You needn’t take me to be saying that the inductive premises are obviously correct; I’m saying that conventional speech is such that “evidence” is implicitly always relative to certain ubiquitous background assumptions unless otherwise specified, “justification” is always implicitly always relative to those, etc., and that the inductive premises are part of that conventional background theory. I’m saying, as far as determining what counts as evidence, justification, etc., it doesn’t matter whether that background theory is correct or not, since we are always implicitly only evaluating evidentiary status relative to it.
That having been said, it is woven into the fabric of our fundamental (extra-deductive) logic to speak this way, and there is no more use quibbling over the hesitation that those particular background premises (or equivalently, methods of inference) haven’t been validated by some nebulous external criterion than there is in worrying about the possibility that, say, “A and (B or C) implies (A and B) or (A and C)” or any other aspect of our logic isn’t actually legitimate. It is legitimate by fiat, legitimated by definition, by the empirically observable nature of the language-game of reasoning we play, the same way it is legitimate to use the word “raven” for a particular kind of bird rather than a barstool for no more reason than that that is the customary employment of that term; all reasoning must come to an end somewhere, and acceptance of inductive reasoning is, in the contexts of concern, part of that starting point.