Deep waters indeed. I am inclined to hold that it cannot, although it certainly can strongly suggest it, contingent on the strength of the logical system. A lot of folks smarter than I am have tried it (Leibniz and Godel to name only two). But I don't think that faith would be necessary if it were actually possible to nail the thing down with human logic. A lot of folks smarter than I am disagree... ;-)
I guess we’re sort of saying the same thing from different perspectives. The Bible says that God has concealed himself from those considered wise in a worldly way. Now I think that in part He does this in exactly the way you allude to, that logic, pursued honestly, strongly suggests His existence, but it’s equivocal enough so that if you want to deny it, you can. So that’s one way he calls humble ones. That’s my story and I’m sticking to it.
I guess I shouldn’t have said “demonstrate”. From another perspective, how about the fact the the very existence and beauty of logic leave imply an awesome God, just like a beautiful field of flowers or a starry night sky? Not persuasive, I know, to someone who doesn’t want to be persuaded, but it was another slightly related facet of our discussion that occurred to me.