Bear with me, this is kind of complicated:
A. God is, by definition, illogical.
Therefore B. God can not (and does not) exist.
Wow, mind blowing huh? It's just so involved and complex. And it assumes so many premises! (It assumes one premise) But I assure you, the logic is perfectly valid and irrefutable.
Now, while the logic is perfect and irrefutable, you could possibly argue with the premise (Anybody have a definition of a god that is not illogical? If you do, then please do share.), but provided A, then B. God can not and does not exist. If A. is true, B. is true. If A. is not true, B. may or may not be true.
If you have an alternative definition of a god which isn't illogical, I would:
1. Like to shake your hand. You are a very rare breed: a logical theist. I can count on one hand how many I have met.
2. Be happy to have a civil and logical conversation with you on the matter.
Here are some common sayings of theists that provide evidence for A:
"God is, by definition, beyond logic" What's beyond logic? Oh, yes! Illogic.
"God is omnipotent, and immune to logic" The only things that *need* people to claim they are immune to logic are illogical things. Omnipotence is not a logically coherent concept (particularly if it contains those things which are illogical).
"God is outside of reality, and so logic does not apply to him" What's outside of reality? Oh, right- things that aren't real.
"God created logic, and is not subject to his creation." The only use of logic is to falsify things that are illogical (separating what is real from what is false). Did this god create something that doesn't work? Interesting! Does this god break the rules that he himself made? What a dirty cheat!
"Logic is a human invention, it doesn't apply to God." Logic is a human discovery along with mathematics (one plus one was still two in concept before we had words for those concepts), and it only applies to things that are logical (real things). If it doesn't apply to a god, that is only because said god is not real.
If you don't understand why logical contradiction is a problem, study logic, and specifically study the principle of explosion (why logic is not tolerant of illogic).
To be fair, here are some (very rarely seen) sayings of theists that provide some evidence against A. (though seemingly outweighed by those in opposition to these claims):
"God can not defy his own nature" This implies that this god can not be contradictory, which implies that this god might be logical in some way.
"God is perfectly logical and natural" More overt- I see something like this once in a blue moon.
Logical replies will be appreciated and responded to in so far as I have the time to do so.
Replies that overtly reject logic will be hidden (I will assume such replies to be trolling).
EDIT: And unfortunately I have to add another condition: No semantic tomfoolery.
That is, use a dictionary. No dishonest twisting of definitions to suit your agenda, as demonstrated by "DarkQasim" below in the comments.