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.
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.
Common causes and common common causes Proposition 3.9 tells us that every common cause incomplete classical probability space can be strongly common cause extended with respect to any (finite) set of correlations; Proposition 4.19 states that every classical probability space is even common cause completable. Note that what these propositions say is not that for a set of correlations between ( A i , B i )( i = 1, 2 … n ) there exists a single, common common cause C in the extension (or completion) for the whole set of correlations; in fact, the common causes C i constructed explicitly in the proof of Proposition 3.9 are all different: C i ≠ C j ( i ≠ j ). This observation leads to the following question. Let ( A i , B i )( i = 1, … n )be a finite set of pairs of events in( X, S, p )that are correlated [ Corr p ( A i , B i ) ＞ 0 for every i ]. We say that C is a common common cause of these correlations if C is a Reichenbachian common cause of the correlated pair ( A i , B i )for every i . Does every set of correlations in a classical probability space have a common common cause? In view of the generality of this question one may surmise that the answer to it is “yes”; that is to say, one may conjecture that given any two correlations there can always exist a Reichenbachian common cause which is a common cause for both correlations, since, one may reason, we just have to refine our picture of the world by adding more and more events to the original event structure in a consistent manner, and finally we shall find a single common cause that explains both correlations.
Common cause partitions Confronted with a common cause incomplete probability space ( X, S, p )in which a direct causal influence between the correlated events is excluded, one can have in principle two strategies aiming at saving the Common Cause Principle: one may try to argue that S is not “rich enough” to contain a common cause, but there might exist a larger probability space ( X ′, S ′, p ′) that already contains a common cause of the correlation. As we have seen in Chapter 3 this strategy always works in the sense that it is always possible to enlarge ( X, S, p ) in such a way that the enlarged probability space already contains an event C that satisfies the Reichenbachian conditions. Another natural idea is to suspect that the correlation between A and B is not due to a single factor but may be the cumulative result of a (possibly large) number of different “partial common causes,” none of which can in and by itself yield a complete common-cause-type explanation of the correlation, all of which, taken together, can however account for the entire correlation. In this chapter we elaborate this idea by formulating precisely a notion of the Reichenbachian Common Cause System (RCCS) and prove propositions on the existence and features of such systems. As we have seen in Chapter 2, if the events A, B, C satisfy the Reichenbachian conditions (2.5)–(2.8) then there is a positive correlation between A and B (Proposition 2.5).
Reichenbach's Common Cause Principle is the claim that if two events are correlated, then either there is a causal connection between the correlated events that is responsible for the correlation or there is a third event, a so called (Reichenbachian) common cause, which brings about the correlation. The paper reviews some results concerning Reichenbach's notion of common cause, results that are directly relevant to the problem of how one can falsifY Reichenbach's Common Cause Principle. Special emphasis will be put on the question of whether EPR-type correlations can have an explanation in terms of Reichenbachian common causes. Most of the results to be recalled indicate that falsifYing Reichenbach's Common Cause Principle is much more tricky than one may have thought and that, contrary to some claims in the literature, there is no conclusive proof yet that the EPR correlations predicted by ordinary, non-relativistic quantum mechanics cannot have a common cause; furthermore, recent results about possible Reichenbachian common causes of correlations predicted by quantum field states between spacelike separated local observable algebras in algebraic quantum field theory strongly indicate that there may very well exist Reichenbachian common causes of superluminal correlations predicted by quantum field theory.
Common cause (in)completeness and extendability In principle, there are two ways to interpret Reichenbach's Common Cause Principle in general, each determined by how one views the status of the Principle with respect to the conditions of its validity: the Principle can be viewed as a falsifiable or a nonfalsifiable principle. In the falsificationist interpretation, the Common Cause Principle is a claim that can possibly and conclusively be shown not to hold for some empirically given events and their correlations; in the nonfalsificationaist interpretation, the Common Cause Principle cannot be falsified conclusively – whatever the actual circumstances. Is the Common Cause Principle falsifiable or nonfalsifiable? We shall argue that the Common Cause Principle, as formulated in Chapter 2, is not falsifiable conclusively, but the argument cannot be trivial, since the Common Cause Principle is certainly not trivially nonfalsifiable: it is not true that every classical probability space ( X, S, p ) is provably common cause complete in the sense that for any A, B ϵ S that are correlated in p there exists a C ϵ S that is a ( proper ) common cause of the correlation between A and B . There exists common cause incomplete probability spaces (for instance the probability space described in Figure 3.1 is common cause incomplete). This makes the following definition nonempty: Definition 3.1 The probability space ( X, S, p )is common cause incomplete if there exist events A, B ϵ S such that Corr p ( A, B ) ＞ 0 but S does not contain a common cause of the correlation Corr p ( A, B ).
Given a probabilistic correlation between two events, this correlation might be explained in terms of a common cause. In [1] Reichenbach defines the notion of common cause and shows that the definition is consistent with the explicable correlation, i.e. if two events have a common cause then they do correlate. In this paper we generalize the notion of common cause to Hilbert lattices in two different ways according to the two different definitions of conditional probability in the quantum case, and show that Reichenbach's theorem does not hold in either case. There will be given counter-examples when a common cause 'causes' correlation, anticorrelation and independence, respectively.
It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains events that can be regarded as common causes of the correlations in the sense of Reichenbach's definition of common cause. It is shown, further, that, given any quantum probability space and any set of commuting events in it which are correlated with respect to a fixed quantum state, the quantum probability space can be extended in such a way that the extension contains common causes of all the selected correlations, where common cause is again taken in the sense of Reichenbachs definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle.