Logic will only tell you what follows from your choice of axioms, not how to choose them, and only if you can trust your ability to apply it correctly. Absent t

Logic will only tell you what follows from your choice of axioms, not how to choose them, and only if you can trust your ability to apply it correctly. Absent that, your only option appears to be to put your trust in other people - which is, I suppose, what you are saying in your final paragraph.