The problem of induction is a philosophical problem that questions the rationality of predictions about unobserved things based on previous observations. These inferences from the observed to the unobserved are known as "inductive inferences". David Hume, who first formulated the problem in 1739, argued that there is no non-circular way to justify inductive inferences, while he acknowledged that everyone does and must make such inferences. The traditional inductivist view is that all claimed empirical laws, either in everyday life or through the scientific method, can be justified through some form of reasoning. The problem is that many philosophers tried to find such a justification but their proposals were not accepted by others. Identifying the inductivist view as the scientific view, C. D. Broad once said that induction is "the glory of science and the scandal of philosophy". In contrast, Karl Popper's critical rationalism claimed that inductive justifications are never used in science and proposed instead that science is based on the procedure of conjecturing hypotheses, deductively calculating consequences, and then empirically attempting to falsify them.
Formulation of the problem In inductive reasoning, one makes a series of observations and infers a claim based on them. For instance, from a series of observations that a woman walks her dog by the market at 8 am on Monday, it seems valid to infer that next Monday she will do the same, or that, in general, the woman walks her dog by the market every Monday. That next Monday the woman walks by the market merely adds to the series of observations, but it does not prove she will walk by the market every Monday. First of all, it is not certain, regardless of the number of observations, that the woman always walks by the market at 8 am on Monday. In fact, David Hume even argued that we cannot claim it is "more probable", since this still requires the assumption that the past predicts the future. Second, the observations themselves do not establish the validity of inductive reasoning, except inductively. Bertrand Russell illustrated this point in The Problems of Philosophy:
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.
Ancient and early modern origins
Pyrrhonism The works of the Pyrrhonist philosopher Sextus Empiricus contain the oldest surviving questioning of the validity of inductive reasoning. He wrote:
It is also easy, I consider, to set aside the method of induction. For, when they propose to establish the universal from the particulars by means of induction, they will effect this by a review either of all or of some of the particular instances. But if they review some, the induction will be insecure, since some of the particulars omitted in the induction may contravene the universal; while if they are to review all, they will be toiling at the impossible, since the particulars are infinite and indefinite. Thus on both grounds, as I think, the consequence is that induction is invalidated. The focus upon the gap between the premises and conclusion present in the above passage appears different from Hume's focus upon the circular reasoning of induction. However, Weintraub claims in The Philosophical Quarterly that although Sextus's approach to the problem appears different, Hume's approach was actually an application of another argument raised by Sextus:
Those who claim for themselves to judge the truth are bound to possess a criterion of truth. This criterion, then, either is without a judge's approval or has been approved. But if it is without approval, whence comes it that it is truthworthy? For no matter of dispute is to be trusted without judging. And, if it has been approved, that which approves it, in turn, either has been approved or has not been approved, and so on ad infinitum. Although the criterion argument applies to both deduction and induction, Weintraub believes that Sextus's argument "is precisely the strategy Hume invokes against induction: it cannot be justified, because the purported justification, being inductive, is circular." She concludes that "Hume's most important legacy is the supposition that the justification of induction is not analogous to that of deduction." She ends with a discussion of Hume's implicit sanction of the validity of deduction, which Hume describes as intuitive in a manner analogous to modern foundationalism.
Indian philosophy The Cārvāka, a materialist and skeptic school of Indian philosophy, used the problem of induction to point out the flaws in using inference as a way to gain valid knowledge. They held that since inference needed an invariable connection between the middle term and the predicate, and further, that since there was no way to establish this invariable connection, that the efficacy of inference as a means of valid knowledge could never be stated. The 9th century Indian skeptic, Jayarasi Bhatta, also made an attack on inference, along with all means of knowledge, and showed by a type of reductio argument that there was no way to conclude universal relations from the observation of particular instances.
Medieval philosophy Medieval writers such as al-Ghazali and William of Ockham connected the problem with God's absolute power, asking how we can be certain that the world will continue behaving as expected when God could at any moment miraculously cause the opposite. Duns Scotus, however, argued that inductive inference from a finite number of particulars to a universal generalization was justified by "a proposition reposing in the soul, 'Whatever occurs in a great many instances by a cause that is not free, is the natural effect of that cause.'" Some 17th-century Jesuits argued that although God could create the end of the world at any moment, it was necessarily a rare event and hence our confidence that it would not happen very soon was largely justified.
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