The Problem of Induction

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 (people use the adverb ‘of course’ if and only if they have no argument). 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.

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? The argument to follow says NO.

Clarification: Our question is not whether all forms of inductive reasoning are reliable. We know that they are not. Our question is whether there is some form of inductive argument, such as simple enumerative induction, 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. Simple enumerative induction is recognizable.

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

Suppose: Induction is rationally justifiable if and only if there exist i-arguments that can 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?

One way to defend the reliability of this argument is to appeal the presupposition that nature is uniform. But nature is not uniform is all respects (c.f. the example of the Thanksgiving turkey who has been fed for the 100 days before Thanksgiving). So, in what respects is nature uniform? And, assuming that this uniformity does imply that our argument is reliable, how do we show that nature is uniform in this way. If our ‘friend’ challenges us on this, what do we say? That nature has been uniform in the past, therefore nature will be uniform in this way in the future? When our ‘friend’ asks us to show that this argument is inductively strong, we cannot appeal to the uniformity of nature again unless it is a ‘higher level’ of uniformity. But then we are going to be asked the same question: Show that nature is uniform in this sense. At no stage have we shown that i-arguments are reliable.

Summary: Hume presents us with a 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.

Remark: The argument does not show that there are no reliable i-arguments. It shows that we cannot show that i-arguments are reliable.


The previous section is an elaboration on what is found in Hume (1748), where in the bread example, he points out that "the consequence seems nowise necessary." But Hume begins with another argument, which goes like this:

  1. Our conviction that some forms of inductive reasoning are strong is a psychological conviction.
  2. The conviction is the result of a habit, which does not depend on the future in any way.
  3. Rational justifiability requires the future reliability of induction.
  4. Therefore, the strength of our conviction is not evidence for the rational justifiability of induction.

Hume assumes that many readers will have a different view of the psychological origin of inductive habits; namely that it arises from a chain of reasoning. "But if you insist that the inference is made by a chain of reasoning, I desire you to produce that reasoning." Apart from pointing out that we are not aware of such reasoning, Hume now argues that this point of view must be wrong because no such chain of reasoning exists. Our previous argument is an elaboration of his.

A look ahead: In a second argument, Hume claims that "It is only after a long course of uniform experiments in any kind that we attain a firm reliance and security with regard to a particular event." How could this fact be explained if our inductive habits were the result of reasoning? "[W]here is that process of reasoning which, from one instance, draws a conclusion so different from that which it infers a hundred instances that nowise different from that single one?" Hume’s idea is that any principle that could bridge the logical gap between many instances and the conclusion could also bridge the gap between a single instance and conclusion. Certainly, that is true of a principle such as "like causes produce like effects." But there are forms of statistical inference where there is good reason for drawing a conclusion from many instance not drawn from a single instance.

So, what is the problem of induction? Hume has taught us, I think, that we cannot prove that scientific inference is reliable. Does not leave us, as Russell suggested, with no way of separating us from the lunatic who believes he is a poached egg? Is the lunatic to be condemned solely on the ground that he is in the minority?

Goodman’s New Riddle of Induction

Definition: Object x is grue at time t if and only if x is green at time t and t < 2100, or x is blue at time t and t 2100.

The following inference both follow the pattern of simple enumerative induction:

  1. All emeralds observed to date have been green. Therefore, all emeralds are green (at all times).
  2. All emeralds observed to date have been grue. Therefore, all emeralds are grue (at all times).

Note: The conclusion of these arguments cannot both be true. The second predicts that any emerald observed after the year 2100 will be blue at that time, while the green hypothesis predicts that they will be green at that time. One of these predictions will be false.

Remark: The idea that Newton’s theory of motion may be true when stated in French, but false when stated in English is absurd. However, we don’t have this kind of language dependence in the grue example, for each hypothesis says the same thing in the color and the grolor languages. It is the language dependence of inductive inference that is at issue, although this dependence is equally unacceptable.

Argument: The inference is reliable in the first instance, but not in the second. There are as many inferences like the second as there are the first. Therefore, simple enumerative induction is unreliable in most cases.

Conclusion: Simple enumerative induction must be restricted in its use if it is to be reliable.

Suggestion 1: Simple enumerative induction should not be applied to terms that are unfamiliar.

Problem: This suggestion will not work because there are many cases in science in which unfamiliar terms are used. ‘Electron’, ‘volt’, ‘mass’, and so on, were all unfamiliar at one time.

Suggestion 2: Simple enumerative induction should not be applied to terms that are that are concocted and artificial.

Problem: Relative to our language, ‘grue’ does appear this way, but relative to a language in which ‘grue’ is entrenched, then ‘green’ will appear concocted.

Definition: Object x is bleen at time t if and only if x is blue at time t and t < 2100, or x is green at time t and t 2100.

Now we have a new ‘grolor’ language, consisting of {grue, bleen}. Color terms appear concocted in the grolor language:

Definition: Object x is green at time t if and only if x is grue at time t and t < 2100, or x is bleen at time t and t 2100.

Definition: Object x is blue at time t if and only if x is bleen at time t and t < 2100, or x is grue at time t and t 2100.

Suggestion 3: The green hypothesis is simple and the grue hypothesis is complex. So, by the principle of parsimony, which says that we should choose the simpler hypothesis, other things being equal (fit with past observations in this case).

Problem: Again, the notion of simplicity appealed to here is language dependent. In the grolor language, it is the grue hypothesis that looks simple, and the green hypothesis is complex.

Note: This also suggests that the idea that nature is uniform is language dependent.

Puzzle: Suppose we come across a group of people who use the grolor language. They will claim that our language is concocted and artificial. We will restrict our use of simple enumerative induction in different ways, and it is impossible that both our methods are reliable. How do we convince them our language is the correct one? In fact, how can we prove to ourselves that we are not the "grolor people". We cannot prove that in the strong sense that Hume demanded.

Summary: The use of simple enumerative induction is restricted to ‘normal’ entrenched, established, vocabulary. But we have no proof of the reliability of this form of induction.

Remark: Even if simple enumerative induction is understood as restricted in a way, there are good reasons for denying it is the method of science. It does not fit curve-fitting, for example. Nor does it fit the cases in which the inductive conclusion introduces new concepts not contained in the observation statements, like ‘gravity’ (c.f. the Whewell-Mill debate).

A look ahead: The grue problem shows that the same observational evidence may be described in different ways. Mill thought that the description is determined by the observational facts, whereas Whewell disagreed. The grue puzzle shows that Mill is wrong.

Popper’s Solution of the Problem of Induction

The commonsense problem of induction is based on the ‘bucket theory of the mind’—roughly, the assertion that ‘there is nothing in our mind which has not entered through our senses.’ But we do have expectations and we strongly believe in regularities. How can these have arisen? Answer: Through repeated observations. The commonsense view takes for granted that the resulting expectations are justified.

Popper has three theses:

  1. There is no rationally justifiable method of induction
  2. There is no reliable method of induction.
  3. Nevertheless, there is a critical method of science that is rational.

Popper distinguishes Hume’s logical problem of induction —whether we are justified in reasoning from repeated instances—from Hume’s psychological problem of induction — Why do we have expectations in which we have great confidence? But for Popper, there is no such thing as induction by repetition (simple enumerative induction), as is shown by the fact that it is false that "The sun will rise and set once in 24 hours" (counterexample: the midnight sun at the Earth’s poles) and "All bread nourishes" (counterexample: ergotism in a French village). However, this does not show that simple enumerative induction fails to lead to true, or approximately true, conclusions most of the time. So, Popper has not even shown that simple enumerative induction is unreliable, yet alone that there is not reliable method of induction.

Note: Goodman presents the view that simple enumerative induction is restricted to entrenched terms, like ‘green’ and ‘blue’. This is his solution to the psychological problem of induction. So, we do know that simple enumerative induction must be restricted in some way if it is to be reliable.

Popper rejects Hume’s assumption (that if there is a reliable method of induction then it is simple enumerative induction), so he much reformulate the logical problem of induction:

L1: Can an explanatory universal theory be justified by assuming the truth of observation statements?

Note: Talk of explanatory theories alludes to the idea of induction as inference to the best explanation.

Here he agrees with the answer Hume would give: NO. To understand the basis of Popper’s claim, consider a very Mill’s example of a universal ‘theory’: "All swans are white." This ‘theory’ is not proven by any number of swans that have been observed to be white because the claim applies to swans that have not been observed.

However, Popper’s answer to the following question is YES.

L2: Can the claim that an explanatory universal theory is true or is false be justified by assuming the truth of observation statements?

For we can prove that "All swans are white" by observing a black Australian swan.

If T then O

This is a deductively valid argument. Let T be a theory, and O be an observational prediction from the theory, which proves to be false.

There is a logical asymmetry between proof and refutation, which rests only on deductive logic. There is no need to appeal to the contentious facets of inductive logic and epistemic probability.

An affirmative answer to L2 is a weaker than an affirmative answer to Hume’s question. But it is sufficient to separate us from Russell’s lunatic who believes he is a poached egg.

Popper’s picture of rational science is one in which competing theories ‘stick their necks out’, and are subject to severe tests. Some are refuted, but some survive, and the survivors are rationally justified theories in this weaker sense.

Eliminative induction: If there is a finite list of competing theories T1, T2, …, Tn such that all except one contradicts the known observational statements, then conclude that the surviving theory is true. This is actually a deductive inference. The weakness of the argument is the truth of the premise that says that one of the theories in the list is true.

In other cases, not all theories but one on the list are eliminated. If there is more than one unrefuted theory, how do we choose between them?

Problems with Popper’s Solution

Eliminative induction (which goes back to Francis Bacon) assumes that observational evidence can eliminate, falsify, or refute theories. Popper’s solution to the problem of induction appeals to the same idea. But the Quine-Duhem thesis denies that this is possible.

Quine-Duhem Thesis: Any seemingly disconfirming observational evidence can always be accommodated by any theory.

The idea behind this thesis is that theories, like Newton’s theory of gravitation, are never tested in isolation. It is only when they are conjoined with auxiliary assumptions—about the number of planets, the distribution of the mass within a planet, the shape of the a planet, and the absence of other forces such as electromagnetic forces—that the theory makes testable predictions. If those predictions are false, then logic forces us to reject either the theory or one of the auxiliary assumptions. It does not force us to reject the theory.

Example: Prior to its discovery, Uranus was the outermost planet that was known. Under the auxiliary assumption that there were no other planets, Newton’s theory could not account for observed wobbles in the orbit of Uranus. Leverrier and Adams postulated the existence of another planet, and even calculated (under the assumption that Newton’s theory was true) where the planet would have to be. When telescopes were pointed to that place, Neptune was discovered.

The Quine-Duhem thesis asserts that there are always auxiliary assumptions that will rescue a theory from refutation.

Note: The Quine-Duhem thesis, if accepted, does not show that there is no rational way of accepting one theory in favor of another. But it does undermine Popper’s idea about how this can be done.

The Falsifiability of Models

The Quine-Duhem thesis allows the following application of modus tollens. Let A stand for a set of auxiliary assumptions. Then the fact that observation O follows from T combined with A shows that "If (T and A) then O" is true. Further suppose that the prediction O is observed to be false. Then, the premises of the following argument are true.

If (T and A) then O
Not-(T and A)

Moreover, this argument is valid because it is a modus tollens argument.

Definition: Call the combination of a theory T with a set of auxiliary assumptions A a model of the theory.

Popper could then ask the following question:

L3: Can the claim that a model of a theory is true or is false be justified by assuming the truth of observation statements?

Now he could answer ‘yes’ without being vulnerable to the Quine-Duhem thesis.

Note: This leads to a picture of science as a series of models, starting with the simplest, followed by the next simplest when the simplest is falsified, and so on. It is quite an accurate picture of the history of science, although it does not address the question of how we science changes from one theory to the next. This that Kuhn addresses in his book The Structure of Scientific Revolutions and that Lakatos tackles in his methodology of scientific research programs.