define it in this way, then we apply the name induction to different operations, we include under it both in- duction and deduction. If induction is both induction and de- duction, then what is the process called induction, which with deduction constitutes induction? Of course, we may, if we choose, apply the term induction to scientific methods in gen- eral, to the method which everybody uses in the pursuit of truth, and which embraces all the operations of the mind that lead to truth. But in that case what is the process called induction proper? And why should we use one term for two processes, first for a combination of induction and deduction, then for in- duction itself? The logical thing to do is to restrict the term in- duction to induction proper, to the process of inferring a general truth from particular instances, and to use another name for the combination of this process with deduction. In his smaller book Jevons calls this method, which he designates as induction in his Principles of Science, the combined or complete method. "What Mr. Mill has called the deductive method, but which I think might more appropriately be called the combined or complete method, consists in the alternate use of induction and deduction. It may be said to have three steps, as follows: — (i) Direct in- duction ; (2) Deduction, or, as Mr. Mill calls it, ratiocination; (3) 183] THE PROCESS OF INDUCTIVE INFERENCE 35 Verification. The first process consists in such a rough and simple appeal to experience as may give us a glimpse of the laws which operate, without being sufficient to establish their truth. Assum- ing them as provisionally true, we then proceed to argue to their effects in other cases, and a further appeal to experience either verifies or negatives the truth of the laws assumed." 20 5. There is another point to be observed. It is held that when I infer from one or more cases to all like them, I base myself either consciously or unconsciously on the principle of the uniformity of nature. That is, I reason thus : This is true of these cases; what is true of some cases is true of all like them; hence this is true of all. In other words, induction is really deduction. This, however, does not seem to me to be the case. In fact, the statement that what is true in some cases is true in every case like them, is the very thing that is inferred in induction. We infer that this will always happen because it has happened. As soon as we observe the co-existence or sequence of certain qualities several times, we naturally draw our conclusion, we make the inductive leap. We say, sometimes, hence, always. Why we do so, it is impossible to say; it is one of those inexplicable facts, a natural function of the human mind, a way we have of thinking, that is all. We expect repetition. We may have no right to expect it, but the fact remains that we do expect it and conclude that it will come. We infer when we find a ground or reason for our proposition. Everything is a ground for us that really satisfies us. Closer thinking may destroy our satisfaction, but so long as we have grounded our proposition upon some other proposition and are satisfied, we have reasoned. We may have reasoned wrong, but we have reasoned. Inductive inference is a function of the mind aroused by the experience of recurrence, in which we regard the par- ^Lessons in Logic, pp. 258 f. Compare page 14 of this paper. ■l6 UNIVERSITY OF MISSOURI STUDIES [184 ticular as a type, as having universal significance. It is fre- quently hasty and its results are frequently discovered to be false, but that does not aflfect its nature. The point to be emphasized here is that induction consists in making the leap spoken of, re- gardless of whether we have any warrant for doing so or not. We say, what is true of these particular instances is true of their class, and, after having made many such inferences, we finally reach the belief that nature at large is uniform. The belief in the general uniformity of nature is a late product in the history of civilization, and is not even universally accepted to-day. It is preceded by, and grows out of, the belief that a particular in- stance will repeat itself. § 3. This brings us to our second fundamental ques- tion : What is the validity of the process of induction ? What is its warrant? Here we may discuss two problems, (a) How can we reach the greatest possible certainty in particular in- ductions? (b) How can we prove induction in general? (a) Certainty is a feeling. We feel certain that a proposi- tion is true; the proposition is certain because it arouses in us the feeling of certainty. What must we do to reach such cer- tainty in a particular induction? We increase our feeling of certainty in many ways. We notice that qualities go together. The more often we observe it, the more certain we feel that they will continue to go together. When we observe that one fails to appear the other fails to appear also, and that when one varies the other varies, we feel still more certain that they go together, that our induction is true. The purpose of the so-called induc- tive methods is to bring this certainty to the highest possible de- gree. We feel most certain of propositions which have been verified countless times, and of which we have experienced no contradictory instances. It is for this reason that we strive to 185] THE PROCESS OF INDUCTIVE INFERENCE 37 subsume all other propositions under such propositions, that we try to consider them as instances of these. We have had a great deal of experience with motion, for example; hence, if we can reduce a phenomenon to motion, we feel that we know something about it. In other words, we reach the greatest possible cer- tainty for our particular inductions when we subsume them un- der generally accepted principles, or prove them deductively. That is why sciences become more and more deductive in the course of time. It is also to be noted here that, wherever the connection is believed to be a causal connection, one case is as good as a thou- sand. When I believe that two phenomena are causally related, I am sure that one will always follow the other, because causal connection means a necessary connection, because the notion of cause implies that when one phenomenon appears the other must somehow appear also. When I conceive of a particular case as a case of causality, when I say in this particular case a was the cause of h, I do not need any other cases to convince me that there is a universal relation. I conclude from one to many, be- cause I have already assumed uniformity by assuming causality. Similarly, whenever I conceive of phenomena as necessarily re- lated in any other way, one case is as good as a thousand. When I see that the sum of the angles of a triangle is equal to two right angles, having proved it for a particular triangle by showing that it follows necessarily from the definition of a tri- angle, then I am satisfied that it will be true of all triangles ; and there is no need of my examining any more. These cases, however, are not cases of induction. When I say, this phenomenon caused that one in this particular case, therefore whenever I have this phenomenon in other cases I will have the other also, I am reasoning deductively. By saying that a particular relation is a causal relation, I am implying that it has 38 UNIVEUSITY OF MISSOURI STUDIES [186 universal validity. I reason: If a and b are causally related, then when a appears b will appear also. Now a and b are caus- ally related. Hence, when a appears, b will appear also. This is deduction. (b) How can we prove induction? By proof we mean deduction. Our question therefore means: What must we do in order to deduce a conclusion which has already been de- rived inductively? In deduction we consciously draw a propo- sition from premises in which it is already implied ; we explicate it. Here our conclusion will give us a feeling of absolute cer- tainty, that is, we will feel that if the premises are true, the con- clusion must be true, unless we have made a mistake in our rea- soning. It is not difficult to construct a syllogism in which the in- ductive proposition forms the conclusion. For example, if it is true that nature is uniform, that nature repeats itself, that it is a reign of law, then we have a proof for induction. One should remember, however, that this does not make induction deduction. Induction is induction; by proving a proposition that has been derived inductively, we do not make induction deduction, we simply apply another process, deduction, to a proposition that has already been derived inductively. The process of proving the inductive proposition is not induction, but deduction. Here the certainty of the proof will, as always, depend upon the cer- tainty of the principle of uniformity. The more we believe in this principle, the more certain we shall be of our inductions, the more satisfied we shall be with them. Induction, therefore, may be proved by assuming the law of uniformity. We are warranted in leaping from part to whole by the regularity, or orderliness, or uniformity of nature. If it is true that nature is uniform, that nature repeats itself, we have the right to conclude from a few instances to all Hke them. The only problem here is to discover the particular combinations, the co-existences and sequences in nature. 187] THE PROCESS OF INDUCTIVE INFERENCE 39 But the question at once arises: What warrant have we for saying that nature is uniform? It may perhaps be said that this principle is a postulate of thought, and that it carries its war- rant in itself. We cannot prove its truth, but we feel certain that it is true ; we accept it without cavil. But is it really a pos- tulate of thought? Does everybody really accept it? Does it inhere so in the nature of our thought that we must accept it ? That depends entirely upon what we mean by it. If we mean by it the clearly conscious thought that nature at large, internal and external nature, is governed by law, that it is a uni- fied system, then we cannot regard the principle as a postulate of thought. In this sense, it is plainly a product of development, the result of much reflection upon the world, and even then not at all universally accepted. There are many persons who will not admit that external nature is a closed system, exempt from interference, and there are still more who will not admit that the mental realm is subject to law. Interpreted in the above sense, the principle of uniformity must be regarded as the result of re- flection upon our experiences. We have noticed many particu- lar uniformities; we conclude that nature at large is uniform, that is, we consciously ground our proposition upon our past ex- periences. In this sense, the principle of uniformity is an induc- tion : Because there are uniformities, there is uniformity. And if we try to base the inductive process upon the principle thus understood, we are really reasoning in a circle, as has been so often pointed out. We prove the uniformities by the uniformity, and the uniformity by the uniformities. We say we are war- ranted in inferring from the particular to the universal, because nature repeats itself, because nature is uniform ; and we say we know nature is uniform, because we discover particular uniform- ities and conclude from these that there is general uniformity. We may, however, mean by the principle of uniformity of 40 UNIVERSITY OF MISSOURI STUDIES [l88 nature as a postulate of thought, not a clear conviction that na- ture as a whole is a unified system, subject to law, but the feel- ing in every particular case that this particular experience, will come again. Here we form no conception of nature as a whole ; but every time we have a particular experience, we expect it to recur. After having a particular experience a number of times, we feel that it will come again, we expect particular things to repeat themselves. Our feeling of expectation here may be called a postulate of thought, and it becomes the psychological ground of our inductive inference. That is, there is no reason for inferring that a particular co-existence or sequence of quali- ties will recur except the expectation that it will recur. We feel that what happens in this particular case will happen so again, we expect it to happen so again ; we therefore infer or conclude that, because it happened once, it will happen again. That is, I have no other warrant for inferring that a combination of qual- ities will recur than the feeling of expectation that it will do so. UNIVERSITY OF MISSOURI STUDIES VOLUME I Number i Contributions to a Psychological Theory of Music, by Max Meyer, Ph. D., Professor of Experimental Psychology, pp. VI, 80. 75 cents. Number 2 Origin of the Covenant Vivien, by Raymond Weeks, Ph. D., Professor of Romance Languages, pp. Vril, 64. 75 cents. Number 3 The Evolution of the Northern Part of the Inlands of Southeastern Missouri, by C. F. Marbut, A. M., .Professor of Geology, pp. VIII, 63. $1.25. Number 4 Eileithyia, by Paul V. C. Baur, Ph. D., Acting Professor of Classical Archaeology, pp. VI, 90. $1.00. Number 5 The Right of Sanctuary in England, by Norman Maclaren Trenholme, Ph. 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