Ben,

The reason QM does not produce a 4×N table has nothing to do with non locality. It is the same reason why a LHV model of the EPRB experiment does not produce one -- is, it is impossible to measure a particle more than once. In QM terms, the E(a,b) measurement does not commute with the E(a,c) measurement on the same set of particles.

As Adenier explained, there are two possible interpretations for the QM E(a,b), E(a,c).

a) strongly objective: outcomes from a single set of particles ( the same set of lambdas in each expectation value).

b) weakly objective: outcomes from two separate sets of particles (different sets of lambdas)

Bell's inequalities can only be derived for the strongly objective case since you need the same set of lambdas in both for the algebra to proceed. Yet for the strongly objective case the QM expectations do not commute /are incompatible, so you can not simply add outcomes from two separate individual measurements and expect to get the same result as if you had measured both together.

By trivially adding the QM expectations as Bell does, when demonstrating violation by QM, he is assuming that they commute, which is the weakly objective interpretation. So the schism is due to erroneously mixing inequalities derived under a strongly objective interpretation with expectation values from a weakly objective interpretation.

This is the same error von Neumann made in his no-go theorem, which Bell described as silly.

Some here even go as far as to claim that there is no difference between the strongly objective and the weakly objective views. An idea I already demolished using a simple coin toss example(

viewtopic.php?f=6&t=44#p1439), since it implies that counterfactual probabilities are the same as actual ones.

Note: All EPRB experimental results are necessarily weakly objective. The weakly objective inequalities have different upper bounds which have never been violated.

What Watson shows is that Bell's inequalities can not be derived under the weakly objective interpretation, which we must use if the inequalities should be relevant for actual experimental results.