The Theory of Induction

Thilly, Frank. “The Theory of Induction.” The Philosophical Review 12, no. 4 (1903): 401.

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. 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 induction with scientific method in general, including under it the forming of hypotheses, deducing their consequences, and verifying them.

The second question also receives various answers. According to some thinkers, only so-called perfect induction is certain; 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 differently by 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 saying 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 aa postulate of our thinking, by others as the product of experience.

Let us now attempt to answer the first question: What is the nature of induction? Induction is a process of inference. We must be careful to distinguish between inference and association of ideas. The perception of fire may arouse in the child's consciousness 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. Ianthe 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 established between them." There are two kinds of reasoning, deduction, and induction. Both are processes of inference, and therefore essentially the same, that is, both consciously relate with other judgments. In both cases a certain judgment 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 difference 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 proposition, that is, we start from a universal proposition and draw from its other propositions according to the principle of identity. "In induction, then, we conclude that all A is B, because we have observed that a1 and a2 (all essentially alike and capable of being 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 reasoning inductively. In induction we leap from a particular case or cases tall; 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 inductive 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 consists 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. Wherever we infer a universal statement from a particular case or cases, leap from the particular to the universal, we have induction. We do not strive to know merely the causes of things; we are interested in other relations also, for instance 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 infrequences and co-existences as causally related, then we cannot define induction as the quest for causal relations; for, as was already said, we are interested in all kinds of regularity or orderliness. 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 relations existing between things. In a certain sense, this is true. The thinker is always eager to find out what qualities are connected 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 between 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 satisfy 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, by showing that it is only one case of the general tendency of heat to expand substances. The crack is caused by the successful effort of the heated glass to expand in spite of the colder glass with 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 between objects than the other, and therefore impresses us as necessary. He seeks for a simple formula under which he can embrace a great many cases that seem to have nothing at all in common. "Suppose someone 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 assumption that every one of these three phenomena are a cause of the increase 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 flames increased, something was added to the fire which, according to analysis, contains oxygen, he will reduce the manifold experiences to the simple formula: All things which contain oxygen and are added to fire increase the flame. He will probably go farther and say: Oxygen is the cause of the increase of the flame."

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 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 animals 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 an 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 observation, but we can give no other reason for their being true, that is, we cannot bring them into harmony with any other scientific 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 scientific 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 circumstances because heat expands substances. If heat expands substances, it must expand glass; and if the colder parts of the glass 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 define induction as forming hypotheses, drawing their consequences, 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 induction and deduction. If induction is both induction and deduction, 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 general, 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 induction itself? The logical thing to do is to restrict the term induction 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 induction; (2) Deduction, or, as Mr. Mill calls it, ratiocination; (3) 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. Assuming them as provisionally true, we then proceed to argue to their effects in other cases, and a further appeal to experience either verifies or negates the truth of the laws assumed."

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 bathe 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 coexistence 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 aground 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, wave 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 particular as a type, as having universal significance. It is frequently hasty, and its results are frequently discovered to be false, but that does not affect its nature. The point to be emphasized here is that induction consists in making the leap spoken of, regardless 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 instance will repeat itself.

This brings us to our second fundamental question: 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 inductions? (b) How can we prove induction in general?

(a) Certainty is a feeling. We feel certain that a proposition is true; the proposition is certain because it arouses in us the feeling of certainty. What must we do to reach such certainty 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 when 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 inductive methods is to bring this certainty to the highest possible degree. 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 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 certainty for our particular inductions when we subsume them under 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 thousand. 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 b, I do not need any other cases to convince me that there is a universal relation. I conclude from one to many, because I have already assumed uniformity by assuming causality. Similarly, wherever I conceive of phenomena as necessarily related 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 triangle, 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 universal validity. I reason: If a and b are causally related, then when a appears b will appear also. Now a and b are causally related. Hence, when a appears, b will appear also. This is a 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 derived inductively? In deduction we consciously draw a proposition from premises in which it is already implied; we explicate it. Here our conclusion will give us a feeling of absolute certainty, that is, we will feel that if the premises are true, the conclusion must be true, unless we have made a mistake in our reasoning. It is not difficult to construct a syllogism in which the inductive proposition forms the conclusion. For example, if it is true that nature is uniform, that nature repeats itself, that it is a rule of law, then we have a proof for induction. One should remember, however, that this does not make induction deduction. Inductions 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 propositions not induction, but deduction. Here the certainty of the proof will, as always, depend upon the certainty 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 like them. The only problem here is to discover the particular combinations, theca-existences, and sequences in nature.

But the question at once arises: What warrant have we for saying that nature is uniform? It may perhaps be said that there is a postulate of thought, and that it carries its warranting itself. We cannot prove its truth, but we. Feel certain that it is true; we accept it without cavil. But is it really a postulate 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 unified 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 reflection upon our experiences. We have noticed many particular uniformities; we conclude that nature at large is uniform, that is, we consciously ground our proposition upon our past experiences. In this sense, the principle of uniformity is an induction: 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 warranted 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 uniformities and conclude from these that there is general uniformity.

We may, however, mean by the principle of uniformity of nature as a postulate of thought, not a clear conviction that nature as a whole is a unified system, subject to law, but the feeling 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 apostolate 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 qualities 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 qualities will recur than the feeling of expectation that it will do so.

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