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It's impossible to "prove" within the rigors of formal logic the existence or non-existence of God or gods.
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This, I agree, is completely unacceptable. For me, however, this is for a completely different reason. I believe it is possible to prove (no quotation marks) the existence of God using logic from the standpoint of the impossibility to the contrary (transcendental argument). I will argue that only theism offers the preconditions of intelligibility for reality.
This is not to say atheists are not intelligent or committed to intelligence and rationality. Quite the contrary. It is only to say that though atheists can count, they cannot "account for their counting." There is nothing in the atheist worldview that can account for using certain philosophical tools (laws of logic, absolute abstracts, etc.). I propose that in so doing, they are borrowing from the theist worldview, which is able to account for the absolutes everyone uses.
Two asides before the onslaught begins. I'll be defending Christian theism because 1) I find other theistic religions to be philosophically indefensible and 2) since I am, by the grace of God, a Christian, I cannot ethically argue for a position with which I disagree.
I apologize for everyone with loads of scientific evidence. I wasn't blessed with that tendency (and, yes, I have empirically observed that). I also apologize to any Christians reading this who can argue better than I. Poorly argued truth is a sad thing to see.
Send out the lions.