I'd like to point out that there could be more than only one
logic.
A logical system can be thought as a set of foundative axioms (which are statements), and a set of rules to deduce a new statement from already existing ones.
Actually, there could be different sets of rules for the same set of axioms, generating different logics.
Anyway, keeping this in mind it could be quite difficult to find out what are the implicit axioms of a man reasoning system, because with don't know neither the deducing rules.
In some respect is the same as with
geometry: you can stick to
euclidean geometry, but it's not the only one, and, actually,
relativistic theory is not based on
euclidean geometry.