To satisfy itself upon the problem, the mind appeals to actual experience either by ordinary observation or through experimentation. These observations or experiments, which necessarily deal with particular instances, are supposed to provide a number of particular judgments, by examining which a satisfactory conclusion is ultimately reached.
Example of Induction.—As an example of induction, may be taken the solution of such a problem as, "Does air exert pressure?" To meet this hypothesis we must evidently do more than merely abstract the manifest properties of an object, as is done in ordinary conception, or appeal directly to some known general principle, as is done in deduction. The work of induction demands rather to examine the two at present known but disconnected things, air and pressure, and by scientific observation seek to discover a relation between them. For this purpose the investigator may place a card over a glass filled with water, and on inverting it find that the card is held to the glass. Taking a glass tube and putting one end in water, he may place his finger over the other end and, on raising the tube, find that water remains in the tube. Soaking a heavy piece of leather in water and pressing it upon the smooth surface of a stone or other object, he finds the stone can be lifted by means of the leather. Reflecting upon each of these circumstances the mind comes to the following conclusions:
Air pressure holds this card to the glass,
Air pressure keeps the water in the tube,
Air pressure holds together the leather and the stone,
∴ Air exerts pressure.
How Distinguished from, A. Deduction, and B. Conception.—Such a process as the above constitutes a process of reasoning, first, because the conclusion gives a new affirmation, or judgment, "Air exerts pressure," and secondly, because the judgment is supposed to be arrived at by comparing other judgments. As a process of reasoning, however, it differs from deduction in that the final judgment is a general judgment, or truth, which seems to be based upon a number of particular judgments obtained from actual experience, while in deduction the conclusion was particular and the major premise general. It is for this reason that induction is defined as a process of going from the particular to the general. Moreover, since induction leads to the formation of a universal judgment, or general truth, it differs from the generalizing process known as conception, which leads to the formation of a concept, or general idea. It is evident, however, that the process will enrich the concept involved in the new judgment. When the mind is able to affirm that air exerts pressure, the property, exerting-pressure, is at once synthesised into the notion air. This point will again be referred to in comparing induction and conception as generalizing processes.
In speaking of induction as a process of going from the particular to the general, this does not signify that the process deals with individual notions. The particulars in an inductive process are particular cases giving rise to particular judgments, and judgments involve concepts, or general ideas. When, in the inductive process, it is asserted that air holds the card to the glass, the mind is seeking to establish a relation between the notions air and pressure, and is, therefore, thinking in concepts. For this reason, it is usually said that induction takes for granted ordinary relations as involved in our everyday concepts, and concerns itself only with the more hidden relations of things. The significance of induction as a process of going from the particular to the general, therefore, consists in the fact that the conclusion is held to be a wider judgment than is contained in any of the premises.
Particular Truth Implies the General.—Describing the premises of an inductive process as particular truths, and the conclusion as a universal truth, however, involves the same fiction as was noted in separating the percept and the concept into two distinct types of notions. In the first place, my particular judgment, that air presses the card against the glass, is itself a deduction resting upon other general principles. Secondly, if the judgment that air presses the card against the glass contains no element of universal truth, then a thousand such judgments could give no universal truth. Moreover, if the mind approaches a process of induction with a problem, or hypothesis, before it, the general truth is already apprehended hypothetically in thought even before the particular instances are examined. When we set out, for instance, to investigate whether the line joining the bisecting points of the sides of a triangle is parallel with the base, we have accepted hypothetically the general principle that such lines are parallel with the base. The fact is, therefore, that when the mind examines the particular case and finds it to agree with the hypothesis, so far as it accepts this case as a truth, it also accepts it as a universal truth. Although, therefore, induction may involve going from one particular experiment or observation to another, it is in a sense a process of going from the general to the general.
That accepting the truth of a particular judgment may imply a universal judgment is very evident in the case of geometrical demonstrations. When it is shown, for instance, that in the case of the particular isosceles triangle ABC, the angles at the base are equal, the mind does not require to examine other particular triangles for verification, but at once asserts that in every isosceles triangle the angles at the base are equal.
Induction and Conception Interrelated.—Although as a process, induction is to be distinguished from conception, it either leads to an enriching of some concept, or may in fact be the only means by which certain scientific concepts are formed. While the images obtained by ordinary sense perception will enable a child to gain a notion of water, to add to the notion the property, boiling-at-a-certain-temperature, or able-to-be-converted-into-two-parts-hydrogen-and-one-part-oxygen, will demand a process of induction. The development of such scientific notions as oxide, equation, predicate adjective, etc., is also dependent upon a regular inductive process. For this reason many lessons may be viewed both as conceptual and as inductive lessons. To teach the adverb implies a conceptual process, because the child must synthesise certain attributes into his notion adverb. It is also an inductive lesson, because these attributes being formulated as definite judgments are, therefore, obtained inductively. The double character of such a lesson is fully indicated by the two results obtained. The lesson ends with the acquisition of a new term, adverb, which represents the result of the conceptual process. It also ends with the definition: "An adverb is a word which modifies a verb, adjective, or other adverb," which indicates the general truth or truths resulting from the inductive process.
Deduction and Induction Interrelated.—In our actual teaching processes there is a very close inter-relation between the two processes of reasoning. We have already noted on page [322] that, in such inductive lessons as teaching the definition of a noun or the rule for the addition of fractions, both the preparatory step and the application involve deduction. It is to be noted further, however, that even in the development of an inductive lesson there is a continual interplay between induction and deduction. This will be readily seen in the case of a pupil seeking to discover the rule for determining the number of repeaters in the addition of recurring decimals. When he notes that adding three numbers with one, one, and two repeaters respectively, gives him two repeaters in his answer, he is more than likely to infer that the rule is to have in the answer the highest number found among the addenda. So far as he makes this inference, he undoubtedly will apply it in interpreting the next problem, and if the next numbers have one, one, and three repeaters respectively, he will likely be quite convinced that his former inference is correct. When, however, he meets a question with one, two, and three repeaters respectively, he finds his former inference is incorrect, and may, thereupon, draw a new inference, which he will now proceed to apply to further examples. The general fact to be noted here, however, is that, so far as the mind during the examination of the particular examples reaches any conclusion in an inductive lesson, it evidently applies this conclusion to some degree in the study of the further examples, or thinks deductively, even during the inductive process.
Development of Reasoning Power.—Since reasoning is essentially a purposive form of thinking, it is evident that any reasoning process will depend largely upon the presence of some problem which shall stimulate the mind to seek out relations necessary to its solution. Power to reason, therefore, is conditioned by the ability to attend voluntarily to the problem and discover the necessary relations. It is further evident that the accuracy of any reasoning process must be dependent upon the accuracy of the judgments upon which the conclusions are based. But these judgments in turn depend for their accuracy upon the accuracy of the concepts involved. Correct reasoning, therefore, must depend largely upon the accuracy of our concepts, or, in other words, upon the old knowledge at our command. On the other hand, however, it has been seen that both deductive and inductive reasoning follow to some degree a systematic form. For this reason it may be assumed that the practice of these forms should have some effect in giving control of the processes. The child, for instance, who habituates himself to such thought processes as AB equals BC, and AC equals BC, therefore AB equals AC, no doubt becomes able thereby to grasp such relations more easily. Granting so much, however, it is still evident that close attention to, and accurate knowledge of, the various terms involved in the reasoning process is the sure foundation of correct reasoning.