Dirty little secret about logic: If induction has a justification problem (and it does), then so does deduction. Why? Because deductions rely on inductive conclusions imported into their premises. Here are a few examples. A. Aristotelian Syllogism: All men are mortal Socrates is a man C: Socrates is mortal Look at premise 1. What gives us the right to say that this is a true premise? Well, because we cast our gaze over a range of humans, and we see that they have all grown old and died.
Susan Haack nicely diagrammed the problem of circularity in her 1976 paper, The Justification of Deduction. In that diagram, she drew a direct parallel to the circularity of the inductive justification of induction, as outlined originally by Hume. Haack argues that justification must mean syntactic justification, and offers an illustrative example argument to show why semantic justification fails – namely, that it is an axiomatic dogmatism: deduction is justified by virtue of the fact that we have defined it to be truth preserving.