Let’s see. I have to think back, but I think how this was how it was explaned to me in class.
Hume catagorises truths in two catagories, analytic and synthetic.
Analytic truths can be know by only looking at the meaning of the terms. For example, “All bachelors are male.” Because bachelor means unmarried male, we don’t even have to look at a bachelor to know all bachelors are male.
Synthetic truths requre you to look at the world. For example, “That swan is white.” Nothing about swan means white, so to see if it is white we must use our senses.
Ok?
Well then we get to problematic statements like “All swans are white.” First of all, it’s synthetic. So we must need sense data. What sense data do we have?
We have, “All 100 swans I’ve seen are white.” But does that lead directly to “All swans in the world are white,”? Not quite we need something else. We need the Principle of Universality of Nature.
The PUN: Things I haven’t seen are similar to things I have seen. (Tommarow is like today, ect.)
Ok, well how do we know the PUN is true. First of all it too is synthetic. There is nothing in the definition of nature that means universal. So whats the sense evidence?
“Things I’ve seen for the first time in the past have been similar to things I had already seen before.” But agian this is insufficent, so we need then PUN.
But then we are using the PUN to prove the PUN. That is circular. So here, according to Hume, inductive logic is not justified by deductive logic.
Another way to put it is that Hume sees Deduction as the logic of Analytic truths, and Induction as the logic of Synthetic truths, and that the two systems are seperate forms of logic that we use.
I think this interpretation of Hume is pretty uncontroversal- I’m sure I’ll be corrected if it’s not.
As for arguments to overcome Hume, look for Quine’s paper Two Dogmas of Empiricism.