Here's the argument from logic as best I could reconstruct it from everything I've heard about it.
1. The laws of logic are abstract entities.
2. Abstract entities are concepts.
3. Concepts can only exist in minds.
4. Therefore, the laws of logic can only exist in minds (or a mind).
6. If the mind(s) that contains the laws of logic were not necessary, then it could fail to exist.
7. If the mind(s) could fail to exist, then logic could fail to exist.
8. But logic cannot fail to exist.
9. Therefore, the mind(s) that contains the laws of logic is necessary.
10. Therefore, at least one necessary mind exists and is the ground of logic.
The only premise I have any reservation about at all is the second premise. I have run across people who accept the second premise but reject the eighth premise. They reject it precisely because they accept the second premise. If the laws of logic are only concepts, then if nobody is around to conceive of them, then the laws of logic wouldn't exist. The laws of logic are just products of human thought.
When people say that the laws of logic are just products of human thought, I can't help but wonder if there's some misunderstanding going on. If the laws of logic are just products of human thought, then it would seem to follow that before there were ever humans, the world could've been a topsy turvy place. There could've been square circles. Things could exist and not exist at the same time. All sorts of absurdities could've taken place. The universe would not have operated according to any laws at all since all laws presuppose the laws of logic.
I've always suspected that when people say the laws of logic are human concepts, what they are referring to is not the laws themselves, but either the articulation of them in language or the propositions that describe them. So they're not really talking about whether or not there could have been square circles during the Jurassic period. They're talking about whether or not propositions like "If two statements contradict each other, they cannot both be true" were just floating around with nobody thinking them.
When I talk about the laws of logic, I'm not talking merely about the concepts or propositions that we think when we think of the laws of logic. I'm talking about the way reality actually is. I'm talking the fact that in reality, there cannot actually be things that both exist and don't exist at the same time and in the same sense. I'm talking about the fact that if A is bigger than B, and B is bigger than C, then A is bigger than C. It seems to me that would be the case whether anybody was thinking about it or not.
So I wonder if there's some equivocation going on when people make the argument for God from logic. On the one hand, the articulation of the laws of logic are propositions that can only exist in a mind. But on the other hand, the reality captured by these propositions exists independently of minds. While one is not necessary, the other is necessary, but they get conflated in the argument resulting in the conclusion that there are necessary concepts that exist in a necessary mind.
In the eighth premise, I mean that the reality described by what we call the laws of logic cannot be otherwise. I mean that there are no possible worlds in which things can both be and not be at the same time and in the same sense.
It seems to me that the basic laws of logic are about as fundamental to reality as you can get. As Aristotle once pointed out, there can be no significant speech or action without the law of non-contradiction. But I would go a step further and say that there can be no significant (or meaningful) existence without the law of non-contradiction. Without the law of non-contradiction, there would be no difference between existing and not existing. The statement, "Bob exists" wouldn't communicate anything if it did not exclude the statement, "Bob does not exist." If Bob could both exist and not exist at the same time and in the same sense, then if I told you he exists, I would not be ruling out his non-existence. He may not exist even if it's true that he exists. So me telling you he exists would be meaningless. And it's not just me telling you that he exists. His existing would, itself, be meaningless.
That's why I have a hard time believing that the laws of logic depend on the mind of God. It seems to me to be the other way around. Unless the laws of logic were true, God couldn't definitely exist. God's existence would be a meaningless thing if not for the law of non-contradiction.
I do agree that the laws of logic are abstract. They are not concrete things. But they don't appear to be concepts either in the sense that they are simply ideas that only exist in thought. I agree with what Ronald Nash said: "The law of noncontradiction is not simply a law of thought. It is a law of thought because it is first a law of being" (Worldviews In Conflict, p. 84). That's why I don't think the second premise is true.
There's another category of people who either deny the laws of logic, or deny the universal validity of the laws of logic, or deny the necessity of logic. Quite frankly, I think those people are either nuts or they just haven't been able to wrap their heads around what the laws of logic actually are.
But I'm curious what you think. I'd like to know what you think of the argument as a whole, but I'd especially like to know what you think of the second premise. Also, what do you think the laws of logic are?