You apparently don't think it's self evident that induction works. Do you think that it's self evident that deduction works (like Descartes did)? If you do, why? If you met someone who was willing to accept the premises of a deductive argument but not the conclusion, like the tortoise in this parable, how would you convince the person they were wrong? The only way to do it would be using deductive arguments, but that's circular! So it seems that deductive arguments are just as "unjustifiable" as inductive arguments.
But if you reject deductive arguments as well, then you can't do anything. Even if you start with self-evident premises, you won't be able to conclude anything from them. Perhaps this is a hint that your standards for justification are so high that they're effectively useless.
A premise isn't self-evident because anybody whatsoever would accept it, but because it must be true in any possible universe.
Deductive arguments aren't self-evident, but for a different reason than you think- the Evil Demon Argument, which shows that even if it looks completely solid it could easily be mistaken. There may be some way to deal with it, but I can't think of any. That's why I came here for ideas.
You claim my standards of justification are too high because you want to rule skepticism out- you are implicitly appealing to the fact skepticism results as a reason for me to lower my standards. Isn't that bias against skepticism, lowering standards specifically so it does not result?
I've raised arguments for philosophical scepticism before, which have mostly been argued against in a Popper-esque manner of arguing that even if we don't know anything with certainty, we can have legitimate knowledge on probabilities.
The problem with this, however, is how you answer a sceptic about the notion of probability having a correlation with reality. Probability depends upon axioms of probability- how are said axioms to be justified? It can't be by definition, or it has no correlation to reality.