Induction

In almost all our previous discussions we have been concerned in the attempt to get clear as to our data in the way of knowledge of existence. What things are there in the universe whose existence is known to us owing to our being acquainted with them? So far, our answer has been that we are acquainted with our sense-data, and, probably, with ourselves. These we know to exist. And past sense-data which are remembered are known to have existed in the past. This knowledge supplies our data.

In almost all our earlier discussions, we have tried to clarify the evidence for our knowledge of what exists. What things in the universe do we know exist because we are directly acquainted with them? So far, our answer is that we are acquainted with our sense-data and probably with ourselves. We know these exist. We also know that past sense-data we remember existed in the past. This knowledge provides our evidence.

But if we are to be able to draw inferences from these data—if we are to know of the existence of matter, of other people, of the past before our individual memory begins, or of the future, we must know general principles of some kind by means of which such inferences can be drawn. It must be known to us that the existence of some one sort of thing, A, is a sign of the existence of some other sort of thing, B, either at the same time as A or at some earlier or later time, as, for example, thunder is a sign of the earlier existence of lightning. If this were not known to us, we could never extend our knowledge beyond the sphere of our private experience; and this sphere, as we have seen, is exceedingly limited. The question we have now to consider is whether such an extension is possible, and if so, how it is effected.

But to draw conclusions from these data, we must know some general principles that allow us to do so. We need these principles to know that matter and other people exist. We also need them to know about the past before our own memories begin, or about the future. We must know that the existence of one kind of thing, A, signals the existence of another kind, B. B may exist at the same time as A, or at an earlier or later time. For example, thunder is a sign that lightning occurred earlier. Without knowing such principles, we could never extend our knowledge beyond our own private experience. And as we have seen, our private experience is extremely limited. We must now consider whether we can extend our knowledge in this way and, if so, how.

Let us take as an illustration a matter about which none of us, in fact, feel the slightest doubt. We are all convinced that the sun will rise to-morrow. Why? Is this belief a mere blind outcome of past experience, or can it be justified as a reasonable belief? It is not easy to find a test by which to judge whether a belief of this kind is reasonable or not, but we can at least ascertain what sort of general beliefs would suffice, if true, to justify the judgement that the sun will rise to-morrow, and the many other similar judgements upon which our actions are based.

Let us consider something that none of us, in fact, doubts in the slightest. We are all convinced that the sun will rise tomorrow. Why? Is this belief simply a blind result of past experience, or can we justify it as reasonable? Finding a way to judge whether such a belief is reasonable is not easy. But we can at least identify the general beliefs that, if true, would justify our belief about tomorrow’s sunrise. Those general beliefs would also justify the many similar judgments on which we base our actions.

It is obvious that if we are asked why we believe that the sun will rise to-morrow, we shall naturally answer 'Because it always has risen every day'. We have a firm belief that it will rise in the future, because it has risen in the past. If we are challenged as to why we believe that it will continue to rise as heretofore, we may appeal to the laws of motion: the earth, we shall say, is a freely rotating body, and such bodies do not cease to rotate unless something interferes from outside, and there is nothing outside to interfere with the earth between now and to-morrow. Of course it might be doubted whether we are quite certain that there is nothing outside to interfere, but this is not the interesting doubt. The interesting doubt is as to whether the laws of motion will remain in operation until to-morrow. If this doubt is raised, we find ourselves in the same position as when the doubt about the sunrise was first raised.

If asked why we believe the sun will rise tomorrow, we will obviously give the natural answer: 'Because it has always risen every day'. We firmly believe the sun will rise in the future because it has risen in the past. If asked why we believe the sun will keep rising as before, we may appeal to the laws of motion. The earth, we will say, is a freely rotating body. Such bodies do not stop rotating unless something outside them interferes. And nothing outside the earth will interfere with its rotation between now and tomorrow. Of course, we might question whether we are quite certain that nothing outside the earth will interfere. But that is not the interesting doubt. The interesting doubt is whether the laws of motion will continue to apply until tomorrow. If that question is raised, we are back where we were when we first questioned whether the sun would rise.

The only reason for believing that the laws of motion will remain in operation is that they have operated hitherto, so far as our knowledge of the past enables us to judge. It is true that we have a greater body of evidence from the past in favour of the laws of motion than we have in favour of the sunrise, because the sunrise is merely a particular case of fulfilment of the laws of motion, and there are countless other particular cases. But the real question is: Do any number of cases of a law being fulfilled in the past afford evidence that it will be fulfilled in the future? If not, it becomes plain that we have no ground whatever for expecting the sun to rise to-morrow, or for expecting the bread we shall eat at our next meal not to poison us, or for any of the other scarcely conscious expectations that control our daily lives. It is to be observed that all such expectations are only probable; thus we have not to seek for a proof that they must be fulfilled, but only for some reason in favour of the view that they are likely to be fulfilled.

The only reason to believe the laws of motion will keep working is that they have worked so far. At least, that is what our knowledge of the past tells us. It is true that past evidence for the laws of motion is stronger than past evidence for the sunrise. This is because the sunrise is just one example of the laws of motion holding true. There are countless other examples. But the real question is: Can any number of past examples of a law holding true provide evidence that it will hold true in the future? If not, we have no reason at all to expect the sun to rise tomorrow. Nor can we expect the bread at our next meal not to poison us. The same goes for all the other expectations that guide our daily lives, often without our noticing. We should note that all such expectations are only probable. So we do not need proof that these expectations must come true. We need only some reason to think these expectations are likely to come true.

Now in dealing with this question we must, to begin with, make an important distinction, without which we should soon become involved in hopeless confusions. Experience has shown us that, hitherto, the frequent repetition of some uniform succession or coexistence has been a cause of our expecting the same succession or coexistence on the next occasion. Food that has a certain appearance generally has a certain taste, and it is a severe shock to our expectations when the familiar appearance is found to be associated with an unusual taste. Things which we see become associated, by habit, with certain tactile sensations which we expect if we touch them; one of the horrors of a ghost (in many ghost-stories) is that it fails to give us any sensations of touch. Uneducated people who go abroad for the first time are so surprised as to be incredulous when they find their native language not understood.

To address this question, we must first make an important distinction. Without it, we would soon become hopelessly confused. Experience has shown us that repeated patterns have so far been a cause of our expectations. When events frequently happen in the same order or occur together, we expect the pattern to repeat next time. Food with a certain appearance generally has a certain taste. When a familiar-looking food has an unusual taste, the surprise is a severe shock. Through habit, we associate things we see with certain sensations of touch. We expect those sensations if we touch those things. One of the horrors of a ghost (in many ghost stories) is that we cannot feel it by touch. Uneducated people going abroad for the first time are surprised when others do not understand their native language. Their surprise is so great that they can hardly believe it.

And this kind of association is not confined to men; in animals also it is very strong. A horse which has been often driven along a certain road resists the attempt to drive him in a different direction. Domestic animals expect food when they see the person who usually feeds them. We know that all these rather crude expectations of uniformity are liable to be misleading. The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.

And this kind of association is not limited to humans. It is also very strong in animals. A horse often driven along a particular road resists attempts to lead it in a different direction. Domestic animals expect food when they see the person who usually feeds them. We know that all these rather crude expectations of consistent patterns can be misleading. The man who has fed a chicken every day of its life finally wrings its neck instead. This shows that a more refined understanding of nature’s consistency would have been useful to the chicken.

But in spite of the misleadingness of such expectations, they nevertheless exist. The mere fact that something has happened a certain number of times causes animals and men to expect that it will happen again. Thus our instincts certainly cause us to believe that the sun will rise to-morrow, but we may be in no better a position than the chicken which unexpectedly has its neck wrung. We have therefore to distinguish the fact that past uniformities cause expectations as to the future, from the question whether there is any reasonable ground for giving weight to such expectations after the question of their validity has been raised.

But even though such expectations are misleading, they still exist. Simply because something has happened a certain number of times, animals and humans expect it to happen again. So our instincts certainly lead us to believe that the sun will rise tomorrow. But we may be no better off than the chicken whose neck is unexpectedly wrung. We must therefore distinguish two things. One is the fact that past patterns cause expectations about the future. The other is whether we have any reasonable grounds for trusting those expectations once their validity has been questioned.

The problem we have to discuss is whether there is any reason for believing in what is called 'the uniformity of nature'. The belief in the uniformity of nature is the belief that everything that has happened or will happen is an instance of some general law to which there are no exceptions. The crude expectations which we have been considering are all subject to exceptions, and therefore liable to disappoint those who entertain them. But science habitually assumes, at least as a working hypothesis, that general rules which have exceptions can be replaced by general rules which have no exceptions. 'Unsupported bodies in air fall' is a general rule to which balloons and aeroplanes are exceptions. But the laws of motion and the law of gravitation, which account for the fact that most bodies fall, also account for the fact that balloons and aeroplanes can rise; thus the laws of motion and the law of gravitation are not subject to these exceptions.

We must discuss whether there is any reason to believe in what is called 'the uniformity of nature'. This means believing that everything that has happened or will happen follows some general law with no exceptions. The rough expectations we have been considering all have exceptions. They can therefore disappoint those who hold them. But science usually assumes, at least as a working hypothesis, that rules with exceptions can be replaced by rules without exceptions. 'Unsupported objects in the air fall' is a general rule with exceptions, such as balloons and airplanes. But the laws of motion and gravity explain both why most objects fall and why balloons and airplanes can rise. So these cases are not exceptions to the laws of motion and gravity.

The belief that the sun will rise to-morrow might be falsified if the earth came suddenly into contact with a large body which destroyed its rotation; but the laws of motion and the law of gravitation would not be infringed by such an event. The business of science is to find uniformities, such as the laws of motion and the law of gravitation, to which, so far as our experience extends, there are no exceptions. In this search science has been remarkably successful, and it may be conceded that such uniformities have held hitherto. This brings us back to the question: Have we any reason, assuming that they have always held in the past, to suppose that they will hold in the future?

The belief that the sun will rise tomorrow might prove false if Earth suddenly collided with a large body. Such a collision might stop Earth’s rotation. But such an event would not violate the laws of motion or the law of gravitation. The task of science is to find consistent patterns, such as the laws of motion and the law of gravitation. As far as our experience goes, these patterns have no exceptions. Science has been remarkably successful in this search. We may accept that these patterns have held until now. But we return to the question: Assuming these patterns have always held in the past, do we have any reason to expect them to hold in the future?

It has been argued that we have reason to know that the future will resemble the past, because what was the future has constantly become the past, and has always been found to resemble the past, so that we really have experience of the future, namely of times which were formerly future, which we may call past futures. But such an argument really begs the very question at issue. We have experience of past futures, but not of future futures, and the question is: Will future futures resemble past futures? This question is not to be answered by an argument which starts from past futures alone. We have therefore still to seek for some principle which shall enable us to know that the future will follow the same laws as the past.

Some have argued that we have reason to know the future will resemble the past. Their reason is that what was once future has continually become past and has always resembled the past. So, they argue, we really do have experience of the future: times that were once future. We may call these times past futures. But this argument assumes the very point it is meant to establish. We have experience of past futures, but not of future futures. The question is: Will future futures resemble past futures? An argument based only on past futures cannot answer this question. So we must still look for a principle that lets us know the future will follow the same laws as the past.

The reference to the future in this question is not essential. The same question arises when we apply the laws that work in our experience to past things of which we have no experience—as, for example, in geology, or in theories as to the origin of the Solar System. The question we really have to ask is: 'When two things have been found to be often associated, and no instance is known of the one occurring without the other, does the occurrence of one of the two, in a fresh instance, give any good ground for expecting the other?' On our answer to this question must depend the validity of the whole of our expectations as to the future, the whole of the results obtained by induction, and in fact practically all the beliefs upon which our daily life is based.

The reference to the future in this question is not essential. The same question arises when we apply laws that hold in our experience to past events we have not experienced. Examples include geology and theories about the origin of the Solar System. The question we really need to ask is: 'Suppose two things have often been found together, and neither has ever been known to occur without the other. If one occurs in a new case, does that give us good reason to expect the other?' Our answer determines whether all our expectations about the future and all the results of induction are justified. In fact, practically all the beliefs on which our daily life depends rest on our answer.

It must be conceded, to begin with, that the fact that two things have been found often together and never apart does not, by itself, suffice to prove demonstratively that they will be found together in the next case we examine. The most we can hope is that the oftener things are found together, the more probable it becomes that they will be found together another time, and that, if they have been found together often enough, the probability will amount almost to certainty. It can never quite reach certainty, because we know that in spite of frequent repetitions there sometimes is a failure at the last, as in the case of the chicken whose neck is wrung. Thus probability is all we ought to seek.

We must first admit that two things may have been found together often and never apart. But that fact alone cannot prove that the two will be found together in the next case we examine. At most, we can hope that finding things together more often makes finding them together again more likely. And if the things have been found together often enough, the probability will come almost to certainty. The probability can never quite reach certainty, because even after many repetitions, the pattern sometimes fails in the end. The chicken whose neck is wrung is an example of such a failure. So probability is all we should seek.

It might be urged, as against the view we are advocating, that we know all natural phenomena to be subject to the reign of law, and that sometimes, on the basis of observation, we can see that only one law can possibly fit the facts of the case. Now to this view there are two answers. The first is that, even if some law which has no exceptions applies to our case, we can never, in practice, be sure that we have discovered that law and not one to which there are exceptions. The second is that the reign of law would seem to be itself only probable, and that our belief that it will hold in the future, or in unexamined cases in the past, is itself based upon the very principle we are examining.

Someone might object to our view by saying that we know all natural phenomena follow laws. They might add that observation sometimes shows that only one law can possibly fit the facts. There are two answers to this objection. First, even if some law without exceptions applies to our case, we can never be sure in practice that we have found it. We may instead have found a law with exceptions. Second, the idea that all natural phenomena follow laws would itself seem only probable. We believe that laws will continue to hold in the future, or held in past cases we have not examined. But this belief itself rests on the very principle we are examining.

The principle we are examining may be called the principle of induction, and its two parts may be stated as follows:

The principle we are examining may be called the principle of induction. Its two parts may be stated as follows:

(a) When a thing of a certain sort A has been found to be associated with a thing of a certain other sort B, and has never been found dissociated from a thing of the sort B, the greater the number of cases in which A and B have been associated, the greater is the probability that they will be associated in a fresh case in which one of them is known to be present;

(a) Suppose things of type A have been found together with things of type B. Suppose also that A has never been found without B. The more cases in which A and B have been found together, the more likely they are to occur together in a new case where either is known to be present.

(b) Under the same circumstances, a sufficient number of cases of association will make the probability of a fresh association nearly a certainty, and will make it approach certainty without limit.

(b) Under the same circumstances, enough cases of association will make another association almost certain. And as the number of cases increases, the probability of another association will approach certainty without limit.

As just stated, the principle applies only to the verification of our expectation in a single fresh instance. But we want also to know that there is a probability in favour of the general law that things of the sort A are always associated with things of the sort B, provided a sufficient number of cases of association are known, and no cases of failure of association are known. The probability of the general law is obviously less than the probability of the particular case, since if the general law is true, the particular case must also be true, whereas the particular case may be true without the general law being true. Nevertheless the probability of the general law is increased by repetitions, just as the probability of the particular case is. We may therefore repeat the two parts of our principle as regards the general law, thus:

As just stated, the principle applies only to confirming our expectation in a single new case. But we also want to know whether the general law is probably true, given enough supporting cases and no exceptions. The general law is that things of type A are always associated with things of type B. This general law is clearly less probable than the particular case. If the general law is true, the particular case must also be true. But the particular case may be true without the general law being true. Nevertheless, repeated observations increase the probability of the general law, just as they increase the probability of the particular case. We can therefore restate the two parts of our principle for the general law as follows:

(a) The greater the number of cases in which a thing of the sort A has been found associated with a thing of the sort B, the more probable it is (if no cases of failure of association are known) that A is always associated with B;

(a) The more often things of type A have been found associated with things of type B, the more likely A is always associated with B (if no exceptions are known).

b) Under the same circumstances, a sufficient number of cases of the association of A with B will make it nearly certain that A is always associated with B, and will make this general law approach certainty without limit.

b) Under the same circumstances, enough cases of A being associated with B will make the following general law nearly certain: A is always associated with B. As the number of cases increases, the certainty of this law will come closer to absolute certainty, without limit.

It should be noted that probability is always relative to certain data. In our case, the data are merely the known cases of coexistence of A and B. There may be other data, which might be taken into account, which would gravely alter the probability. For example, a man who had seen a great many white swans might argue, by our principle, that on the data it was probable that all swans were white, and this might be a perfectly sound argument. The argument is not disproved by the fact that some swans are black, because a thing may very well happen in spite of the fact that some data render it improbable. In the case of the swans, a man might know that colour is a very variable characteristic in many species of animals, and that, therefore, an induction as to colour is peculiarly liable to error. But this knowledge would be a fresh datum, by no means proving that the probability relatively to our previous data had been wrongly estimated. The fact, therefore, that things often fail to fulfil our expectations is no evidence that our expectations will not probably be fulfilled in a given case or a given class of cases. Thus our inductive principle is at any rate not capable of being disproved by an appeal to experience.

Probability is always relative to particular evidence. In our case, the evidence consists only of known cases where A and B occur together. There may be other evidence we might consider that would greatly change the probability. For example, a man who had seen many white swans might reason according to our principle. He might conclude that, given his evidence, all swans were probably white. His argument might be perfectly sound. The fact that some swans are black does not disprove the argument. Something may well happen even though some evidence makes it unlikely. In the swan example, a man might know that colour varies greatly in many animal species. He might therefore know that an inductive conclusion about colour is especially likely to be wrong. But this knowledge would be new evidence. It would by no means prove that the probability based on the earlier evidence had been wrongly estimated. So the fact that events often fail to meet our expectations does not establish that those expectations are unlikely to be met. Our expectations may still probably be met in a particular case or class of cases. Thus, at least, our inductive principle cannot be disproved by appealing to experience.

The inductive principle, however, is equally incapable of being proved by an appeal to experience. Experience might conceivably confirm the inductive principle as regards the cases that have been already examined; but as regards unexamined cases, it is the inductive principle alone that can justify any inference from what has been examined to what has not been examined. All arguments which, on the basis of experience, argue as to the future or the unexperienced parts of the past or present, assume the inductive principle; hence we can never use experience to prove the inductive principle without begging the question. Thus we must either accept the inductive principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the future. If the principle is unsound, we have no reason to expect the sun to rise to-morrow, to expect bread to be more nourishing than a stone, or to expect that if we throw ourselves off the roof we shall fall. When we see what looks like our best friend approaching us, we shall have no reason to suppose that his body is not inhabited by the mind of our worst enemy or of some total stranger. All our conduct is based upon associations which have worked in the past, and which we therefore regard as likely to work in the future; and this likelihood is dependent for its validity upon the inductive principle.

The inductive principle, however, is equally impossible to prove by appealing to experience. Experience might confirm the inductive principle for cases we have already examined. But only the inductive principle can justify drawing conclusions about unexamined cases from cases we have examined. All arguments based on experience assume the inductive principle when drawing conclusions about the future. The same applies to conclusions about parts of the past or present we have not experienced. So we can never use experience to prove the inductive principle without assuming what we are trying to prove. We must therefore either accept the inductive principle as self-evident or give up all justification for our expectations about the future. If the principle is unsound, we have no reason to expect the sun to rise tomorrow. Nor can we expect bread to be more nourishing than a stone, or expect to fall if we jump off a roof. When someone who looks like our best friend approaches, we have no reason to assume whose mind inhabits his body. It could be the mind of our worst enemy or a total stranger rather than our friend's. All our actions are based on connections that have held in the past. We therefore regard those connections as likely to hold in the future. But the validity of that likelihood depends on the inductive principle.

The general principles of science, such as the belief in the reign of law, and the belief that every event must have a cause, are as completely dependent upon the inductive principle as are the beliefs of daily life All such general principles are believed because mankind have found innumerable instances of their truth and no instances of their falsehood. But this affords no evidence for their truth in the future, unless the inductive principle is assumed.

General scientific principles depend just as fully on the inductive principle as everyday beliefs do. Such principles include the belief that laws govern nature and that every event must have a cause. People believe all such general principles because they have found countless cases where these principles hold and none where they fail. But these cases provide no evidence that the principles will hold in the future unless the inductive principle is assumed.

Thus all knowledge which, on a basis of experience tells us something about what is not experienced, is based upon a belief which experience can neither confirm nor confute, yet which, at least in its more concrete applications, appears to be as firmly rooted in us as many of the facts of experience. The existence and justification of such beliefs—for the inductive principle, as we shall see, is not the only example—raises some of the most difficult and most debated problems of philosophy. We will, in the next chapter, consider briefly what may be said to account for such knowledge, and what is its scope and its degree of certainty.

Thus, all knowledge that uses experience to tell us about things we have not experienced rests on a belief. Experience can neither confirm nor disprove that belief. Yet, at least in its more concrete applications, that belief seems as firmly rooted in us as many experienced facts. The existence and justification of such beliefs raise some of philosophy’s most difficult and most debated problems. As we shall see, the principle of induction is not the only example of such beliefs. In the next chapter, we will briefly consider how such knowledge might be explained. We will also consider its scope and how certain it is.