Would not.
If I insist God does not exist, then (by your own standard) I am not bound by logic. You, on the other hand, believing He DOES exist, ARE so bound. So if I am going to speak to you in a language you understand, it has to be Logic. That doesn’t mean I believe in it (or its Creator), simply that it serves as a convenient medium for the exchange of ideas.
In MY world, it would be just as meaningful to pound on the keyboard like a monkey. But in this exercise, if I wish to communicate with you, I have to speak your language.