Generalised Reichenbachian common cause systems

Abstract

Chancy coincidences happen everyday, but sometimes coincidences are just too striking, or too improbable, not to reveal the presence of some coordinating process. To wit, if all the electrical appliances in a building were to shut down at exactly the same time, it would not be unreasonable to search for a breakdown in their common power supply. Similarly, if the price of petrol were to simultaneously rise in all oil importing countries, it would be a fair bet that exporters had concertedly decided to reduce extraction. The principle of the common cause is the inferential rule governing instances of this kind: informally stated, it asserts that improbable coincidences are to be put down to the action of a common cause.Reichenbach [15] was the first to provide the principle of the common cause with a matematical characterisation. His treatment relied on three major ingredients: first, he represented improbable coincidences as positive probabilistic correlations between random events; second, he demanded that common causes should increase the probability of their effects; and third, he further required that conditioning on the presence, or on the absence, of a common cause should make its effects probabilistically independent from one another.Reichnenbach's treatment, however, was overly restrictive, as it rested on a too narrow conception of improbable coincidences, and on a correspondingly narrow understanding on the explanatory function of common causes. In [11], I accordingly proposed an improved interpretation of the principle, along with a suitably revided probabilistic model for common causes, which generalises Reichenbach's original formulation in two respects. On the one hand, it represents improbable coincidences not as positive correlations, but rather as positive differences between the correlation actually exhibited by a speficied pair of events, and the correlation that they should exhibit according to historical data, background beliefs, or established theory. On the other hand, and correspondingly, it demands that conditioning on the presence or on the absence of a common causes should restore the expected correlation between its effects.Reichenbach's understanding of the principle is demonstrably a special case of this interpretation, applying when the expected correlation between the events of interest is null. Nevertheless, there is one respect in which the probabilistic model proposed in [11] is still not general enough. Like Reichenbach's original account, in fact, it depicts the action of a single common cause, and it is accordingly inadequate to capture