The Problem of Induction
(This post is inspired by a tweet by Nathan Oseroff in which he says that we should "retire" the word induction because the word has come to mean so many different things in so many different contexts that the so-called problem of induction is just a useless collection of vaguely related concepts. This is my attempt to give a clear definition of induction and the problem everyone has with it. Yes, I'm aware of this comic.)
Deductive inference is a process by which we come to know that some proposition, Q, is true on the basis of other propositions P_1,...,P_n. The hallmark feature of deductive inference is that if it's employed in the "right" way (e.g. in a valid inference) then we can be absolutely certain that Q is true, given P_1,...,P_n.
Inductive inference (or "induction" for short) is a process by which we come to know that some proposition, Q, is true on the basis of other propositions P_1,...,P_n. In that way, deduction and induction are identical. However, unlike deductive inference, inductive inference never guarantees the truth of Q.
That's where we get the infamous "problem" of induction (which isn't actually a problem but a feature of inductive inference). The problem of induction is a challenge to tell me under what conditions, if any, I can know that Q on the basis of some propositions P_1,...,P_n. We take ourselves to know how this works for valid inferences. What about invalid inferences? Trying to answer that question is what generates the "problem" of induction.
The problem of induction is just the problem of evidential support for invalid inferences. The problem of evidential support asks why are some propositions in a relation of evidential support with others and can we tell when this happens and when it doesn't? We generally take ourselves to have a good answer to this question for deductive inference (most people think the evidential support relation reduces to the relation of validity), so the "problem" with induction is that we don't have a good answer for inductive inference.
As I understand it, there are only two plausible possible answers to the question of when we can know Q on the basis of P_1,...,P_n (in the case of induction): never and sometimes. (I'm ignoring people who asy "always" because I don't think any such people exist.)
People who say "never" are skeptics about induction. People who say "sometimes" are statisticians.
The further, and I think really interesting, question here is whether an "all-purpose" answer to this question exists. Is there a unity of "confirmation" (like Carnap thought) such that deduction and induction all fall under the same broad process? If not, is there at least a good, subject-neutral answer to the question for induction (separate from the one for deduction) that we can apply in cases where deduction won't help us?