Online research in philosophy

To infer the state of a cause from the states of its effects, independent lines of evidence are preferable to dependent ones. This familiar idea is here investigated, the goal being to identify its presuppositions. Connections are drawn with Reichenbach's (1956) and Salmon's (1984) discussions of the principle of the common cause.

The common cause principle states that correlations have prior common causes which screen off those correlations. I argue that the common cause principle is false in many circumstances, some of which are very general. I then suggest that more restricted versions of the common cause principle might hold, and I prove such a restricted version.

Russell (1948), Reichenbach (1956), and Salmon (1975, 1979) have argued that a fundamental principle of science and common sense is that "matching" events should not be chalked up to coincidence, but should be explained by postulating a common cause. Reichenbach and Salmon provided this intuitive idea with a probabilistic formulation, which Salmon used to argue for a version of scientific realism. Van Fraassen (1980, 1982) showed that the principle, so construed, runs afoul of certain results in quantum mechanics. In this paper a new formulation of the principle is offered that emerges from its use in evolutionary theory. This characterization identifies fairly general conditions in which postulating common causes will be more explanatory than postulating separate causes.

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.

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).

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.

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.