Here, I explain the problem of defeasible justification.* This is a philosophical puzzle that applies to almost all our knowledge & justified beliefs (all those that have fallible justification); Hume’s problem of induction and the problem of external-world skepticism are special cases.
I think the second solution still leads to infinite regress. Take n to be the necessary truth that provides reasons for rejecting h'. Now, n must provide either defeasible or indefeasible reasons for that. If first, we need to provide a defeasible justification for (n entails ~h'). If second, both n and (n entails ~h') are necessary truths and thus ~h' is a necessary truth, but we assumed it is contingent.
I think the second solution still leads to infinite regress. Take n to be the necessary truth that provides reasons for rejecting h'. Now, n must provide either defeasible or indefeasible reasons for that. If first, we need to provide a defeasible justification for (n entails ~h'). If second, both n and (n entails ~h') are necessary truths and thus ~h' is a necessary truth, but we assumed it is contingent.
Why would we need a justification for (n entails ~h')? (n entails ~h') is false, since n only defeasibly supports ~h'.
Ah, I see my mistake. Now, (n defeasibly supports ~h') can be necessary while ~h' is contingent.