Induction

We have already defined induction as the process by which our minds proceed from a sufficient number of instances to a universal truth. This passage from the less universal, or particular, to the more universal is called the inductive ascent. Induction, then, proceeds from I to A. Now when we studied oppositional inference we were told that it was illicit to go from I to A. Notice, though, that we were then treating of formal inference, whereas induction is material inference. In induction we proceed from I to A by reason of the special character of the matter, or thought content, and not by reason of the form, or structure, of our argument.

The induction that we are chiefly concerned with begins with a knowledge of concrete individual material beings, which we know through sense experience. From these concrete individuals, we rise by this kind of induction to a universal truth. For instance, from the fact that this piece of copper conducts electricity and that pieces two, three, and four also conduct it, we might infer (rightly or wrongly) that all copper conducts electricity. But induction can also begin with universal truths and proceed to still more universal truths, as when we proceed from what is true of various species to a statement about the genus that these species belong to. For instance, from the fact that copper, iron, silver, and gold, which are species of the genus “metal,” conduct electricity, we might infer (rightly or wrongly) that all metal conducts electricity.

We must distinguish (a) between incomplete induction, which some logicians misleadingly call imperfect induction, and complete, or perfect, induction and (b) between intellective and rational induction. These two divisions overlap—that is, both incomplete and complete induction can be either intellective or rational.

First we shall briefly explain the distinction between incomplete and complete induction. Then, because of their very great importance, we shall treat of intellective and rational induction under separate headings. Finally we shall treat very briefly of the argument from analogy and make some further comments on induction.

Incomplete, or imperfect, induction proceeds from what is known of individual subjects having a nature to an assertion about a nature as such. It proceeds from I to A—from a limited number of instances to a universal statement—as when we infer that all copper conducts electricity because pieces one, two, three, and four conduct it.

We made extensive use of incomplete induction when we established the validity or invalidity of various logical forms. For instance, by insight into a single example (“Every dog is an animal; therefore some animal is a dog”) we clearly understand the validity of the logical form “Every S is a P; therefore some P is an S. Like¬ wise by insight into a single example ( Every dog is an animal; therefore every animal is a dog”) we clearly understand the invalidity of the logical form “Every S is a P; therefore every P is an S.

When we speak of induction without qualification, we generally have in mind incomplete induction.

We shall now explain the nature of complete induction. According to certain logicians who misconstrue its nature, it consists in affirming something of all the individuals of a class, one by one, and then affirming it of the entire class. Suppose, for instance, that you want to find out whether all in this room are wearing shoes. You examine every individual and find out that he is wearing shoes. And then, from your assertions about each individual—from your assertions, that is, that Peter is wearing shoes, that John is wearing shoes, that Mary is wearing shoes, and so on—you proceed to an assertion about the whole class; namely, that all in this room are wearing shoes. Notice that the statement that all in this room are wearing shoes merely summarizes what has already been said and involves no advance in knowledge. It is a mere enumerative universal—a. mere statement of fact—and does not even suggest that to wear shoes is a necessary attribute of everyone in this room or that being in this room is the reason why its occupants are wearing shoes. We mention this as an example of what induction is not.

Complete induction, rightly understood, consists in proceeding from what is true of each species of a genus to an assertion about the genus itself, as when we assert that copper, iron, silver, gold, and so on, and so on, conduct electricity and therefore all metal (not “all metals”) conducts it. Notice that this is a true inference and that there is a true advance in knowledge (supposing, of course, that the inference is made correctly). To know that metal as such conducts electricity is more perfect knowledge than to know that various kinds of metal conduct it without also knowing that this is due to their generic nature as metals.

We also made extensive use of complete induction in establishing the validity or invalidity of various logical forms. Recall, for instance, how we showed that the mood I-E of a categorical syllogism is always invalid because it always involves an illicit process of the major term. First we showed by incomplete induction that I-E is invalid in each of the four figures. We clearly understood how these four figures include all possible arrangements of the syllogistic terms and that therefore I-E is always invalid. We proceeded from what is true of each of the four species of the genus “categorical syllogism in the mood I-E” to an assertion about the genus “categorical syllogism in the mood I-E” itself.

We shall lead up to a definition of intellective induction by analyzing a few examples and reflecting on how our minds proceed in regard to each of them

a. Some Examples

First we shall examine two contingent propositions from which it is impossible for us to ascend to a universal truth. Then we shall examine a proposition from which we shall immediately see that we can ascend to a universal truth. While examining these propositions, we shall not pay attention to their form, but to the reality that they present to the mind.

1) “THIS HOUSE IS RED.” Consider the following propositions, and suppose that what is asserted in them is true.

• 1. This house is red.

  2. John is running down the street.

When you consider these propositions, you see no necessary connection between the subject and predicate. Of course, if the house is red, it cannot not be red in the respect in which it is red; and if John is running down the street, he cannot at the same time not be running down the street. But this is the only kind of necessity present; except on the supposition that the house is actually red or that John is actually running down the street, it is equally possible for the house not to be red or for John not to be running down the street. The actualities understood in these judgments are contingent; hence, it is impossible to ascend from them to universal statements about house as such or man as such. In the following example, however, from a consideration of a single instance, our minds spontaneously ascend to a universal truth.

2) ‘THIS WHOLE IS GREATER THAN THIS PART.” The entire rectangle represents a whole card; the dotted lines mark off a part of the whole card. On seeing the whole card and the part, we know what is expressed in the proposition,

This whole card is greater than this part.

Insofar as it would be possible for this card and this part not to exist at all, this too is a contingent proposition. Yet there is an element of necessity in this proposition that is not found in the two previous examples. On looking at this card and its part, we clearly understand that the whole card is not greater than this part only as a matter of fact, but that it must be greater and cannot be otherwise. In other words, we grasp the intelligibility of the fact that this whole card is greater than this part and clearly perceive the intrinsic reason why it is impossible for the whole card not to be greater than its part. When we see a red house, we see nothing in the nature of the house requiring that it be red; and when we see a man running down the street, we see nothing in the nature of the man requiring that he run down the street. But on seeing this whole and this part, we clearly understand that the very nature of this whole and this part requires that the whole be greater than the part. Moreover, we know and clearly understand that the reason why this whole is greater than this part is not that it is this whole and this part in these particular circumstances, but simply that it is a whole and a part. Once we have grasped this necessary relationship of a whole to its parts, our minds spontaneously pass from this concrete instance to the universal truth,

A whole is greater than any of its parts.

By insight into the particular example we know that this must be true. We cannot withhold our assent to the proposition “A whole is greater than any of its parts” because we clearly understand that it cannot be otherwise—we grasp the intelligible, necessary relationship between a whole and its parts so completely that we know with absolute certainty that, if a whole and a part exist at all, the whole must be greater than a part.

b. Definition of Intellective Induction

Our definition of intellective induction is nothing but a description of what we do when, on looking at the whole card and the part of the card, we pass from the proposition “This whole card is greater than this part” to the proposition “A whole as such is greater than any of its parts.”

Intellective induction, then, is the process whereby our minds rise from a consideration of particular cases to a universal truth because we understand through insight into the particular case that the universal is necessarily true.

This definition will be understood better when, after illustrating, defining, and explaining rational induction, we make a detailed comparison of intellective and rational induction.

The rules of formal inference are established exclusively by intellective induction. In explaining these rules we always begin with examples whose validity or invalidity is obvious and then through insight into these examples we draw up our universal rules. For instance, to show the invalidity of the simple conversion of an A proposition we use an example like Every dog is an animal; therefore every animal is a dog. The mere fact that the antecedent is true but the consequent (or pseudo consequent) is false shows that this logical form is invalid and then through insight into the quantitative relationship of the subject and predicate as illustrated in the terms “dog” and “animal” we clearly see the reason for the invalidity.

First we shall re-examine the first two examples given in the last section and compare them with a third example. Then we shall give a definition of rational induction. Finally we shall compare intellective and rational induction in such a way as to throw further light on the nature of each of them.

a. Some Examples

Consider the following propositions, and suppose that what is asserted in them is true. Notice, again, that we are not now concerned with the form of these propositions, but with the reality that they present to the mind.

• 1. This house is red.

  2. John is running down the street.

  3. The apple, unsupported, falls toward the earth.

If we look at this third example in itself as a single concrete statement of fact and without reference to other knowledge or other instances, we see no necessities other than those seen in the first two propositions. But when we multiply instances in our experience—for example, the wind blows a tile loose and it falls, the man steps out of the window and he falls, the plate slips and it falls, ripe fruit breaks from the branch and it falls—when we multiply such instances, we perceive a similarity among the instances and, with or without accuracy, we generalize in some such fashion as follows:

Unsupported things fall toward the earth.

In such a generalization we do not see the intelligibility of the fact any more clearly than in a single instance. Still, we are convinced that within the complexus of concrete realities there must be some factor, or combination of factors, that renders it necessary for unsupported things to fall toward the earth. In other words, we are sure that the only sufficient reason for the constancy with which unsupported things fall toward the earth is the presence of some necessity by reason of which they must fall toward the earth.

Such a generalization involves an implicit deduction, since the conclusion (“All unsupported things fall towards the earth”) can be drawn only on the assumption that determined effects require a determined and intelligible ground.

b. Definition of Rational Induction

Our definition of rational induction is a description of what we do when, after experiencing, for instance, that countless things fall toward the earth when unsupported, we make a generalization such as “Unsupported things fall toward the earth.”

Rational induction, then, is the process whereby our minds rise from a consideration of particular cases to a universal judgment because we know, or at least have reason to think, that the judgment is necessary, although we do not see the reason for this necessity.

c. Comparison of Intellective and Rational Induction

A comparison of intellective and rational induction will throw further light on the nature of each of them.

The most basic difference between intellective and rational induction is this: in intellective induction, while considering the particular instance, we see and understand the intrinsic necessity, and therefore the intelligibility, of the universal proposition, whereas in rational induction we do not see this intrinsic necessity, but are induced to admit the presence of some kind of necessity as the only sufficient reason for the constancy of the effects we have observed.

This difference implies two other differences. Intellective induction depends on no previous judgments and gives absolute certainty, because in it we see and clearly understand the full intelligibility of a truth. But rational induction rests, at least implicitly, on previous judgments (such as the principle of sufficient reason or intelligibility, the principle of causality, the principle of uniformity of nature, and so on) and does not by itself give absolute certainty.

There is a fourth difference between intellective and rational induction. In intellective induction a generalization can be made, in some cases at least, from a single instance. A multiplicity of instances may often be necessary, but only accidentally, for it serves only to direct our attention to the proper intelligibility of some complex situation and to stimulate our minds to proceed to generalization. But once our attention has been directed to the proper intelligibility of a concrete situation, there is no need whatsoever for further instances, since the generalization is implicit in the knowledge of each single instance. In rational induction, on the other hand, the multiplicity of instances is usually a formal part of the evidence—until a stage is reached (if it ever is reached) in which the intelligibility itself is displayed.

The aim of science is always full intelligibility; therefore the first type of induction (induction by insight, or intellective induction) remains the ideal. It sometimes happens in the course of scientific observation or experiment that a truth which was previously known only by rational induction suddenly becomes subject to immediate insight. At this point our assent no longer depends on previous observation and deduction except accidentally. In itself, the insight is immediate, and the previous process (observation, experiment, and so on) has merely displayed the factors involved so that the insight could be gained in immediate understanding.

We have examples of such development in astronomy. For instance, the first investigations of eclipses and other celestial phenomena interrelated certain factors with their occurrence; but when the structure and motion of the heavens was sufficiently understood, it became immediately evident why the occurrence of certain phenomena was necessarily related to the occurrence of an eclipse. The validity of the law formulating the functional relationship between these phenomena and the occurrence of eclipses no longer depended on the previous rational induction but was now self-evident.

In this section we shall treat very briefly of the argument from analogy. First we shall define, or rather describe, it; then we shall call attention to its use and limitations; and finally we shall give two rules or cautions regarding the use of arguments from analogy.

a. Definition

The argument from analogy is a probable argument based on a resemblance. Suppose that X is known to resemble Y in the attributes a, b, c, d, and e; suppose, too, that X is also known to have the attribute f. If on the basis of Y’s similarity to X in the attributes a, b, c, d, and e, we argue that, since X has f, Y most likely also has f, we are using an argument from analogy. For instance, we might argue as follows: Deer are similar to cows, goats, and sheep in that they have horns and are cloven-footed and chew cud. Cows, goats', and sheep have stomachs with many chambers. Therefore, it seems, deer must have stomachs with many chambers.

The argument from analogy proceeds from one or more particular instances through an unexpressed universal to another particular instance that is similar to the former but not (logically) identical with them. The passage from the one or more particular instances to an unexpressed universal involves an implicit induction. The application of this universal to another instance involves an implicit deduction. If it is expressed completely, the argument given above will be stated as follows:

Cows, goats, and sheep have horns and are cloven-footed and chew cud.

But cows, goats, and sheep have stomachs with many chambers.

(Therefore, it seems, animals that have horns and are cloven-footed and chew cud have stomachs with many chambers.)

But deer have horns and are cloven-footed and chew cud.

Therefore, it seems, deer must have stomachs with many chambers.

From the particular instances of the cows, goats, and sheep we pass by induction to the implicit universal proposition stated in the parentheses; and then by an implicit deduction we apply this universal proposition to the particular case of the deer.

b. Use and Limitations

Arguments from analogy are of very great importance both in the practical concerns of every-day life and in scientific investigation. Think, for instance, of the advances that the science of medicine has made by experimenting on mice, guinea pigs, dogs, and so on, and then applying the lessons of these experiments to man on the basis of man’s similarity to these animals. The historian s attempts to interpret present events in the light of the past are arguments from analogy. The housewife uses analogy when she argues that cooking herring in vinegar will improve its flavor because this method of cooking improves the flavor of smelt. In every-day life we continually solve our problems by reflecting on what we ourselves, or others, have done in situations similar to our present circumstances.

In itself, analogy does not lead to certainty but merely points the way to probable answers to our problems or suggests the direction that our investigations might take. What cured a guinea pig, for instance, might cure a man; but we will ordinarily not know for certain whether or not it will cure a man until we have tried it on a man—or maybe on many men. Unless arguments from analogy are used with extreme caution, they are as likely to lead to error as to truth.

c. Rules or Cautions

In the first place, if an argument from analogy is to be legitimate, the resemblance on which it is based must be significant; that is, there must be good reason to think that there is a necessary connection between the attributes in which two subjects are similar and the attribute we wish to predicate of the one subject because of its resemblances to the other. From the fact that John and James are of equal weight and height and of similar build, we can legitimately infer that they have about equal strength. All these factors have significance in relation to strength. But from the equality of their weight and height and the similarity of their builds we cannot legitimately infer that they are of equal intelligence, because these factors have no significance whatsoever in regard to intelligence.

Once we are certain that there is a causal connection between the attributes in which two subjects are similar and the attribute we wish to predicate of the second subject because of its similarity to the first, our argument ceases to be a mere probable argument from analogy and becomes a perfect syllogism that leads to a certain conclusion. This takes place whenever the implicit universal proposition is a necessary proposition.

Secondly, when we use arguments from analogy we must take into account important differences. As an example of how this caution is sometimes ignored we cite the tendency of certain evolutionists to concentrate solely on man’s somatic resemblances to the other animals and to ignore the many ways in which he differs from them.

In this section we shall indicate the extent to which the study of induction belongs to logic and explain why there are no rules for induction as there are for deduction.

a. The Extent to which Induction Belongs to Logic

To what extent does the study of induction belong to logic? To answer this question we must anticipate what we shall explain at length in Chapter 16, where we shall define logic in terms of its formal object. Logic, as we shall see in Chapter 16 (and as we have already indicated briefly in Chapter 1), is not a science of real beings but of second intentions. The scope of logic is therefore limited to what we can know by reflecting on our knowledge and on the attributes and relationships that things have as they exist in the mind and that they get as a result of being thought of. Now, when we reflect on our knowledge of various kinds of things, we advert to the fact that we know different things in different ways—and we discover the types of induction that we have just described. At the same time we clearly understand that their validity (unlike the validity of syllogisms) depends on the matter under consideration and not at all on the structure, or form, of our propositions and arguments.

Now, all that logic can do about the various kinds of induction is to acknowledge their existence, describe their nature, and confess its inability to make rules to regulate them.

b. No Rules for the Inductive Ascent

There are no rules governing induction as the rules of the syllogism, and so on, govern deduction. The reason for this is that inductive arguments are not reducible to logical forms that are valid regardless of their matter, but depend for their validity on the special character of the matter under consideration.

Directives on scientific investigation tell us how to conduct experiments, how to test hypotheses, and so on, but are not rules for making the inductive ascent itself. Besides, such directives lie outside the scope of logic inasmuch as they are instructions on the handling, not of second intentions, but of real beings.