The problem of induction has always bothered me. It has never seemed to me like a problem at all.
And now I can prove it.
Deduction works by elucidating the relationships between concepts. The problem is, for creatures;born, there's no way to obtain concepts except through induction. This includes evolving to be born with certain concepts, as the genetic material does not inherently know anything, and must learn entirely through trial and error.
As a result, in practice, deduction is always a sub-discipline of induction. Thus the 'problem of induction' was itself found using induction. As a result, to throw out induction is to throw out the problem of induction, allowing induction once more...and there's a barber I'd like you to meet.
Also as a result of this fact, a priori is clearly an invalid concept. All concepts are a posteriori, including a priori.
No, the only way the problem of induction is an actual problem is if you're actually some kind of infinite consciousness, in which case you'd know things just because you're an infinite consciousness. Presumably, you'd then use your mighty infinitude to solve the problem in an infinitesimal instant.
Because deduction is in effect a species of induction, a priori knowledge in the usual definition doesn't exist.
Therefore, I will redefine it.
A conclusion reached from first principles or a priori was reached through the logical consequences of the definitions of the concepts involved.
Such a proof always applies if the concepts apply; this is why a priori proofs are so valuable, which is why they are so prized.
My favorite type of proof is what I call an a priori hammer of Thor. Such a proof is true a priori, and even if you disprove the existence of some of the premise-concepts, you end up at the same conclusion. While the details of the proof will change under various assumptions, the conclusion will not. Such a proof is a thorough examination of a full tree of yes/no questions, where each terminating twig is identical.
Such proofs are transcendentally true. They are true regardless of concepts, and depend only on the rules of logic themselves. Such conclusions are probably true in not only all universes, but all possible universes. (Including that infinite consciousness.)