This question is one that has been bothering me for some time now. We have assumed, based on our observations, that the universe is subservient to the principles of classical logic. But is it necessarily so? Could it be that the universe is not logically classical but only behaves in that manner under the limits we can observe?
Additionally, there are known to be several gaps in modern theory. Is it possible that some (though not necessarily all) of them arise out of applying the limits of classical logic on systems that do not adhere to classical logic?
It seems highly unlikely, and the question feels extremely isolated from physical reality, but I cannot convince myself that it is trivial enough to be straightforwardly dismissed. So, to sum up:
Can we be absolutely certain that our universe is a system that adheres to the laws of classical logic?
Edit: Just to be clear, I do not refer to classical physics but to classical logic, the system of logic in which the principle of explosion, the principle of noncontradiction, and the law of excluded middle holds.