Reichenbachian Common Cause Systems

Abstract

A partition Ci i∈ I of a Boolean algebra $$\mathcal{S}$$ in a probability measure space $$(\mathcal{S},p)$$ is called a Reichenbachian common cause system for the correlated pair A,B of events in $$\mathcal{S}$$ if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in $$(\mathcal{S},p)$$ , and given any finite size n>2, the probability space $$(\mathcal{S},p)$$ can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of $$\mathcal{S}$$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.