THE DEFINITION

It has been noted that, when man discovers common characteristics in a number of objects, he tends on this basis to unite such objects into a cla.s.s. It is to be noted in addition, however, that in the same manner he is also able, by examining the characteristics of a large cla.s.s of objects, to divide these into smaller sub-cla.s.ses. Although, for example, we may place all three-sided figures into one cla.s.s and call them triangles, we are further able to divide these into three sub-cla.s.ses owing to certain differences that may be noted among them.

Thus an important fact regarding cla.s.sification is that while a cla.s.s may possess some common quality or qualities, yet its members may be further divided into sub-cla.s.ses and each of these smaller cla.s.ses distinguished from the others by points of difference. Owing to this fact, there are two important elements entering into a scientific knowledge of any cla.s.s, first, to know of what larger cla.s.s it forms a part, and secondly, to know what characteristics distinguish it from the other cla.s.ses which go with it to make up this larger cla.s.s. To know the cla.s.s equilateral triangle, for instance, we must know, first, that it belongs to the larger cla.s.s triangle, and secondly, that it differs from other cla.s.ses of triangles by having its three sides equal. For this reason a person is able to know a cla.s.s scientifically without knowing all of its common characteristics. For instance, the large cla.s.s of objects known as words is subdivided into smaller cla.s.ses known as parts of speech. Taking one of these cla.s.ses, the verb, we find that all verbs agree in possessing at least three common characteristics, they have power to a.s.sert, to denote manner, and to express time. To distinguish the verb, however, it is necessary to note only that it is a word used to a.s.sert, since this is the only characteristic which distinguishes it from the other cla.s.ses of words. When, therefore, we describe any cla.s.s of objects by first naming the larger cla.s.s to which it belongs, and then stating the characteristics which distinguish it from the other co-ordinate cla.s.ses, we are said to give a definition of the cla.s.s, or to define it. The statement, "A trimeter is a verse of three measures,"

is a definition because it gives, first, the larger cla.s.s (verse) to which the trimeters belong, and secondly, the difference (of three measures) which distinguishes the trimeter from all other verses. The statement, "A binomial is an algebraic expression consisting of two terms," is a definition, because it gives, first, the larger cla.s.s (algebraic expression) to which binomials belong, and secondly, the difference (consisting of two terms) which distinguishes binomials from other algebraic expressions.

JUDGMENT

=Nature of Judgment.=--A second form, or mode, of thinking is known as judgment. Our different concepts were seen to vary in their intension, or meaning, according to the number of attributes suggested by each. My notion _triangle_ may denote the attributes three-sided and three-angled; my notion _isosceles triangle_ will in that case include at least these two qualities plus equality of two of the sides. This indicates that various relations exist between our ideas and may be apprehended by the mind. When a relation between two concepts is distinctly apprehended in thought, or, in other words, when there is a mental a.s.sertion of a union between two ideas, or objects of thought, the process is known as _judgment_. Judgment may be defined, therefore, as the apprehension, or mental affirmation, of a relation between two ideas. If the idea, or concept, _heaviness_ enters as a mental element into my idea _stone_, then the mind is able to affirm a relation between these concepts in the form, "Stone is heavy." In like manner when the mind a.s.serts, "Gla.s.s is transparent" or "Horses are animals," there is a distinct apprehension of a relation between the concepts involved.

=Judgment Distinguished from Statement.=--It should be noted that judgment is the mental apprehension of a relation between ideas. When this relation is expressed in actual words, it is spoken of as a proposition, or a predication. A proposition is, therefore, the statement of a judgment. The proposition is composed of two terms and the copula, one term const.i.tuting the subject of the proposition and the other the predicate. Although a judgment may often be expressed in some other form, it can usually be converted into the above form. The proposition, "Horses eat oats," may be expressed in the form, "Horses are oat-eaters"; the proposition, "The sun melts the snow," into the form, "The sun is a-thing-which-melts-snow."

=Relation of Judgment to Conception.=--It would appear from the above examples that a judgment expresses in an explicit form the relations involved within the concept, and is, therefore, merely a direct way of indicating the state of development of any idea. If my concept of a dog, for example, is a synthesis of the qualities four-footed, hairy, fierce, and barking, then an a.n.a.lysis of the concept will furnish the following judgments:

{ A four-footed thing.

{ A hairy thing.

A dog is { A fierce thing.

{ A barking thing.

Because in these cases a concept seems necessary for an act of judgment, it is said that judgment is a more advanced form of thinking than conception. On the other hand, however, judgment is implied in the formation of a concept. When the child apprehends the dog as a four-footed object, his mind has grasped four-footedness as a quality pertaining to the strange object, and has, in a sense, brought the two ideas into relation. But while judgment is implied in the formation of the concept, the concept does not bring explicitly to the mind the judgments it implies. The concept snow, for instance, implies the property of whiteness, but whiteness must be apprehended as a distinct idea and related mentally with the idea snow before we can be said to have formed, or thought, the judgment, "Snow is white." Judgment is a form of thinking separate from conception, therefore, because it does thus bring into definite relief relations only implied in our general notions, or concepts. One value of judgment is, in fact, that it enables us to a.n.a.lyse our concepts, and thus note more explicitly the relations included in them.

=Universal and Particular Judgments.=--Judgments are found to differ also as to the universality of their affirmation. In such a judgment as "Man is mortal," since mortality is viewed as a quality always joined to manhood, the affirmation is accepted as a universal judgment. In such a judgment as "Men strive to subdue the air," the two objects of thought are not considered as always and necessarily joined together. The judgment is therefore particular in character. All of our laws of nature, as "Air has weight," "Pressure on liquids is transmitted in every direction," or "Heat is conducted by metals," are accepted as universal judgments.

=Errors in Judgment due to: A. Faulty Concepts.=--It may be seen from the foregoing that our judgments, when explicitly grasped by the mind and predicated in language, reflect the accuracy or inaccuracy of our concepts. Whatever relations are, as it were, wrapped up in a concept may merge at any time in the form of explicit judgments. If the fact that the only Chinamen seen by a child are engaged in laundry work causes this attribute to enter into his concept Chinaman, this will lead him to affirm that the restaurant keeper, Wan Lee, is a laundry-man. The republican who finds two or three cases of corruption among democrats, may conceive corruption as a quality common to democrats and affirm that honest John Smith is corrupt. Faulty concepts, therefore, are very likely to lead to faulty judgments. A first duty in education is evidently to see that children are forming correct cla.s.s concepts. For this it must be seen that they always distinguish the essential features of the cla.s.s of objects they are studying. They must learn, also, not to conclude on account of superficial likeness that really unlike objects belong to the same cla.s.s. The child, for instance, in parsing the sentence, "The swing broke down," must be taught to look for essential characteristics, and not call the word _swing_ a gerund because it ends in "ing"; which, though a common characteristic of gerunds, does not differentiate it from other cla.s.ses of words. So, also, when the young nature student notes that the head of the spider is somewhat separated from the abdomen, he must not falsely conclude that the spider belongs to the cla.s.s insects. In like manner, the pupil must not imagine, on account of superficial differences, that objects really the same belong to different cla.s.ses, as for example, that a certain object is not a fish, but a bird, because it is flying through the air; or that a whale is a fish and not an animal, because it lives in water. The pupil must also learn to distinguish carefully between the particular and universal judgment. To affirm that "Men strive to subdue the air," does not imply that "John Smith strives to subdue the air." The importance of this distinction will be considered more fully in our next section.

=B. Feeling.=--Faulty concepts are not, however, the only causes for wrong judgments. It has been noted already that feeling enters largely as a factor in our conscious life. Man, therefore, in forming his judgments, is always in danger of being swayed by his feelings. Our likes and dislikes, in other words, interfere with our thinking, and prevent us from a.n.a.lysing our knowledge as we should. Instead, therefore, of striving to develop true concepts concerning men and events and basing our judgments upon these, we are inclined in many cases to allow our judgments to be swayed by mere feeling.

=C. Laziness.=--Indifference is likewise a common source of faulty judgments. To attend to the concept and discover its intension as a means for correct judgment evidently demands mental effort. Many people, however, prefer either to jump at conclusions or let others do their judging for them.

=Sound Judgments Based on Scientific Concepts.=--To be able to form correct judgments regarding the members of any cla.s.s, however, the child should know, not only its common characteristics, but also the essential features which distinguish its members from those of co-ordinate cla.s.ses. To know adequately the equilateral triangle, for instance, the pupil must know both the features which distinguish it from other triangles and also those in which it agrees with all triangles. To know fully the mentha family of plants, he must know both the characteristic qualities of the family and also those of the larger genus l.a.b.i.atae.

From this it will be seen that a large share of school work must be devoted to building up scientific cla.s.s notions in the minds of the pupils. Without this, many of their judgments must necessarily be faulty. To form such scientific concepts, however, it is necessary to relate one concept with another in more indirect ways than is done through the formation of judgments. This brings us to a consideration of _reasoning_, the third and last form of thinking.

REASONING

=Nature of Reasoning.=--Reasoning is defined as a mental process in which the mind arrives at a new judgment by comparing other judgments.

The mind, for instance, is in possession of the two judgments, "Stones are heavy" and "Flint is a stone." By bringing these two judgments under the eye of attention and comparing them, the mind is able to arrive at the new judgment, "Flint is heavy." Here the new judgment, expressing a relation between the notions, _flint_ and _heavy_, is supposed to be arrived at, neither by direct experience, nor by an immediate a.n.a.lysis of the concept _flint_, but more indirectly by comparing the other judgments. The judgment, or conclusion, is said, therefore, to be arrived at mediately, or by a process of reasoning. Reasoning is of two forms, deductive, or syllogistic, reasoning, and inductive reasoning.

DEDUCTION

=Nature of Deduction.=--In deduction the mind is said to start with a general truth, or judgment, and by a process of reasoning to arrive at a more particular truth, or judgment, thus:

Stone is heavy; Flint is a stone; .'. Flint is heavy.

Expressed in this form, the reasoning process, as already mentioned, is known as a syllogism. The whole syllogism is made up of three parts, major premise, minor premise, and conclusion. The three concepts involved in the syllogism are known as the major, the minor, and the middle term. In the above syllogism, _heavy_, the predicate of the major premise, is the major term; _flint_, the subject of the minor premise, is the minor term; and _stone_, to which the other two are related in the premises, is known as the middle term. Because of this previous comparison of the major and the minor terms with the middle term, deduction is sometimes said to be a process by which the mind discovers a relation between two concepts by comparing them each with a third concept.

=Purpose of Deduction.=--It is to be noted, however, as pointed out in Chapter XV, that deductive reasoning takes place normally only when the mind is faced with a difficulty which demands solution. Take the case of the boy and his lost coin referred to in Chapter II. As he faces the problem, different methods of solution may present themselves. It may enter his mind, for instance, to tear up the grate, but this is rejected on account of possible damage to the brickwork. Finally he thinks of the tar and resorts to this method of recovery. In both of the above cases the boy based his conclusions upon known principles. As he considered the question of tearing up the grate, the thought came to his mind, "Lifting-a-grate is a-thing-which-may-cause-damage." As he considered the use of the tar, he had in mind the judgment, "Adhesion is a property of tar," and at once inferred that tar would solve his problem. In such practical cases, however, the mind seems to go directly from the problem in hand to a conclusion by means of a general principle. When a woman wishes to remove a stain, she at once says, "Gasoline will remove it."

Here the mind, in arriving at its conclusion, seems to apply the principle, "Gasoline removes spots," directly to the particular problem. Thus the reasoning might seem to run as follows:

Problem: What will remove this stain?

Principle: Gasoline will remove stains.

Conclusion: Gasoline will remove this stain.

Here the middle term of the syllogism seems to disappear. It is to be noted, however, that our thought changes from the universal idea "stains," mentioned in the statement of the principle, to the particular idea "this stain" mentioned in the problem and in the conclusion. But this implies a middle term, which could be expressed thus:

Gasoline will remove stains; This is a stain; .'. Gasoline will remove _this_.

The syllogism is valuable, therefore, because it displays fully and clearly each element in the reasoning process, and thus a.s.sures the validity of the conclusion.

=Deduction in School Recitation.=--It will be recalled from what was noted in our study of general method, that deduction usually plays an important part during an ordinary developing lesson. In the step of preparation, when the pupil is given a particular example in order to recall old knowledge, the example suggests a problem which is intended to call up certain principles which are designed to be used during the presentation. In a lesson on the "Conjunctive p.r.o.noun," for instance, if we have the pupil recall his knowledge of the conjunction by examining the particular word "if" in such a sentence as, "I shall go if they come," he interprets the word as a conjunction simply because he possesses a general rule applicable to it, or is able to go through a process of deduction. In the presentation also, when the pupil is called on to examine the word _who_ in such a sentence as, "The man who met us is very old," and decides that it is both a conjunction and a p.r.o.noun, he is again making deductions, since it is by his general knowledge of conjunctions and p.r.o.nouns that he is able to interpret the two functions of the particular word _who_. Finally, as already noted, the application of an ordinary recitation frequently involves deductive processes.

INDUCTION

=Nature of Induction.=--Induction is described as a process of reasoning in which the mind arrives at a conclusion by an examination of particular cases, or judgments. A further distinguishing feature of the inductive process is that, while the known judgments are particular in character, the conclusion is accepted as a general law, or truth. As in deduction, the reasoning process arises on account of some difficulty, or problem, presented to the mind, as for example:

What is the effect of heat upon air?

Will gla.s.s conduct electricity?

Why do certain bodies refract light?

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 gla.s.s filled with water, and on inverting it find that the card is held to the gla.s.s. Taking a gla.s.s 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 circ.u.mstances the mind comes to the following conclusions:

Air pressure holds this card to the gla.s.s, 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 const.i.tutes 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 a.s.serted that air holds the card to the gla.s.s, 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 gla.s.s, is itself a deduction resting upon other general principles. Secondly, if the judgment that air presses the card against the gla.s.s 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 a.s.serts 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 a.s.sumed 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.

CHAPTER XXIX