# Causal completeness of probability theories — Results and open problems

# Causal completeness of probability theories — Results and open problems

A classical (Kolmogorovian) probability measure space is defined to be causally closed with respect to a causal independence relation between pairs of random events if the probability space contains a Reichenbachian common cause of every correlation between causally independent random events. A number of propositions are presented that characterize causal closedness. Generalizing the notion of Reichenbachian common cause in terms of non‐classical probability spaces, where the Boolean algebra of random events is replaced by a non‐distributive orthocomplemented lattice, the notion of causal closedness is defined for non‐classical probability spaces and propositions are presented that state causal closedness of certain non‐classical probability spaces as well. Based on the generalization of the notion of common cause to a common cause system containing *N* random events, causal *N*‐closedness is defined with respect to a common cause system both in classical and non‐classical probability spaces, and the problem of causal *N*‐closedness is formulated. Characterizing causal *N*‐closedness remains a largely open problem.

*Keywords:*
common cause, probabilistic causality, Reichenbach

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