For the record, logical, analytic, probabilistically correct thinking is only a way of reaching goals. The goals themselves cannot be established by means of logic, only deconflicted/clarified/etc. To think otherwise is illogical.
On the latter point, that's a pretty major dispute in philosophy and mathematics. The axiomatic Hilbert-style approach probably has the upper hand in mathematics, but a Russell-style logicist approach that sees foundations themselves as subject to rational analysis is still a major position, and among philosophers, the idea that axioms are arbitrary and rationality only applies past that is probably the minority view (though existentialists hold a view somewhat like that). Not that that's necessarily practical advice.