The problem of induction is a dispute in the philosophy of science concerning how the use of induction can be justified. The problem is essentially that there seem to be only two options: a deductive justification or an inductive justification. However induction is clearly not justified deductively, as if it were we wouldn’t need to bother with induction in the first place; we could simply substitute it with deduction. In other words, the whole point of inductive inference is that the conclusion is never rendered true with deductive certainty, but only to some degree of probability depending on the strength of inductive support, and as such deduction will never be able to justify the use of induction. On the other hand, using an inductive argument to justify the use of induction seems problematic as this would be circular reasoning, assuming the very thing we are trying to prove, namely that induction is a valid or effective form of reasoning.
A humorous way of illustrating this is to consider a possible alternative method called ‘anti-induction’, in which one always assumes that the future will be totally different to how things were in the past. A proponent of anti-induction could attempt to defend their method of inference by arguing “its never worked in the past, therefore it is likely to work in the future”. This reasoning seems absurd, however it is arguably no more question-begging than the argument that “induction will probably work in the future because it has worked plenty of times in the past”.
How to deal with this apparent conundrum is a puzzle that many philosophers have considered over the centuries. Various approaches have been proposed, including attempting to understand induction as part of what we intuitively mean when we talk about a belief being justified, or alternatively to argue that the type of circularity involved in justifying induction by the use of induction is not necessarily a problem. Regardless of what one thinks of these or other responses, an understanding of the problem of induction is vital to be able to appreciate many writings and debates that occur in epistemology and philosophy of science.
The problem of induction: a detailed analysis of the issue from the Stanford Encyclopedia of Philosophy
The problem of induction: useful discussion and resources from a university course