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Hume's problem of induction is that any attempt to justify induction would lead to a circular argument. Can someone give an example to illustrate this and maybe explain the problem more?

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Induction is a “Generalization from Experience” (Mill, p. 223, §1). But Hume sees a serious problem. “[T]here is no justification for regarding what has been observed to happen in the past as any sort of reliable guide to the future” (Howson, p. 1).

Nor can such an inference [from the observed to the as-yet unobserved] be justified, without circularity…. Since ‘the course of nature may change’, indeed proceed from here in a virtually uncountable number of different ways, the inference that the future will proceed or even probably proceed in any one of them must beg the question. (Howson, p. 10).

If no result is reliable, then reliability must be assumed; thus the circularity problem. The Uniformity Principle, that the future will resemble the past (Mill, pp. 224-25), becomes the unspoken assumption.

Assume the goal is identification of that Object P which confirms “if P then Q”. After a large number of tries, the result is always exactly the predicted event: Object P followed by Result Q. So does the collection of these observations, called “simple enumeration”, confirm the generalization? No.

The syllogism that describes simple enumeration is invalid. Aside from the fatal problem of assuming what the syllogism sets out to prove, a further problem is the undistributed middle term. In a valid syllogism the middle term must be distributed in at least one premise. The following syllogism is that used in simple enumeration. It is AAA in the second figure:

All emeralds are green.

This particular emerald is green.

Therefore: All emeralds are green.

The premises are both A statements (All S are P), where the subject is distributed but the predicate is not. In both premises, the middle term (green) is the predicate and undistributed. Nothing links the two statements. The conclusion is invalid, and will remain so after every observation.

References:

Howson, Colin. 2000. Hume’s problem: induction and the justification of belief. Clarendon Press, Oxford.

Mill, John Stuart. 1882. A system of logic, 8th Ed. New York: Harper & Brothers. http://www.gutenberg.org/files/27942/27942-h/27942-h.html#toc47

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Induction is the move from data to rules. Obviously, we can observe correlations. But we cannot know what correlations constitute causes unless we compare them to one another. But on what basis would you compare them? You would need rules that indicate which kinds of correlation look causal and which kinds of correlation look derived or spurious. How would you determine those rules? It obviously involves observation. But that assumes you can get from observation to rules.

The only way out of this loop is grounding in some basic assumption about cause and effect which does not originate in observation.

Ultimately, in order to get traction, we have to appeal to our own emotions - some things just feel more basic than others, and we consider those more reliable and build from there. But even discerning the pattern in our own reactions is ultimately observation. So for individual reactions to build up into a rule, we are already making the assumption that one can get from data to rules. We have just pushed that assumption into unconsciousness.

It is altogether possible, given this difficulty, that rules just are not the way the universe works, but are only the way our minds work, and the notion of causation itself nothing more than a useful illusion humans naturally share.

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In terms of prima facie validity, it does seem like justifying induction on the basis of induction having worked in the past might involve circularity. Hence the logical foundations of inductions have to lie somewhere else.

Possibly an induction is rooted in an abduction; we “steal” a hypothesis without explicit grounding, retroactively justifying based on experiments. (This seems to have connections with the sort of ideation scientists are doing when they’re inventing hypotheses.)

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