Online research in philosophy

A partition $\{C_i\}_{i\in I}$ of a Boolean algebra $\cS$ in a probability measure space $(\cS,p)$ is called a Reichenbachian common cause system for the correlated pair $A,B$ of events in $\cS$ 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 $(\cS,p)$, and given any finite size $n>2$, the probability space $(\cS,p)$ 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 \cS$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.

I consider the problem of extending Reichenbach's principle of the common cause to more than two events, vis-a-vis an example posed by Bernstein. It is argued that the only reasonable extension of Reichenbach's principle stands in conflict with a recent proposal due to Horwich. I also discuss prospects of the principle of the common cause in the light of these and other difficulties known in the literature and argue that a more viable version of the principle is the one provided by Penrose and Percival (1962).

Hofer-Szabo, Redei and Szabo (Int. J. Theor. Phys. 39:913–919, 2000) defined Reichenbach’s common cause of two correlated events in an orthomodular lattice. In the present paper it is shown that if logical independent elements in an atomless and complete orthomodular lattice correlate, a common cause of the correlated elements always exists.

Suppose that two geysers, about one mile apart, erupt at irregular intervals, but usually erupt almost exactly at the same time. One would suspect that they come from a common source, or at least that there is a common cause of their eruptions. And this common cause surely acts before both eruptions take place. This idea, that simultaneous correlated events must have prior common causes, was first made precise by Hans Reichenbach (Reichenbach 1956). It can be used to infer the existence of unobserved and unobservable events, and to infer causal relations from statistical relations. Unfortunately it does not appear to be universally valid, nor is there agreement as to the circumstances in which it is valid.

The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach's Common Cause Principle.

A classical probability measure space was defined in earlier papers \cite{Hofer-Redei-Szabo1999}, \cite{Gyenis-Redei2004} to be common cause closed if it contains a Reichenbachian common cause of every correlation in it, and common cause incomplete otherwise. It is shown that a classical probability measure space is common cause incomplete if and only if it contains more than one atom. Furthermore, it is shown that every probability space can be embedded into a common cause closed one; which entails that every classical probability space is common cause completable with respect to any set of correlated events. The implications of these results for Reichenbach's Common Cause Principle are discussed, and it is argued that the Principle is only falsifiable if conditions on the common cause are imposed that go beyond the requirements formulated by Reichenbach in the definition of common cause.

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.

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.