Induction

Induction is a type of reasoning that extrapolates knowledge about the observed to make conclusions about the unobserved. An inductive argument usually takes the following form:

All observed X have been Y.
Therefore,
All X are Y.

This is a very important method of reasoning in science and in common sense. In science, it is used to formulate hypotheses from data, and to make predictions. In common sense, it is used to make safe judgements about the world around us. Despite our reliance on induction, it has been attacked several times. The following arguments illustrate some of the problems of induction.

The Pyrrhonian Critique

Arguments against the validity of induction can be traced back to the Pyrrhonian skeptic Sextus Empiricus, who argued that a universal law cannot be established from from an incomplete set of individual instances. This can be because of two reasons. The first is the use of a poor method. An induction may be based on a sample that is not representative of the general population, and so the conclusion may be false. For example, a doctor may observe that all of the patients in the oncology ward have cancer, and make the following claim:

All observed people in the ward have cancer.
Therefore,
All people have cancer.

Clearly, the doctor has made an erroneous claim. His sample is not representative of the general population, and so the induction drawn from it is unsound.

The second reason why a universal law cannot be established from an incomplete set of individual instances is that in an inductive argument, there is no logical entailment between the premises and the conclusion. Induction takes a finite set of observations and infers a new claim based. This new claim is a conjecture based on the observations, and is not logically entailed by them.

Consider the following inductive argument:

The sun has risen every morning so far.
Therefore,
The sun will rise every morning.

Although the sun is likely to rise every morning, there is no logical entailment from the fact that it always has to the claim that it will continue to do so. It is logically conceivable that in a possible world in which the sun has risen every day so far, the sun will not rise tomorrow.

The Problem of Justification

The problem of justification was described by David Hume in An Enquiry Concerning Human Understanding (1748). Hume argued that one cannot justify the method of induction without using induction. Therefore, all justifications are circular.

Consider the previous argument concerning the rising sun. One could that whenever I had previously induced that the sun would rise the following day, my experience of it actually rising the following day confirmed the validity of my induction. Therefore, I continue to assume that the sun will rise tomorrow, because this assumption has been right so far. However, as Hume shrewdly argues, this provides no justification whatsoever, for what is being said is that induction has been valid so far, and, on this basis, I conclude that induction will continue be valid. We can see that this statement, itself, is an inductive inference:

Induction has so far been successful.
Therefore,
Induction will always be successful.

This argument attempts to justify induction via induction, and so fails due to a vicious circularity. Thus, Hume has shown that we have no metaphysical justification for our inductive practices.

Some argue that this circularity is benign. However, consider the counter-inductivist who uses counter-induction as his preferred method of reasoning. A typical counter-inductive argument has the following form:

All observed X have been Y.
Therefore,
All other X are not Y.

Let us now apply counter-induction to formulate a hypothesis about the success of counter-induction:

Counter-induction so far has been unsuccessful.
Therefore,
Counter-induction will in future be successful.

It appears that if the inductivist can use a circular justification to support induction, the counter-inductivist can also use the same argument to support counter-induction.

The Grue Paradox

Nelson Goodman has presented a different attack against inductive reasoning. He introduces an adjective, grue. Something is grue if it is green before the year 2100 and blue afterwards. Now, consider that it is the year 2009, and we have observed all emeralds so far to be green. The following inductive statement can be made:

All emeralds so far have been green.
Therefore,
All emeralds are green.

However, since it is before 2100, our observations also support the following statement:

All emeralds so far have been grue.
Therefore,
All emeralds are grue.

This suggests that the same observations support two incompatible hypotheses about emeralds observed after 2100. The first hypothesis is that all emeralds will be green after 2100. The second hypothesis is that all emeralds are grue, and so will be blue after 2100. Therefore, induction cannot select one hypothesis over another.

The Raven Paradox

Proposed by Carl Gustav Hempel, this paradox illustrates how inductive logic can violate intuition. Consider the following hypothesis:

All ravens are black.

Evidence supporting this hypothesis would be a specific observable instance of the general class, or, in other words, a black raven.

Note that the hypothesis stated above is logically equivalent to the following hypothesis:

All non-black things are not ravens.

Evidence supporting this hypothesis would be a non-black thing that is not a raven. For example, a piece of white paper is non-black and not a raven. Therefore a piece of white paper supports the above hypothesis. However, since both hypotheses are logically equivalent, it follows that a piece of white paper provides supportive evidence for the hypothesis that all ravens are black.

Conclusions

Induction is a crucial method of reasoning in everyday life. Furthermore, the construction of physical theories and postulation of laws in science also presupposes the validity of inductive reasoning. However, the problems of induction presented here reveal that inductive reasoning has no metaphysical justification. This raises questions about the nature of the knowledge generated through inductive reasoning.

Perhaps a sensible position to take is reliabilism. Although induction has no justification, it is a reliable method, and so its conclusions can be trusted, at least from a pragmatic point of view. Indeed, there is no logical entailment that the sun has to rise tomorrow, but I can assume that it will, because the method I have used to make this assumption is reliable. Furthermore, it is more pragmatic for me at this present time to assume that the sun will rise tomorrow than to make extensive preparations for the forthcoming darkness.

References

• Goodman N (1955). Fact, Fiction, and Forecast. Cambridge, Massachusetts: Harvard University Press.
• Hempel CG (1945). "Studies in the Logic of Confirmation". Mind, 54:213, 1.
• Hume D (1748). An Enquiry Concerning Human Unerstanding.
• Sextus Empiricus (c. 200). Outlines of Pyrrhonism.