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2 Screening Off and Explanation: Formal Properties

Leszek Wroński

Published Aug 19, 2014• 0 citations
Publisher

Abstract

Arguably, the most important feature of Reichenbach-style common causes is that they screen o their e ects.In this chapter we will consider a few constructions based on this idea which extend, or transform, Reichenbach's original notion.A part of the literature refers to events de ned as meeting Reichenbach's conjunctive fork conditions as ("Reichenbachian") "common causes"; see e.g.Hofer-Szabó and Rédei (2004).Hofer-Szabó et al. (2000) and Hofer-Szabó and Rédei (2006) even go so far as to state that Reichenbach himself de ned common causes as the middle elements of conjunctive forks with correlated extreme elements; in other words, that ful lling the statistical requirements for being the middle element of a conjunctive fork is su cient to be a common cause for the correlated events.Due to the reasons already presented in section 1.3 we are reluctant to adhere to this tradition.Nevertheless, the main results of this work (see chapter 5) pertain to problems posed in various papers by the above-cited authors.Therefore, some slight terminological changes are in order. De nition 9 [Statistical CommonCause] Let Ω, F, P be a probability space.Let A, B ∈ F. If there exists C ∈ F di erent from both A and B such that P(AB | C) = P(A | C)P(B | C); P(AB | C ⊥ ) = P(A | C ⊥ )P(B | C ⊥ ); P(A | C) > P(A | C ⊥ ); P(B | C) > P(B | C ⊥ ), then C is called a statistical common cause of A and B.

DOI
10.2478/9783110372717.2

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Arguably, the most important feature of Reichenbach-style common causes is that they screen o their e ects.In this chapter we will consider a few constructions based on this idea which extend, or transform, Reichenbach's original notion.A part of the literature refers to events de ned as meeting Reichenbach's conjunctive fork conditions as ("Reichenbachian") "common causes"; see e.g.Hofer-Szabó and Rédei (2004).Hofer-Szabó et al. (2000) and Hofer-Szabó and Rédei (2006) even go so far as to state that Reichenbach himself de ned common causes as the middle elements of conjunctive forks with correlated extreme elements; in other words, that ful lling the statistical requirements for being the middle element of a conjunctive fork is su cient to be a common cause for the correlated events.Due to the reasons already presented in section 1.3 we are reluctant to adhere to this tradition.Nevertheless, the main results of this work (see chapter 5) pertain to problems posed in various papers by the above-cited authors.Therefore, some slight terminological changes are in order. De nition 9 [Statistical CommonCause] Let Ω, F, P be a probability space.Let A, B ∈ F. If there exists C ∈ F di erent from both A and B such that P(AB | C) = P(A | C)P(B | C); P(AB | C ⊥ ) = P(A | C ⊥ )P(B | C ⊥ ); P(A | C) > P(A | C ⊥ ); P(B | C) > P(B | C ⊥ ), then C is called a statistical common cause of A and B.
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 notion of common cause In what follows ( X, S, p ) denotes a classical Kolmogorovian probability space with Boolean algebra S of subsets of a set X (with respect to the set theoretic operations ∩, ∪ and A ⊥ = X\A as Boolean algebra operations) and with the probability measure p on S . (See the Appendix for a concise review of the basic concepts of measure theoretic probability theory.) Elements of S are called random events (elements of X are sometimes called (random) elementary events). It is common to assume in probability theory that p is a countably additive (also called σ-additive) and not just a finitely additive measure but the assumption of countable additivity is somewhat controversial in the philosophical literature. The distinction between countable and finite additivity will not play any role in Chapters 3–5, 7 and 9: the results presented are valid under the assumption of either finite or countable additivity. Countable additivity will play a role in Chapters 6 and 8, however, where the problem of correlations in nonclassical (quantum) probability spaces will be investigated, and where the quantum counterpart of p will be assumed to be countably additive (“normal” in the terminology of the theory von Neumann algebras). If S has a finite number of elements, then it is the power set P(X) of a set X having n ＜ ∞ elements denoted by a i ( i = 1, 2 … n ); in this case we write S n .
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.
No correlation without causation. This is, in its most compact and general formulation, the essence of what has become Reichenbach's Common Cause Principle . More explicitly, the Common Cause Principle says that every correlation is either due to a direct causal effect linking the correlated entities, or is brought about by a third factor, a so-called Reichenbachian common cause that stands in a well-defined probabilistic relation to the correlated events, a relation that explains the correlation in the sense of entailing it. The Common Cause Principle is a nontrivial metaphysical claim about the causal structure of the World and entails that all correlations can (hence, should) be explained causally either by pointing at a causal connection between the correlated entities or by displaying a common cause of the correlation. Thus, the Common Cause Principle licenses one to infer causal connections from probabilistic relations; at the same time the principle does not address whether the causal connection holds between the correlated entities or between the common cause and the elements in the correlation. While the technically explicit notion of common cause of a probabilistic correlation within the framework of classical Kolmogorovian probability theory is due to Reichenbach (1956), the Common Cause Principle was articulated explicitly only later, especially in the works by W. Salmon (see the “Notes and bibliographic remarks” to Chapter 2). The chief aim of this book is to investigate the Common Cause Principle; in particular, the problem of to what extent the Common Cause Principle can explain probabilistic correlations.
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In recent years part of the literature on probabilistic causality concerned notions stemming from Reichenbach’s idea of explaining correlations between not directly causally related events by referring to their common causes. A few related notions have been introduced, e.g. that of a “common cause system” (Hofer-Szabó and Rédei in Int J Theor Phys 43(7/8):1819–1826, 2004 ) and “causal (N-)closedness” of probability spaces (Gyenis and Rédei in Found Phys 34(9):1284–1303, 2004 ; Hofer-Szabó and Rédei in Found Phys 36(5):745–756, 2006 ). In this paper we introduce a new and natural notion similar to causal closedness and prove a number of theorems which can be seen as extensions of earlier results from the literature. Most notably we prove that a finite probability space is causally closed in our sense iff its measure is uniform. We also present a generalisation of this result to a class of non-classical probability spaces.