Common-causes are not common common-causes

Hofer-Szabo, Gabor and Redei, Miklos and Szabo, Laszlo E. (2001) Common-causes are not common common-causes.

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.