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Reichenbach defined a common cause which explains a correlation between twoevents if either one does not cause the other. Its intuitive idea is that thestatistical ensemble can be divided into two disjoint parts so that thecorrelation disappears in both of the resulting subensembles if there is nocausal connection between these correlated events. These subensembles can beregarded as common causes. Hofer-Szab\'{o} and R\'{e}dei (2004) generalized aReichenbachian common cause, and called it a Reichenbachian common causesystem. In the case of Reichenbachian common cause systems the statisticalensemble is divided more than two, while it is divided into two parts in thecase of Reichenbachian common causes. The number of these subensembles iscalled the size of this system. In the present paper, we examine Reichenbachiancommon cause systems in general probability theories which include classicalprobability theories and quantum probability theories. It is shown that thereis no Reichenbachian common cause system for any correlation between two eventswhich are not logical independent, and that a general probability theory whichis represented by an atomless orthomodular lattice with a faithful$\sigma$-additive probability measure contains Reichenbachian common causesystems of size 3 for any correlation between two compatible and logicalindependent events. Moreover, we discuss a relation between Bell's inequalityand Reichenbachian common cause systems, and point out that this violation ofBell's inequality can be compatible with a Reichenbachian common cause systemalthough it contradicts a `common' common cause system.

Reichenbachian common cause systems of size 3 in general probability theories