The Problem of Induction

Hume himself does not use the word "induction". But what has come to be called "the problem of induction" comes down to us from him. What follows is not a detailed analysis of Hume's text. I will sometimes speak of Hume's problem, Hume's argument and so on; but you should not assume that the details of my reconstruction are explicitly present in Hume. I encourage you to pay careful attention to what Hume actually says. It very often happens that the "received version" of a philosophical problem -- the version that is passed along in introductory courses such as this one -- is very different from its historical model, and in some cases less interesting. I hope you will take the time to ask whether the problem I outline below really is the problem that Hume meant to point out.

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Two Problems

Hume's problem is usefully divided in two. There is first what I shall call the descriptive problem: How do human beings form opinions about unobserved matters fact? And then there is the normative problem: Are beliefs formed in this way justified? Does someone who "reasons" as we normally do really have reason to believe his conclusions about the parts of nature he has not observed?


A Formulation of the Descriptive Problem

Let's suppose, for now, that perception and memory are clearly sources of knowledge. If you see an apple on the table, you know that there is an apple on the table; If you saw an apple on the table yesterday and you now remember that you saw it, then you know now that yesterday there was an apple on the table. We will raise questions about these matters later on. But let us set them aside for now in order to focus instead on our knowledge of things we have not observed.

Some of this knowledge is trivial. I have not examined every triangle in the universe. But I know in advance that each of them has three sides. I have not examined every bachelor in the universe. But I know in advance that none of them is married. These propositions concern what Hume calls relations of ideas.

A proposition expresses of relation of ideas if and only if its denial is strictly impossible, inconceivable, or self-contradictory. (Hume seems to regard these notions as equivalent.)

For Hume there is no mystery as to how I can know such propositions a priori. I simply consider the proposition, attempt to imagine its falsity, and notice that I am immediately involved in a contradiction or some similar incoherence. This provides one straightforward route to knowledge about things we have not observed. But this is not the knowledge that interests us here.

We are rather concerned with knowledge of unobserved matters of fact.

A matter of fact proposition has the following feature: both it and its denial are fully conceivable, possible, and non-self-contradictory.

Consider the proposition that my house is blue. You can easily conceive of this proposition's being true; but you can just as easily conceive of its falsity. Neither proposition is incoherent or impossible. So both count as claims about matters of fact.

Hume's Descriptive Problem may then be formulated as follows:

How do human beings arrive at their opinions concerning unobserved matters of fact?

There is no doubt that we have a multitude of opinions of this sort, and that practical life would be impossible without them. To take only the most obvious class of examples: every substantive claim about the future fall into this category. But if you had no opinions at all about the future, you would be paralyzed. You might be hungry; but you would have no idea whatsoever where the food was. You might be cold, but you would have no idea what would happen if you turned up the thermostat or put on an extra sweater. There is no doubt that we have opinions about unobserved mattters of fact, and no doubt that we should be grateful for having. The descriptive question is, How do we arrive at them?


Hume's Solution to the Descriptive Problem.

Hume's first claim is negative: Knowledge of unobserved matters fact cannot be derived a priori. Rather it must somehow result from experience. Imagine an adult human being who has neither seen snow nor heard stories about it. He is shown a snowball for the very first time and asked to predict -- before he has touched it -- whether it will be hot or cold. We all spontaneously predict that it will be cold, and we're right. But all he can do is consider the possibilities. He can conceive that it will be cold; he can conceive that it will be hot. Neither supposition involves any kind of internal contradiction, and so long as he is not allowed to investigate the matter, he cannot rule either proposition out of consideration. So a priori -- before he has made any relevant observations -- he has no grounds for an opinion. The difference between us and him is not a difference in intellectual power. He can reason as well as we can. It is rather a difference in experience. And this seems to be completely general. We do not yet know how our experience is relevant to our prediction. But that it is somehow relevant is obvious. On this basis Hume asserts a general proposition:

Opinions about unobserved matters of fact are somehow derived from experience.

But how exactly does the derivation work? In the case of snow, the answer seems straightforward. In our experience the visual appearance of snow has been regularly and invariably associated with a sensation of cold. We have noticed, in other words, that

In our experience thus far, snow has always been cold.

On this basis we have concluded that

In general, snow is always cold,

or at least

The next piece of snow that I examine will be cold.

And this appears to be a general pattern. When we have noticed that

In our experience, all Fs are G

we tend to conclude that

In general, all Fs are G, or at least, the next F I consider will be G.

This is the general pattern of inductive inference or induction. Hume's answer to the Descriptive Problem may then be framed as follows:

All beliefs about unobserved matters of fact are derived from experience by induction.

Before we proceed you should ask yourself: Is it really true that all of my beliefs about the future and the other unobserved parts of nature have been derived in this way from experience? Can you think of a counter-instance?


The Normative Problem

Suppose that Hume is right about how we actually think. So far all we have is a fact about human cognitive psychology. And this fact, however interesting, does not settle the normative question: Is it legitimate for us to proceed in this way? Are the conclusions we reach as a result of inductive inference really justified?

A first pass suggests a negative answer. After all, the inference pattern

(DATA) In my experience, all Fs are Gs
(THEORY) Therefore, in general all Fs are Gs, (or at least, the next F I examine will be G).

is not deductively valid. It is logically possible for the conclusion to be false when the premise is true. So a skeptic might say: In so-called inductive reasoning, human beings commit a fallacy. They accept a general proposition on the basis of an invalid argument. And this means that their acceptance of that general proposition is unjustified.

Now this is not exactly Hume's way of raising skeptical worries. Hume rather takes the invalidity of the inference from DATA to THEORY as evidence that we have failed to make our method fully explicit. That we unheasitatingly pass from DATA to THEORY shows that we accept a principle connecting the two, a principle that normally passes unnoticed because we take it so completely for granted, but which figures implicitly in every instance of inductive reasoning.

Hume formulates this missing premise as the claim that the future will resemble the past. But for our purposes it will be useful to work with a somewhat more precise formulation. What we need to make the inverence from DATA to THEORY valid is a premise of the form:

(UN) For the most part, if a regularity R (e.g., All Fs are Gs) holds in my experience, then it holds in nature generally, or at least in the next instance.

"UN" stands for the "Uniformity of Nature". This is a traditional (post-Humean) label for the missing premise, though in fact it is misleading. For UN is not simply the claim that nature exhibits regularities. It is the claim that the regularities that have emerged in my experience are among the regularities that hold throughout nature. It might better be called a principle or representativeness, for its central message is that my experience, though limited in time and space to a tiny fraction of the universe, is nonetheless a representative sample of the universe.

The inference from DATA + UN to THEORY is valid. Moreover, there is no question for now about our right to accept the DATA. So if we want to know whether we ever have a right to accept a generalization like THEORY, we must ask whether we have reason to believe UN.


The skeptical problem: We have no good reason to accept UN.

This brings us to the heart of the matter. What reason do we have to believe that our experience is a representative sample of nature? What reason is there to believe UN?

Hume argues, in effect, that there can be no good answers to these questions.

(A) UN itself expresses a matter of fact proposition. Its denial is perfectly conceivable. Given any regularity that holds in our experience so far, we can easily conceive that it will be violated tomorrow. So UN is not a priori. If it is known at all, it is known on the basis of experience.
(B) UN is a claim about unobserved matters of fact. It is a claim, in part, about the future. Hence we cannot know it directly on the basis of observation and experience.
(C) But all "knowledge" of unobserved matters of fact is known (if it is known at all) on the basis of an inductive inference. So if there were any reason to believe UN, it would have to take the form of an inductive argument.
(D) But there can be no convincing inductive argument for UN. UN figures as a premise in any inductive argument. An inductive argument for the principle itself would thus be patently circular.
(E) So there can be no non-circular argument for UN.


I take it that (A) and (B) are straightforward. So if there are questions about this line of thought, they will be questions about (C) and (D). What Hume is most concerned to point out is that one obvious, natural and attractive way of defending UN cannot work in the end. I have in mind the so-called inductive defense of induction: the attempt to establish UN by citing our past record of successful predictions. After all, you might think that our track record of commonsensical and scientific success provides us with ample reason to believe that our experience is normally a pretty reliable guide to what we have not yet experienced. If we had never had much luck in predicting the future, we would have no right to believe UN. But we have had an immense amount of luck of this sort. We tend to remember the failures. But the successes are vastly more numerous. Every time you successfully "predict" that the floor will support you, that your food will not poison you, that your friends will not try to kill you, you acquire, it would seem, more and more evidence for your general reliability as a predictor of the future.

But this brilliant track record provides no grounds whatsoever for accepting UN. How would the argument run?

DATA: In the past, regularities that have held in my experience have been found to hold in nature generally, or at least in the next instance.
THEORY (=UN): Therefore in general, if a regularity holds in my experience, it holds in nature generally, or at least in the next instance.

But now we notice: As it stands this argument is invalid; the premise does not imply the conclusion. Moreover the only way to make it valid is to supply UN itself as an additional premise. But that would make the argument circular.

The conclusion seems inescapable: Every inductive argument employs UN as a premise, so no inductive argument can ever justify UN.


The Cautious Martian and the Counterinductivist

In order to make Hume's point more intuitive, I want to imagine an encounter with two useful aliens. These aliens are adult human beings endowed with a language like ours, sensory capacities like ours, and logical powers every bit as good as ours if not better. They can think, speak and record their experiences just as we do. But they make different assumptions about the representativeness of their experiences.

We accept UN, the claim that our experience is a representative sample of the natural world.


The Cautious Martian has no opinion at all about how representative his experience is.

The Counterinductivist holds (somewhat perversely, we think) that his experience is always misleading. That is, he accepts:

(NN) If a regularity R holds in my experience so far, it does not hold in nature generally, not even in the next instance.

Now imagine that you and the aliens share a common body of experience. They have seen exactly what you have seen, touched exactly what you have touched, and so on. So there is complete agreement about which regularities hold in your collective experience thus far. For instance, you all agree that

(DATA) In our extensive experience thus far, snow has always been cold.

You take this to support the general proposition that

(THEORY) Snow is always cold, or at least it will be the next time I come across it.

That is, you spontaneously infer THEORY from DATA, and you assume that this is reasonable. You report this to the Cautious Martian, and he says:

I agree with you about DATA, but THEORY strikes me as a wild leap. THEORY is consistent with DATA; but so are lots of other generalizations. It could be that there are two sorts of snow, the cold kind and the warm kind, but that the warm kind only exists on mars. It could be that snow is cold in some geological periods and warm in others, and that our experience is wholly confined to the first sort of epoch. You rule out all of these real possibilities on the basis of the DATA. But that strikes me as totally unwarranted.

The Counterinductivist is even more critical of your reasoning.

I agree with you about DATA, but it seems to me that, far from supporting THEORY, the DATA actually refute it! I accept as a general rule that regularities that show up in my experience are always falsified elsewhere in nature, indeed in the very next instance. So I reason as follows:
(DATA) In our extensive experience thus far, snow has always been cold.
(NN) If a regularity holds in my experience thus far, it does not hold in nature generally, not even in the next instance.
(THEORY*) Therefore, snow is not always cold, and won't be the next time I come across it.

Now we are all inclined to believe that someone who has seen lots of cold snow and no snow of any other sort would be crazy not to expect that the snow he has not seen is also cold. The Cautious Martian strikes us as irrationally cautious, and the Counterinductivist strikes us as simply perverse -- mad, insane. But what exactly is the difference between us and them? We accept UN or something like it. The Cautious Martian accepts no principle of this sort, and the Counterinductivist accepts an alternative principle. If we alone are being reasonable here, then it ought to be possible for us to display our reasons: to give an argument of some sort for accepting UN over NN and over nothing at all. So let's see the argument. Let's try to persuade the Cautious Martian and the Counterindctivist to accept our prediction.

It should be obvious that simply citing the record of our past success will not work. The Cautious Man agrees that in the past we inductivists have been successful in the past. But that is the past, and his caution consists precisely in his failure to see pas regularities as any sort of guide to the future. The Counterinductivist also agrees that we have been successful in the past; but he takes this as positive evidence that we will fail in the future, and indeed in the next instance. We can give a circular, question-begging argument for our principle, UN. But the Counterinductivist can give a similarly circular argument for NN. The Cautious Martian points out that neither of these arguments is any good, since both are circular. So appeal to past success is no way to break the stalemate.

Hume's Skeptical Solution to the Doubts

You should try to produce arguments to break this symmetry: reasons for accepting UN that might move the Counterinductivist and the Cautious Martian over to our side. But Hume is convinced that this cannot be done. So he rejects the possibility of a straight solution to his skeptical problem. And yet he does not conclude that we should reject inductive reasoning or the principle of the uniformity of nature. He does not urge any change in our ordinary practice of forming expectations about the future based on experience. His claims for this practice and the principle that underlies it are rather as follows:

    1. Our habits of inductive inference and our acceptance of UN are non-rational, in the sense that no positive compelling reason can be provided for doing things our way rather than, say, the counterinductivist's way. But our acceptance of UN is not irrational; it is not contrary to reason. The principle is not logically inconsistent or incoherent, and no positive argument can be raised against it.
    2. Our acceptance of UN is not optional. It is, in Hume's phrase, a matter of custom or habit; but it might better be called a matter of instinct. We do not reason our way to the principle: we do not accept it on the basis of arguments. Rather, to accept the principle is a natural feature of all human and indeed all animal life. Hume does not pretend to know how we came to accept the principle. It might be a trace of divine beneficence, or it might have any of a number of other causes. But wherever it comes from, it is so deeply engrained in us that we have no real choice about whether to accept it. We can temporarily suspend our intellectual assent to the proposition. But nature will soon reassert itself in us and force these doubts from our mind. As Hume famously writes in his first book, the Treatise of Human Nature:

Most fortunately it happens that, since reason is incapable of dispelling these clouds, nature herself suffices to that purpose, and cures me of this philosophical melancholy and delirium, either by relaxing this bent of mind, or by some avocation and lively impression of my senses which obliterate all these chimeras. I dine, I play a game of back-gammon, I converse and I am merry with my friends; and when after three or four hours' amusement, I would return to these speculations, they appear so cold and strain'd, and ridiculous, that I cannot find in my heart to enter into them any further. (Treatise, Book I, Part IV, Section VII, p. 269 in the Selby-Bigge edition)

Hume's "solution" to the normative problem of induction is then this. When we are confronted with a natural belief -- a belief that comes to us as naturally and as inevitably as breathing -- we are rationally justified in acquiescing in it (provided it is not contrary to reason). A belief can be rational in this sense even though we cannot supply any convincing arguments for it, and this is a very important suggestion. The Cautious Martian and the Counterinductivist are different from us; they do not reason as we do, but for all we have said we may all be equally rational. Their ways of proceeding may be as natural and inevitable for them as ours is for us. What this implies is that what we are rationally entitled to believe depends on what sort of beings we are, and not just on the available evidence and argument.

We can think of Hume here as asking the question James asks in "The Will to Believe": When is it reasonable to believe what we cannot prove? Hume's answer is apparently much more restrictive than James's. For James, within this domain, we are free to believe whatever we like. For Hume, by contrast, we are entitled to believe only what it is natural in some basic biological sense for us to believe, or perhaps only what we literally cannot help believing. (These may be different notions.) The difference comes out in their respective attitudes towards religious belief. James's essay is a defense of religious faith. Hume's attitude, expressed in part in his "Dialogues Concerning Natural Religion", seems to have been that religious belief is not natural in the sense that our commitment to UN is natural, nor is it strictly inevitable for us. So it does not meet Hume's conditions for rational belief in the absence of evidence. Indeed, Hume gives every impression of regarding positive religious belief as unreasonable.

If we accept the analysis of inductive reasoning sketched above, it may seem that Hume as done something remarkable and disturbing. He has shown that from a strictly intellectual point of view, there is no real difference between common sense and science on the one hand, and religious belief on the other. In all three cases we find a system of belief based on a fundamental conviction that cannot be justified by argument. The most dramatic way to put the point is to say that Hume has shown that common sense and science are matters of faith. Hume would resist this attempt to rehabilitate religion by "softening up" our picture of common sense and science. The faith that Hume defends is a faith that we cannot possibly avoid or resist, a faith that renders skeptical doubt utterly idle. The religious case is very different, at least on the face of it. What we shall have to ask, as we proceed, is whether this difference really makes a difference.