A Problem for Hume's Problem of Induction

Could Hume consistently believe that his argument to the effect that inductive inferences are not justified is successful? In this post I put forward reasons to think the answer is "no."

Hume, very basically, argued that inductive inferences are not justified because there are only two ways that that they could be supported: Either through a priori reasoning, or through further inductive inferences. A priori reasoning cannot support inductive inferences, because there is no contradiction in the supposition that the course of nature may change, and hence it is possible that it could. Nor could inductive inferences rest on further inductive inferences for their support, for they all rest on the supposition that the course of nature will not change, and cannot support that supposition without begging the question. Hence, inductive inferences are not justified.

Let's see what happens when we apply Hume's Fork to the conclusion of Hume's argument. We get the following result:

           "Inductive inferences are not justified" states either a relation of ideas, or a matter of fact.

If we accept the first alternative, then, if it's true, it only states a relation between certain of our ideas--namely, our ideas of justification and inductive inference--and says nothing significant about matters of fact. It is no more informative, in any real sense, than "a = a". If we accept the second alternative, it states a matter of fact, which according to Hume's own principles could only be confirmed or refuted through induction. If inductive inferences are justified, or justified for the most part, it cannot establish that they are unjustified. And if they are unjustified, they cannot justifiably establish their own lack of justification. Thus it seems that even if inductive inferences are not justified, Hume cannot justifiably believe that they are not. If one is inclined to accept Hume's argument, one should reject Hume's Fork as a false dichotomy; conversely, if one is inclined to accept Hume's Fork, one should either reject Hume's argument as unsound, or conclude that it may be sound, but if so, we cannot know that its premises are true.