I suppose this amounts to, “You have to use logic to show God is required for logic.” It wouldn’t necessarily be circular, because you’d be using the logic God made possible, to prove that God made logic possible. Yet there must be genuine evidence outside this circle, arguments made using logic that are more than pointing fingers or waving hands.
I think logic exists independently from God, but that God was required for anyone to think logically. I don’t think God created the laws of nature, but that what we’re seeing in nature is the result of what God could do, in the process of creating nature. God can leave it alone, to a certain extent, and the laws abide, perhaps with human observers. One and one still equal two, even if someone only checks now and then. Categories of things exist, and true or false propositions can be generated from the categories, as we get around to it.
There are two more things I’d like to relate in this context. First, at no time in history has anyone inquired seriously into the mind, cataloging its powers and identifying the source and use of these powers. This requires an unheralded degree of introspection, even beyond the boasts about enlightenment in the East. There’s even a battle going on that some believe consciousness has a material (biochemical, neurological) cause, and this battle only occurs because the religions never touched this critical topic in serious depths. To think logically is actually a miracle, and one almost certainly not a result of evolution.
Second, God laments that He could not use logic in the religions, for the human entities are not ready. By rights God should merely need to say, “You are all created souls, now in the bodies of intelligence. I expect you to use this intelligence to dwell harmoniously and joyfully, optimizing the globe.” After hearing this and seeing its truth, the wise humans never make war or even quarrel again, mastering their minds and situations with power! In reality humans cannot accept this proposition, despite its inerrant logic. They also cannot accept any of the corollary or supporting propositions, that would make such a harmonious situation practically possible. Men are not logical enough to become good.