Last modified by Malcolm R. Forster on December 1, 1998

Consider any ampliative inference of the form (called simple enumerative induction):

Billiard ball 1 moved when struck at time 1.

Billiard ball 2 moves when struck at time 2.

Billiard ball 100 moves when struck at time 100.

Therefore, all billiard balls move when struck.

Since it is logically possible for the conclusion to be false while past evidence is true, the evidence does not conclusively establish the truth of the prediction. That is to say, the evidence (stated in the premises) fails to determine the conclusion. That is, the evidence logically underdetermines the conclusion. For example, the theory that "Except for those billiard balls observed so far, no billiard balls move when struck" is equally consistent with the observed facts.

Note: Humean underdetermination does not imply that the method of simple enumerative induction fails to reach a unique conclusion. The evidence does not underdetermine the theory in this sense.

You are hungry and you are about the bite into a hot crusty baguette. But a ‘friend’ stops you and says "Don’t do it. That piece of bread will poison you." This is absurd of course. But what do you say in reply? You would probably reply that "Bread that smells this good has not poisoned me in the past, so it will not poison me now." We tend to count such arguments as inductively strong. Moreover, the premise is known to be true. However, no inductively strong argument guarantees the truth of its conclusion. It is possible that this piece will poison you even though bread has not poisoned you in the past. So, you weaken your conclusion, and say "It is highly probable that this piece of bread will not poison me." As a statement of psychological conviction, it is true. But is it rationally justifiable in any objective sense?

Definition: Let us say that a form of inductive argument is reliable if it yields approximately true conclusions most of the time.

Hume’s Question: Is there a form of inductive reasoning, a recipe, or a method of inference, that we can show will lead to true, or approximately true, conclusions most of the time?

Clarification: Hume’s question is not whether there are any reliable inductive arguments. Our question is whether there is any form of inductive argument that we can show to be reliable.

Note: The form of inductive reasoning must be such that we can recognize when we are using it.

Definition: Consider any form of inductive argument which satisfies whatever recognizable criteria you think will guarantee that inductive reasoning of this kind is reliable. Call these i-arguments.

• Recognizable characteristics of an inductive argument might include things like fit between the conclusion and past data, and the simplicity and unification of conclusion.
• We cannot define i-arguments to be reliable. For reliability refers to the past, present, and future, so it is not a recognizable characteristic. So, i-arguments are the arguments that we believe to be reliable.

Note: According to Hume, induction is rationally justifiable if and only if there exist i-arguments that can be shown to be reliable.

• For induction to be rationally justified, it is not enough to show that i-arguments have been reliable in the past. They must be shown to be reliable in the future as well.
• For induction to be rationally justified, it is not enough that i-arguments are reliable. They must be shown to be reliable.

Now suppose that our ‘friend’ says "Stop. i-arguments will not continue to be reliable in the future."

We may reply: "i-arguments have been reliable in the past. Therefore they will be reliable in the future." But, is this argument reliable?

Dilemma: If we try to justify induction by means of a deductively valid argument with premises that we can show to be true (without using induction), then our conclusion will be too weak. If we try to use an inductive argument, we have to show that it is reliable. Any attempt to do that leads to the same dilemma all over again.

The Argument: Hume’s argument may be summarized as follows:

1. If i-arguments can be shown to be reliable, then there is a reliable argument that shows it.
2. Reliable arguments are either deductive arguments, or they are inductive i-arguments.
3. There is no deductively valid arguments that can justify the reliability of i-arguments (because of Humean underdetermination).
4. No i-argument can justify the reliability of i-arguments (because that would be circular).
5. Therefore, i-arguments cannot be shown to be reliable.

It should be clear enough that the argument is valid. It may not be clear, at this stage, why we should accept the premises are true. Here are some preliminary points that may help. I will provide further clarification after introducing the no-free-lunch theorems.

1. To show that i-arguments are reliable, the statement that i-arguments are reliable must be the conclusion of an argument that not only produces true conclusions, but produces conclusions that are seen to be true.
2. Reliable arguments are either deductive or non-deductive. The non-deductive arguments that (we think) are reliable have been called i-arguments. Therefore, premise 2 is acceptable to anyone who believes that induction can be reliability.
3. For a deductive argument to be reliable, it must be deductively valid, and it must have true premises. To show that the conclusion of such an argument is true, the argument must be valid, and we must show that the premises are true. If the premises are reports of past experience, like all i-arguments have been reliable in the past, then we can know them to be true. But, by Humean underdetermination, we know that we must have premises that go beyond past experience if the argument deductively entails the conclusion, for an argument of the form all past instances of i-arguments have been reliable, therefore all i-arguments are reliable is not deductively valid. The dilemma is that either we have premises that we know are true, in which case the argument is invalid, or we add a premise that we do not know to be true. Either way, we cannot show that the conclusion is true.
4. The conclusion that i-arguments are reliable is in dispute. To resolve the dispute, we need an argument that is not only reliable, but whose reliability is not in dispute. If we use an i-argument to argue that i-arguments are reliable, then the reliability of that argument is in dispute. The argument therefore begs the question, as in the argument: God exists, because God caused me to believe that God exists and God would not deceive me.

A similar argument has recently received much attention in field of machine learning. The so-called no-free-lunch theorems conclude that all methods of learning have equal probability of success if probability is measured by giving equal weight to all possibilities. Or, to put it another way, if some learning rules are to have better probabilities of success, then not all possibilities are equally weighted (that is, the lunch is not free). I will illustrate the general idea in the simplest possible example.

Consider an imaginary universe that lasts for exactly 2 days, and on each day there exists exactly one object, which is either a sphere or a cube. The object may or may not be the same shape on both days. There are exactly 4 possible histories that this world may have: (sphere, sphere), (sphere, cube), (cube, sphere) and (cube, cube). In this world, there are exactly 4 learning rules: Same = "same object both days", Diff ="different object both days", Sphere = "sphere no matter what on second day", and Cube = "cube no matter what on second day". Given the four histories have the same probability (1/4), what is the probability that each learning rule will make a correct prediction on the second day? If you work it out, you will find that the probability is 1/2 in all cases. The reason is that, no matter what happens on the first day, and no matter what prediction the rule makes for the second day, the two possibilities for the second day (sphere or cube) are equally probable.

Uniformity of Nature: A natural response is to insist that the world is uniform in some way. In our simple imaginary universe, we might insist that whatever appears on the first day will also appear on the second day. This uniformity of nature assumption implies that there are only two possible states of the universe: (sphere, sphere) and (cube, cube). Assuming that these each have a probability of 1/2, the learning rules Same, Diff, Sphere, and Cube have probabilities of successful predictions on the second day of 1, 0, 1/2 and 1/2, respectively. Now Same is the best learning rule. But this conclusion is only reached by assuming that nature is uniform. But where does this assumption come from? How it is justified?

Hume’s Argument Again: We might attempt to show that the uniformity of nature assumption is true by pointing to the fact that Same has been reliable in the past (suppose that our imaginary universe has many objects and many previous days. This argument would look like: The learning rule Same has been reliable in the past, therefore Same will continue to reliable in the future. This assumes a uniformity assumption of the form: What has been true of Same in the past, will be true of Same in the future. Hume’s point is that we have no way of showing that this assumption is true.

Hume’s argument appears to prove too much. The same argument proves that we cannot show deductively valid arguments, like modus ponens, to be reliable, since we would have to assume the reliability of modus ponens in the process. To see this just replace "i-arguments " with "modus ponens" throughout.

Conclusion: While cannot prove that i-arguments are reliable, perhaps we could explain why they might be reliable. After all, this is the standard of justification that is met in the courtroom, or in other walks of life, so why not in the theory of knowledge.

Any such explanation would depend on what i-arguments are. So, we need to describe them.

• The problem of description is about what the method of scientific induction is, rather than how it is justified.

Popper characterized the psychological problem of induction as the problem of saying how our predictive expectations arise. The problem is not necessarily psychological in the sense of involving the formation of belief formation, because beliefs are not the only form that expectations may take. Expectations, for example, are built into the way that vision works at a level below our conscious awareness. And primitive animals, like bees and sea slugs, have predictive expectations while it is implausible to suppose that they have beliefs in the same sense as us.

• The problem of description is the problem of saying how predictive expectations are formed, or how predictions are made (see the handout Prediction and the Problem of Induction).