Bell's Theorem is a statement of pure classical logic. Out of Bernard d'Espagnat and David Harrison you get this reduction:
(A, B-NOT) plus (B, C-NOT) equals or is greater than (A, C-NOT)
... which is Venn-diagrammable using circles or ovals A, B and C and can't be violated. Add cardinality operators and it becomes the basis for physical experiment using keys, coins, people in a room, cars in a parking lot, dogs in a dog park and so on. Always works as long as the elements in the set are separable.
However, when A, B and C become spin-properties of quanta the situation appears to change. But it's been long argued that the logic of the microscopic world contradicts that of the macroscopic. Does "contradict" translate to "overturn"? Is the glass half-empty or half-full?
Unfortunately, I am not qualified to comment on this issue as I have not even achieved competency in the area of quantum mechanics and its implications for logic. The document I attached under this section was from a question I asked Dr. William Lane Craig which contains his response. He doesn't think Bells Theorem overturns or contradicts classical logic, and I am deferring to him on that.
There may be a bit of confusion here. Bell's Theorem doesn't contradict classical logic -- it can't, because it IS classical logic.
The issue is that Bell's Theorem (or Inequality) itself is believed to be violated (contradicted) in quantum Bell Test experiments --that is, quantum experiments would seem to violate Bell's logical assumptions, which are classical, and by extension classical logic (and classical probability).
Whether this is true or not (and the experiments aren't 100% loophole-free), Bell's Inequality and its logical (physical-world-based) assumptions look to be perfectly correct in the ordinary large-scale, coarse-grained world in which we live.