2. THE INDUCTIVE HAZARD.

Referring to the first inductive syllogism of section one, it is assumed that the robin, crow and sparrow are representative birds, and that we are thus justified in concluding that if these type birds have wings, then all birds must have wings. Of course this is more or less of a conjecture or “a hazard”; since birds without wings may exist in some undiscovered corner of the globe. However, inasmuch as the generalization concerns a representative quality, we deem the assumption fairly well founded. The logical right to take this “leap into the unknown” will be discussed later. It will profit us at this time to realize more fully how essential the “inductive hazard” is to the progress of the world. When the Schoolmen of mediÆval time refused to venture, they failed to progress, and thus came the dark days. Whenever man has ignored this God given instinct which leads to discovery, the world has stood still. This willingness to “take a leap into the dark” with the hope of finding, in the shadow, truth which would enhance man’s power and increase his serviceableness, has given to the world about all that is worth while. It was the spirit of the hazard which pushed Columbus to the discovery of a new world; which gave Newton the secrets of the motions of the universe; which enabled Edison to harness a multitude of lurking forces; and Morse and Bell to reduce distance to its lowest terms. In ordinary affairs with ordinary men those succeed best who manifest most a safe, steady, persistent spirit of discovery. Here, then, in the “inductive hazard” have we a most important phase of school life which, in this day of making the work easy, is being sadly neglected. On the other hand, an unregulated and insane spirit of venture may result in a great waste of energy, and in the development of low ideals of recklessness and inaccuracy. The “inductive hazard” must be cultivated; yet it must be regulated as well, and, as the reader already realizes, logic needs to concern itself mainly with this regulative aspect.

3. THE COMPLEXITY OF THE PROBLEM OF INDUCTION.

The problem of induction is much more complex than that of deduction because of these reasons: First. Deduction as a process of reasoning was the only kind discussed by the logicians for two thousand years. Aristotle is called the father of deductive logic and this Intellectual Giant, the greatest of ancient time and possibly of all time, so perfected the form of deductive reasoning that, up to the time of Francis Bacon, no scholar possessed the temerity to gainsay its supremacy in the field of logical reasoning. For twenty centuries Aristotle’s Deductive Logic was the Logicians’ Bible. On the other hand, inductive reasoning, though it was briefly discussed by Aristotle, received little attention till the versatile Francis Bacon placed it upon the stage of the thinking world. This makes deduction nearly two thousand years older than induction. Time, by eliminating the personal equation and exposing in various ways fallacious thinking, tends to unify and universalize truth. Hence, logicians are agreed so far as the fundamentals of deductive logic are concerned, but are still at odds over the true conception and use of inductive logic.

A second reason for this confused status in the field of inductive logic is the fact of its being more closely related to the events of every day living. Induction is the natural method of childhood; the popular method of the school room; and the most used method of common life. In consequence its ramifications are so varied and multitudinous, that it will take centuries of thinking to reduce the doctrine of induction to that uniformity and definiteness which so distinguishes deduction.

4. THE VARIOUS CONCEPTIONS OF INDUCTION.

The attending quotations will give the student a fair idea of the leading conceptions concerning induction:

(1) “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 under similar circumstances at all times.” “Induction, as above defined, is a process of inference; it proceeds from the known to the unknown.” “Any process in which what seems the conclusion is no wider than the premises from which it is drawn, does not fall within the meaning of the term.”—J.S. Mill, ASystem of Logic, 1892, p.175.

(2) “An induction is a generalization or an inference based upon propositions that state observed facts.” “The truth inferred may be general or particular, but it must be one which we cannot perceive in a single act of observation.”—Ballentine’s Inductive Logic, 1896, p.14.

(3) “Induction is the process of inference by which we get at general truths from particular facts or cases.”—Ryland’s Logic, 1900, p.148.

(4) “Induction may be defined as the legitimate inference of the general from the particular, or, of the more general from the less general.”—Fowler, 1905, p.10, Vol.2.

(5) “The term induction has been used by logicians to denote this leap of the mind from the limitations of its positive knowledge to belief in universal laws.” “In pedagogy, however, the term is applied to the whole process of arriving at general truths or principles.”—Salisbury’s Theory of Teaching, p.156.

5. INDUCTION AND DEDUCTION CONTIGUOUS PROCESSES.

If there is one thing above another which modern logic is emphasizing it is the unity of the mind and the contiguity of thinking. Induction and deduction are dove-tailed processes which characterize all thinking worthy of the name. Where induction ceases, deduction commences, and vice versa. It becomes the function of inductive thinking to establish a connection between what has been experienced and what has not been experienced. Therefore, the conclusion of an induction must always contain more than is implied in the premises. The premises denote facts which have been observed; whereas the conclusion denotes the observed facts of the premises plus analogous facts which have not been observed. Inductive thought ventures into the unknown, and attempts to establish a bond of connection between it and something already known. Induction seeks new knowledge, and does so by taking that “leap into the dark” already referred to as the “inductive hazard.”

As soon as the mind reaches a universal truth, it sets to work to clarify this truth. Such is accomplished by reference to other facts which the universal is supposed to include; and this application of the general to the particular is deduction. Induction discovers the new knowledge while deduction clarifies it.

6. INDUCTION AN ASSUMPTION.

In this treatment induction as a general process has been subdivided into induction as a mode of inference and induction as a method. Induction as a mode of inference is the process of reasoning from less general premises to a more general conclusion; whereas induction as a method is a procedure from the observation of individual facts to a realization of a universal truth. In either case the conclusion of an inductive process always implies more than is contained in the premises. This gives to the conclusion an uncertainty. No induction is absolutely free from doubt except the so-called perfect induction, which form will receive attention in a later section.

9. INDUCTIVE ASSUMPTION JUSTIFIED.

The function of induction seems to be to universalize particulars. The mind of man has ever been engaged in establishing connections among the concrete experiences of daily life. This ability of his to generalize his individual experiences has been one of the chief agencies in elevating him to the position of “King of the animal world.” In this disposition to generalize man has taken it for granted that nature is honest; that what she tells him under given conditions, she will tell him again under identical conditions. To put it in logical terms man can depend upon the invariability of nature’s activities, or upon the uniformity of nature. Here, then, is one of the most fundamental laws not only of induction but of all activity. But this law implies a second quite as fundamental. If every cause is invariably followed by the same effect under like conditions, then it is thereby implied that every cause has an affect and every event is due to some cause. This, too, is invariable. In consequence of these facts man is justified in thinking that nature is not only honest and therefore “she gives me confidence, but her every activity means something and therefore she arouses my curiosity.” “Uniformity of nature” engenders confidence, “universal causation” inspires the spirit of discovery and with these two weapons man is willing to venture into the jungle of the unknown. Why is man eager to undertake the “inductive hazard?” Because, through the laws of universal causation and uniformity of nature, his curiosity is aroused, and he is given confidence in nature’s activities.

10. THREE FORMS OF INDUCTIVE RESEARCH.

Induction is a matter of universalizing less universal experiences. In this the process may assume any one of three forms, namely: (1)Induction by simple enumeration; (inductio per enumerationem); (2)Induction by analogy; (3)Induction by analysis.

THREE FORMS ILLUSTRATED:

(1) Simple enumeration.

Having observed a few instances the generalization is, “All birds have wings.” The certitude of this may now be strengthened by observing more birds and finding without exception that each has wings.

(2) Analogy.

By noting on Mars geometric markings which resemble canals, the generalization is vouchsafed that Mars is inhabited by human beings. Other similarities in atmospheric conditions, existence of land and water, etc., tend to make this generalization more plausible.

(3) Analysis.

By analyzing water taken from a certain spring, it is found to contain hydrogen and oxygen in the proportion of 1 to 8; in consequence a generalization to this effect is posited. Analyses of specimens from other sources yield similar results and thus the generalization is given greater certitude.

As a usual thing the particular form which the induction assumes depends on the nature of the topic under investigation and also on the mental make-up of the investigator. The general statement that all birds have wings could hardly be derived by means of analogy or analysis, but is a matter of a casual observation of many instances. Moreover, that mind given to accurate observation, but not inclined to note resemblances or to carry on experiments, would naturally follow the first inductive type. On the other hand, simple enumeration would be impossible in questions like the habitability of Mars, and would yield no results in cases requiring definite scientific experimentation like electrolysis.

It is worthy of note that some topics lend themselves to all three modes of procedure. To wit: (1)Enumeration. Without being taught the rule the child is given a list of examples involving the dividing of a decimal by a decimal and is asked to solve them. By comparing his answers with those in the book, he somewhat accidentally discovers what seems to be the correct rule for pointing off in the quotient. By following this rule and each time comparing answers he establishes the truth. (2)Analogy. If .24 ÷ .6 is the first example, the child may resort to the well known process of dividing a common fraction by a common fraction, ( 24
100 ÷ 60
100 = 24
60 = 4
10, ) then, because of their close resemblance, he may reason that decimal fractions should yield the same result. (3)Analysis. Here the child reasons that since division of decimals is the inverse of multiplication of decimals, the rule for pointing off might be the inverse of the multiplication rule. By trying this out and proving his answer in each example, he becomes convinced of the correctness of his reasoning.

11. INDUCTION BY SIMPLE ENUMERATION.

As its name implies this type of inductive research consists in observing many instances which may exemplify the particular uniformity under consideration. The process is quantitative rather than qualitative, the certitude of the generalization depending on the mass of facts collected rather than on any striking resemblance or any detailed analysis. The aim is to observe, accurately if not scientifically, instance after instance until all doubt is removed. The outcome of such observation may be three fold. (1)The enumeration may be complete. This gives the so-called “perfect induction” which will receive attention later. (2)The enumeration may be incomplete and without exceptions; generalizing in this way from uncontradicted experience gives what are termed “empirical” truths. (3)The enumeration may be incomplete with exceptions. It is obvious that this type of induction could give no valid generalization; but the result may be put in the form of a ratio between the uniformities and the exceptions. Such a procedure is a mere “calculation of chances” and the result simply an expressed probability.

THE THREE KINDS OF SIMPLE ENUMERATION ILLUSTRATED.

The subject to receive investigation is a school examination.

Briefly, simple enumeration may take the form of (1)a perfect induction, (2)a probable induction, (3)a mere calculation of chances. The first necessitates completed experience, the second uncontradicted experience and the third contradicted experience.

12. INDUCTION BY ANALOGY.

Induction by analogy assumes that if two (or more) things resemble each other in certain respects, they belong to the same type, and, therefore, any fact known of the one may be affirmed of the other.

THE TYPE.

As the definition implies, analogy involves an extensive use of types; let us, therefore, become better acquainted with them as instruments in analogical inductions. Atype is one of a group which embodies the essential characteristics of that group. How easy and natural it is to dismiss a complex topic with the citing of an example which may be regarded as a type; how common is the use of examples in the school room! On second thought it becomes apparent that analogical induction by example or type is the most common of all forms of induction either as a method or a mode of inference. Analogy by example (or type) assumes that if two or more things are of the same type, they resemble each other in every essential property.

Illustrations of analogical inductions by example or type.

(1) Mathematics.

Inductive Inference: The square of the sum of two quantities is equal to the square of the first, plus twice the first by the second, plus the square of the second.

(2) Nature.

This corn sent me as a sample produced heavy, full ears, and many of them; hence (inductive inference), if Iplant corn like this sample under like conditions, Iwill receive in return heavy, full ears, and many of them.

(3) Geography.

Cities like New York, located on the coast, possess a larger foreign element than the inland cities like Philadelphia.

(4) Grammar.

A noun is the name of anything, as the examples, “George Washington” and “house” would indicate.

In deriving a generalization from one or two examples the prime essential is to select types which are truly representative. Often the example used is a special type and in consequence does not exemplify all of the essential characteristics of the group. To teach the nature of a parallelogram by using a rectangle only, is an easy way to commit this error; or one may affirm that the class can easily cover the work, when the judgment is based entirely on knowledge concerning the brightest one of the grade.

Type work when judicially used is a positive time saver and a very present help in times of perplexity. Let the skillful teacher use types and examples extensively yet cautiously.

THE MARK OF SIMILARITY.

As opposed to analogy by type there is a second form; namely, analogy by one or more similar marks or qualities. This form is best described by the definition: When two things resemble each other in a few marks or qualities they resemble each other in other marks or qualities.

Illustrations of analogy by marks.

THE ERRORS OF ANALOGY BY MARKS OF SIMILARITY.

It follows that analogy by example gives generalizations of much greater certitude than analogy by one or two marks of resemblance. Here is a field bespattered from boundary to boundary with erroneous thinking. The principle of resemblance being an innate tendency, this form of error is most common with the immature. The child reasons by analogy when he invests the poodle with the despised cognomen of “kitty”; or honors every man who wears glasses with “papa.” In the childhood of the race natural events were interpreted by means of analogy. The wind blowing through the trees made sounds much like the human voice; hence these noises were attributed to spirits. Primeval man was led to believe by analogy that everything which moved was alive. We may, therefore, think of our revered forbear as engaged in the undignified task of running after his shadow, or chasing a leaf around a stump.

THE VALUE OF ANALOGY.

Analogy being rich in its suggestions is the favored process of the scientist and inventor. Newton reasoned by analogy when he tentatively affirmed of the moon what he positively knew of the apple. Franklin’s reasoning was analogical when he discovered the identity of the electric spark and lightning. Because this form of induction so often leads to error and at best involves a degree of probability far below induction by analysis, some logicians are inclined to ignore its generalizations altogether. Others deem this a mistake because of these reasons: First. Analogy is serviceable to a high degree in suggesting hypotheses which may be advanced either for the purpose of explanation or verification. It has already been indicated that analogy is the common instrument used by the inventor and discoverer. Second. The principle of analogy, in reality, lies at the basis of classification; because in this, things are grouped according to their resemblances. Third. Analogical induction affords valuable training in originality and initiative. Amind which easily and naturally discerns analogies is “fertile in new ideas.”

REQUIREMENTS OF A TRUE ANALOGY.

It has been remarked that the certitude of an induction by simple enumeration depends upon the number of uncontradicted instances. In analogy the case is different as the process emphasizes the weight of the points of resemblance rather than the number. In substance the requirements of a logical analogy are three.

First. The points of resemblance must be representative and not exceptional. For example: The argument that Mars is inhabited because it has two moons is of little worth, since we have no proof that moonshine is essential to life; this point of resemblance is not representative. On the other hand, if the basis of argument is the fact that Mars has an atmosphere, the conclusion carries some weight; as air seems to be essential to life.

Second. The points of resemblance must outweigh the points of difference. That is, the ratio of probability must always be in favor of the resembling instances. Since it is not a matter of numbers but of weight, a numerical proportion like this would be misleading: Resemblances: Differences = 10:6. It is obvious that the six differences might more than outweigh the ten resemblances. The safer way, if it were possible, would be to attach a value to each point of resemblance or difference, and then express the proportion in terms of the sums of these values.

Third. There must be no difference which is absolutely incompatible with the affirmation which we wish to prove. For example, the fact that the moon has no atmosphere renders nugatory any attempt to prove the habitability of the moon.

13. INDUCTION BY ANALYSIS.

This, the third form of inductive research, is by far the most important. Simple enumeration, because it depends upon the number of observed instances, consumes much time; while we have already noted how easy it is for analogy to lead to error. At the best, the conclusion of these methods must be subjected to analytic investigation, if we are seeking universal validity. Induction by analysis is superior to the other forms because it secures a higher degree of probability and is a positive time saver.

Defined. We have learned that analysis is the process of separating a whole into its related parts. We thus define induction by analysis as the process of separating a whole into its parts with a view of deriving a generalization relative to the nature and causal connection of these parts.

ILLUSTRATIONS:

(1) Concerning the generalization that “all birds have wings,” it becomes possible to observe in detail the nature of the wings and advance the hypothesis that these wings are designed for aËrial navigation. This hypothesis may then be strengthened by observing that the entire structure of the bird is adapted to flying.

(2) If it were possible to analyze the atmosphere, water, and soil of Mars, and should such analysis reveal a composition similar to that of the earth, it would illustrate well not only the method of analysis but also its superiority over the other methods of investigation.

(3) The physician, in diagnosing a “case,” observes that the symptoms resemble those of typhoid; but to be positive of the truth of his diagnosis, he takes a blood test. Noting the resemblances is induction by analogy; but the blood test involves induction by analysis.

Induction by analysis concerns hypothesis, observation, and experiment, including Mill’s experimental methods. These topics will receive due attention in the chapters which follow. It will be sufficient to close this discussion with a brief treatment of perfect induction, and traduction.

14. PERFECT INDUCTION.

As has been indicated under simple enumeration, a perfect induction is one in which the premises enumerate all the instances denoted by the conclusion.

ILLUSTRATIONS:

Because the conclusion of a perfect induction gives nothing new—nothing but what is found in the premises, some claim that the process is practically valueless. From the viewpoint of the discoverer this position is well taken; yet to universalize particular observations puts the knowledge in compact, usable form, and saves one the trouble of returning each time to the consideration of each particular. Thus as a process which leads to verified universals, perfect induction is a time saver. In the second place it was the method used by Socrates when he desired to lead up to a definition or some other general truth. The Sophists were given to a careless use of the “inductive hazard”; they were prone to generalize from one or two particulars, or what is worse, to establish a generalization and then attempt to fit the particular instances to it. This led to a superficiality which the Great Pagan Educator abhorred. The fact that perfect induction was the method used by Socrates to counteract the teachings of the Sophists, is sufficient vindication for its use in discouraging the indefensible assumptions of to-day, and in inspiring warrantable generalizations based on accurate observation.

In the school room with classes addicted to careless, inaccurate work, to accept nothing but a perfectly induced generalization, when this is feasible, is a most valuable lesson. For example, the teacher may not accept the generalization that all of the “first class” cities of the U.S. are located on navigable waterways, until the pupils have investigated the waterway conditions of every city belonging to the class. On the other hand, there may be individual cases of “cocksureness” which need attention. The teacher can do little for the “know-it-all youngster” until he pricks the bubble of conceit. This may be accomplished by allowing the youth to draw a generalization, which seems to meet all the requirements of truth arrived at by means of an imperfect induction; then without warning let the teacher give an instance which will show the generalization to be false. This involves what Socrates termed the “torpedo’s shock.” To illustrate: Consider the “prime number” formula given by Jevons. In deriving this, direct the class to add 2 to its square, and to this sum add 41. Give similar directions relative to numbers 3, 4, 7 and 10. Indicating the work as directed, would give the following:

A question or two will make apparent the fact that all the results are prime numbers, and then the generalization may be drawn; namely, X + X² + 41 = prime number. Now without warning, but under the assumption that you desire to test deductively the general formula, let X = 40. This gives (40 + 40² + 41) 1681, which is the square of 41 and is, therefore, not a prime number.

15. TRADUCTION.

It may have been noted by the student that “perfect induction” is not induction at all according to the definition; viz.: Inductive reasoning is reasoning from less general premises to a more general conclusion. Referring to the first illustration of the previous section it is apparent that the conclusion is no broader than the premises. Ostensibly, the conclusion is a mere summary, or a generalization of the facts mentioned in the premises. Moreover perfect induction does not readily conform to the definition of deductive reasoning, as in this the movement must be from the more general to the less. We are thus forced to the conclusion that perfect induction is a form of a third type of reasoning which is known under the cognomen of traduction. This is from the Latin trans, and ducere meaning to lead across. Definition: Traductive reasoning is reasoning to a conclusion which is neither less general nor more general than the premises.

Aside from the case of perfect induction there are other types which well illustrate traduction. These are: First. Reasoning from particular (or individuals) to particular (or individuals).

ILLUSTRATION:

Second. Reasoning from general to general.

ILLUSTRATION:

It may be observed that all of the propositions in traduction are co-extensive “A’s” or “E’s”; hence all the terms are distributed. This eliminates any possibility of committing fallacies of distribution. Further, the propositions may be interchanged at will, without invalidating the particular conclusion selected. To illustrate we may change the last argument to this:

From the viewpoint of authenticity traduction is the most, and induction the least dependable; whereas the certitude of deductive reasoning lies somewhere between the two. On the other hand, when looked at from the ground of serviceableness the order is reversed, induction being the most useful form of inference and traduction the least.