Hume's Problem of Induction

The "Problem of Induction" is the problem of showing that induction is a good inference procedure. What can we say in support of induction? What could we say that would show that induction is a reliable inference procedure? What can we say to show that the conclusion of an inductive argument is likely to be true?

Hume’s problem of induction: The intuitive idea.

It seems like the only thing we can say in support of induction is:
Induction works so well!
But notice the circularity of that reasoning.
(Sure, in the observed cases induction has worked well, but why does that show that induction will always work well. You need to rely on induction to get this conclusion.)

Hume’s problem of induction: Slightly more rigorous.

As we saw, inductive arguments are not valid as they stand.
However, we could make them valid if we specify a key implicit premise.
Let’s call this premise the "Principle of the Uniformity of Nature".
The Principle of the Uniformity of Nature:
We can put the "Principle of the Uniformity of Nature" (PUN) in a few different, but related, ways:

What reason do we have to believe that (PUN) is true?

Hume argues that to rely on induction, we need have some reason for believing (PUN).
But what reason could we give in defense of this principle?

Notice that we can’t give any deductive argument for (PUN). (As I pointed out in class, there’s nothing that guarantees that the future will be like the past.)

So it seems the only way to show that (PUN) is justified is to give an inductive argument.
But this results in a circularity!

We can’t give a good inductive argument for (PUN) because, as we saw, all inductive arguments rely on (PUN).
Hume’s conclusion is that there is no reason we can give in support of (PUN) and thus no reason we can give in support of inductive arguments.