The problem of induction arises due to the circular reasoning that we derive when we deal with inductive inference. The notion of circularity in this statement refers to further proof the legitimacy of the inductive inference can only be gained by (another) induction. This leaves a conclusion that there is no existence of non-circular justification of induction. Hence, further plague all knowledges that obtain by induction.
One can argue why need justification for induction method in the first place. This due to nature of induction process itself, which is generalisation. The inductive inference cannot guarantee the validity of conclusion because it is achieved by generalising its premise. So, in order to justify inductive inference, one must justify the nature of generalisation which in turned can only be made sense by induction again, hence a circular process. To further cemented the understanding of the problem, one can provide an example like below:
Premise 1: every planetary bodies and stars that have been observed has a round shape (spherical symmetry)
Conclusion: all planetary bodies and stars have a round shape.
This argument clearly is an inductive inference, since the conclusion is the generalisation of its premise. If we further investigate this argument, we have an impression that even if the premise is true, the conclusion cannot be said to be true. Or in other words, the validity of the premise doesn’t entail the validity of its conclusion. Because if there exists even one planetary body or star that does not have a round shape, the conclusion is false hence not giving any knowledge.
Nevertheless, we still confidence about the conclusion because given the fact that we had observed for a long time, using astronomical apparatus such as telescope and satellite since Galilean time that indeed every planetary body and star does have a round shape. From the huge range number of data and satellite photos using the most advanced technologies in the twenty first century also confirms this premise. Maybe in reading the premise, we need to put it like this:
Premise 1*: From large amount of observational data, satellite imaging and other sophisticated apparatus that are designed to investigate the shape of planetary bodies and stars since long time proves that indeed all the observed planetary bodies and stars have a round shape.
Conclusion: All planetary bodies and stars have a round shape.
Everything looks fine until this spot, but there is one problem left that we need to tackle: “what gives us the justification to believe we can generalise the premise to the conclusion?” Because if we examine it closely, the premise has a certain dependency on actual condition which is the round shape of planetary bodies and stars depends on whether they have been observed or not and on whether there is no mistake on all the samples that have been observed. Meanwhile, the conclusion seems to have an independent claim that all planetary bodies and stars since past time and future in any circumstances have been and will always be round.
One might think a way to justify inductive inference is by adding a premise that can become a bridge to justify the gap of the premise and the (generalised) conclusion. One of the possibilities is shown below:
Premise 2: A pattern of observing the round shape of planetary bodies and stars has been established from all previous observations involving large number of samples thus the possibility of this pattern to be repeated is justifiable and therefore it is likely to be applied in general.
By adding premise 2 above, it seems there is no problem inferring the conclusion from premise 1* since the issue is fixed in the premise 2. However, if we look closely at premise 2, one can ask “in what ground we can support our claim of premise 2?”. The only way to link between a pattern of observed-round-shape of planetary bodies and stars with the general claim of conclusion is by repeating the same claim: it is representable since it has been proved and applicable to large amount of data thus can be applied in general.
This is just repeating another inductive inference for an inductive argument or like in the earlier explanation, it is a circular reasoning (problem of induction). This method is (at least according to Hume) cannot be accepted hence there is no justification for induction way of acquiring knowledge.