Why the problem of induction is not limited to induction

Ever since Hume started to wonder if inductive arguments are ever justified, a common concern amongst epistemologists arose. How do we know something based on induction if we cannot philosophically justify the method as such? Can science, which is based on induction, give us any knowledge? How do we know that if I jump from the window I will fall? Various authors have tried to solve these questions, but the problems persist, some form of local scepticism seems to find its way. But what if the problem is not only to do with induction? What if the principle of criticizing the method applies to all forms of rational thinking? In the next few paragraphs I will try to give an answer to this question.
Firstly, the problem of induction in a nutshell is about how we justify arguments that involve taking a leap from the premises to the conclusions, arguments that inform us about the future, arguments that generalize that from the observed cases we can know the unobserved cases: inductive arguments. These typo of argumentation assumes the uniformity principle, which states that the world will continue to be as it is now in the future. We constantly use these inductive arguments on a daily basis, so what is wrong with them? Hume claims that the uniformity principle is not self-justificatory. We can picture a world in which the future has nothing in common with the past. So how de we justify it? We cannot justify induction "a priori" since there is no way of jumping directly from premises to conclusions in a deductively valid way and "a priori" arguments do not say anything about future states. We cannot justify it "a posteriori" since that would make us state that the uniformity is justified because up until now it has worked, that of course would be a circular argument. Since we do not want to beg the question, could we use abduction? The answer is no, because it is hard to define which is the best explanation and abduction does involve some kind of leap from premises to conclusion, so it seems that the uniformity principle must remain unjustified, which is far from being ideal. I am not going to go through the answers philosophers have give to the problem, since that is not the purpose of this short article. My question is: does this sceptical method of questioning induction apply to other methods? The answer seems to me to be yes.
Let's go with abduction, how do we justify it? We cannot do so using induction due to the problem I have just explained, we cannot use abduction since that would make the argument circular and we cannot use deduction either because there is no deductive way of doing so, all of our deductive attempts will fail, we can always question the premises and providing an argument that moves directly from premises to conclusion (deduction) to justify a method that does not do so (abduction or induction) does not seem to make much sense to me. What about deduction? We have got the same problem! You cannot justify it by using deduction because that would be circular, you cannot employ induction due to the problem of induction and you cannot use abduction either form the same reasons. The problem of induction can thus be extrapolated to our three forms of reasoning. It seems that philosophy is unable to justify itself.