With logic: one can exhibit contradictions in religious doctrines. (If there are any.)
I've always found it a bad idea to pretend to be an expert in someone else's religion. Pointing out contradictions in someone else's religion 99.9% of the time is just going to make you sound very ignorant as to how people interpret the things that you find at first glance contradictory.
Christianity has had 2,000 years to iron out the obvious contradictions. I am pretty sure that the more mature faiths are consistent if you accept their axioms.
I doubt you can think of anything that Augustine and Aquinas didn't. The dudes had nothing else to do but think.
I've always found it a bad idea to pretend to be an expert in someone else's religion. Pointing out contradictions in someone else's religion 99.9% of the time is just going to make you sound very ignorant as to how people interpret the things that you find at first glance contradictory.