One can argue against (a) religion with logic and facts.
With logic: one can exhibit contradictions in religious doctrines. (If there are any.)
With facts: for example geological/fossil evidence can be used to argue against creationism. (The best one can do in trying to put forward 'facts' is to put forward what is well supported by the available evidence, and better supported than known alternatives. So, here for example, one would also have to try and find fault with whatever evidence can be brought forward in support of creationism.)
Whether or not those on each side of a debate will give due dispassionate consideration to the other sides' arguments, is another matter, however. Presumably, what you meant was that one can't argue successfully against religion with logic or facts.
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 think it's more like: As far as everyone is concerned, their own opinions are more correct than everyone else's. Which is what leads to debate, argument, etc. If everyone believed opinions were all equally valid, they'd just live and let live. There'd be no reason for conflict.
Edit: There's something incongruent about trying to discuss an opinion of how the validity of opinions is determined... Hmm...
As far as everyone is concerned, all opinions pretty much are equally valid, whether they actually are or not doesn't seem to matter.