The conclusion is not drawn from the analogy but from the sastra. We don't use a combination of logic and authority, we use authority. Logic we use to convince someone who doesn't accept the authority. The basic principle is authority. Vedas say that cowdung is pure and we accept it. There is no logic, but when we practically use it we see that it is correct.
Letter to Satsvarupa das Maharaja, 21 October, 1975