Common‐causes are not common common‐causes

Philosophy of Science 69 (4):623-636 (2002)
Abstract
A condition is formulated in terms of the probabilities of two pairs of correlated events in a classical probability space which is necessary for the two correlations to have a single (Reichenbachian) common‐cause and it is shown that there exists pairs of correlated events, probabilities of which violate the necessary condition. It is concluded that different correlations do not in general have a common common‐cause. It is also shown that this conclusion remains valid even if one weakens slightly Reichenbach's definition of common‐cause. The significance of the difference between common‐causes and common common‐causes is emphasized from the perspective of Reichenbach's Common Cause Principle.
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    Jeremy Butterfield (2007). Stochastic Einstein Locality Revisited. British Journal for the Philosophy of Science 58 (4):805 - 867.
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