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It is widely accepted that the violation of Bell inequalities excludes localtheories of the quantum realm. This paper presents a new derivation of theinequalities from non-trivial non-local theories and formulates a stronger Bellargument excluding also these non-local theories. Taking into account allpossible theories, the conclusion of this stronger argument provably is thestrongest possible consequence from the violation of Bell inequalities on aqualitative probabilistic level (given usual background assumptions). Among theforbidden theories is a subset of outcome dependent theories showing thatoutcome dependence is not sufficient for explaining a violation of Bellinequalities. Non-local theories which can violate Bell inequalities (amongthem quantum theory) are rather characterised by the fact that at least one ofthe measurement outcomes in some sense (which is made precise)probabilistically depends both on its local as well as on its distantmeasurement setting ('parameter'). When Bell inequalities are found to beviolated, the true choice is not 'outcome dependence or parameter dependence'but between two kinds of parameter dependences, one of them being what isusually called 'parameter dependence'. Against the received view established byJarrett and Shimony that on a probabilistic level quantum non-locality amountsto outcome dependence, this result confirms and makes precise Maudlin's claimthat some kind of parameter dependence is required.

A Stronger Bell Argument for (Some Kind of) Parameter Dependence