Recently I have read a lot online about the EPR argument and Bell's inequalities and its implications. When comparing what people write there online with the actual research articles of Einstein and Bell, it seems to me - put drastically - that people online talk about something but not about Bell and Einstein. Let me explain, what I mean.
Online most of the times the logical structure of the argument is roughly given in the following way (see, e.g., here in the first few paragraphs):
Assuming locality and pre-existing properties, Bell's inequalities follow. Quantum mechanics' predictions (and experiments) are in contradiction to the inequality. Hence, one can abandon either locality or pre-existing properties.
Reading the EPR paper (available here) however, the structure seems to be quite different. They consider the following criterion of an element of the physical reality (and a criterion is not an assumption!): If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. This, I think, is very reasonable. One can even very well argue that this is an analytical statement because of the phrase 'without in any way disturbing a system'. So one does not have to assume the correctness of the criterion as it is true anyhow!
It is employed in the following situation (Bohm's version of the EPR set-up):
Think of the singlet state of two spin-1/2 particles with total spin zero. If one measures the spin in the x direction of particle A, one can predict with certainty the spin in x direction of particle B, no matter how far the particles are separated (think of a space-like separation). For the sake of the argument it is not needed that we consider different directions for the two particles.
Now assuming locality, they must conclude by their criterion that there really exist pre-existing properties. That is Einstein.
So, the correct reasoning should be (also in Bell's understanding as he pointed out repeatedly) that locality implies the pre-existing properties. The consequence of Bell's inequalities then is that one cannot abandon either locality or pre-existing properties but must abandon locality.
So, who's right? Why are there two different conclusions? Are Einstein and Bell missing an essential point? I often read that counterfactual definiteness is tacitly assumed. But that isn't an assumption right? It is simply a property of the singlet state if one measures the same spin direction for both particles which is sufficient here.
It would be very much appreciated if it could be pointed out in Einstein's and Bell's original papers where the mistakes or debatable passages are (if there are any), since I think they are a very good basis for this discussion. Thanks.