Bell's Theorem was developed on the basis of considerations

involving a linear combination of spin correlation functions, each

of which has a distinct pair of arguments. The simultaneous

presence of these different pairs of arguments in the same

equation is investigated, and the implicit counterfactual

assumption in Bell's theorem is discussed.

We show how an explicit contextuality can arise from a model

displaying unfair sampling, and we discuss it in the

framework of David Mermin's cleverly simple version of Bell's

theore, which pinpoints in a very straightforward way how

interpreting entanglement from a realistic point of view can be

problematic. We present an extended version of Mermin's device

that can actually be given a straightforward realistic

interpretation.

We stress that the low efficiency of detectors in all experiments

with photons makes the use of the fair sampling assumption

unavoidable. Since this very assumption is false in all existing

local realistic models based on inefficient detection, we thus

question its validity. We show that it is no more reasonable to

assume fair sampling than it is impossible to test, and we

actually propose an experimental test which would provides clear

cut results in case of unfair sampling

We then analyze optical EPR experimental data performed by Weihs et al in Innsbruck 1997-1998. We show that for some linear

combinations of the raw coincidence rates, the experimental

results display some anomalous behavior that a more general source state (like non-maximally entangled state) cannot

straightforwardly account for. We use the fair sampling

assumption, and assume explicitly that the detection efficiencies

for the pairs of entangled photons can be written as a product of

the two corresponding detection efficiencies for the single

photons. We show that this explicit use of fair sampling cannot be

maintained to be a reasonable assumption as it leads to an

apparent violation of the no-signalling principle.