OK lets give this a stab, and bear with me because I'm thinkng as I type and I'm not even sure I have the language necessary to properly define the answer yet.
Logic in every place I've looked for a definition is, roughly speaking, a system of reasoning that allows us to better predict objective reality by identifying and highlighting unsound arguments. It is not any sort of measure of objective truth in and of itself. It simply works by saying that "in a logically sound argument if the premises are true the conclusion must be true".
It can also play tricks on us. Many logical paradoxes have been pointed out which mean that technically sound arguments with true premises do not lead to a true conslusion. But we don't view these as being indicative of objective reality containing natural paradoxes, we simply view these as either having flawed premises or flawed reasoning.
Therefore the truth value of any logical statement is not based on it's soundness or it's unsoundness (unsound arguments can also sometimes provide true conclusions), but on the reality it is describing.
To argue that because something is logically sound it must also be objectively true, or to turn it around to argue that if something is logically unsound it is objectively false, you first have to find a way of linking the soundness with the truth. And logic itself doesn't claim that sort of power. Not one logician, in the entire history of mankind, will look at you with a straight face and tell you that a logically sound argument is always true.
Now, necessary truths are things that are always true no matter how you conceive of any possible universe. But we don't know, and we can never know, the limits of all of the different types of universe, or indeed if there is anything non-natural beyond the bounds of the natural universe(s). We don't even KNOW that this universe exists. There is no way to prove the statement "this universe exists" logically without either committing a logical fallacy or causing a logical paradox by doing so. So when speaking metaphysically on that scale, there is no such thing as a necessary truth that limits absolutely everything. There are natural necessary truths that limit everything natural, and would work in any type of universe with the same natural qualities as ours. But that doesn't mean that there are necessary truths that limit everything including the non-natural.
I can't describe a non-natural dimension or universe or whatever to you, because it's like asking me to describe how God is possible. It's a badly formed question that cannot ever be answered. But the fact that the question is badly formed doesn't prove that there is no non-natural dimension or there is no God, it just means that natural reasoning and logic performed by a natural being in a natural universe cannot describe or quantify or ever give any rational answer. The question itself is at fault, not the objective truth needed to properly answer it.
By trying to disprove an illogical God using logic, you're doing the same as trying to argue using baseball rules why a player in a golf match has committed an offence. It just doesn't work. Those rules are simply not relevant. And you don't have to say that the baseball rules are flawed because they can't apply to golfers, because for understanding and regulating baseball games they are still fully consistent and absolutely sound. You just have to keep in mind the limits to which they can be applied.