The violation of Bell-CHSH inequalities leads to different conclusions depending on the description used

Abstract: Since the experimental observation of the violation of the Bell-CHSH inequalities, much has been said about the non-local and contextual character of the underlying system. But the hypothesis from which Bell's inequalities are derived differ according to the probability space used to write them. The violation of Bell's inequalities can, alternatively, be explained assuming that the hidden variables do not exist at all, or that they exist but their values cannot be simultaneously assigned, or that the values can be assigned but joint probabilities cannot be properly defined, or that averages taken in different contexts cannot be combined. All of the above are valid options, selected by different communities to provide support to their particular research program.