It is widely accepted that the violation of Bell inequalities excludes local theories of the quantum realm. This paper presents a stronger Bell argument which even forbids certain non-local theories. The remaining non-local theories, which can violate Bell inequalities, are characterised by the fact that at least one of the outcomes in some sense probabilistically depends both on its distant as well as on its local parameter. While this is not to say that parameter dependence in the usual sense necessarily holds, it shows that the received analysis of quantum non-locality as “outcome dependence or parameter dependence” is deeply misleading about what the violation of Bell inequalities implies
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