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