Cambridge University Press eBooks

The Common Cause Principle

Gábor Hofer‐Szabó, Miklós Rédei, László E. Szabó

Published May 16, 2013• 2 citations

Abstract

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 .

DOI: 10.1017/cbo9781139094344.003

Related papers

On Reichenbach's Common Cause Principle and Reichenbach's Notion of Common Cause

Gábor Hofer‐Szabó, Miklós Rédei, László E. Szabó

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.

arXiv:quant-ph/9805066The British Journal for the Philosophy of ScienceSep 1, 1999105 citations

Where do we stand?

Gábor Hofer‐Szabó, Miklós Rédei, László E. Szabó

The history of philosophy teaches us that metaphysical claims of sweeping generality are neither verifiable nor conclusively falsifiable. One can only aim at assessing their plausibility on the basis of the best available evidence provided by the sciences – both formal and empirical sciences. This is what has been done in this book in connection with the Principle of the Common Cause. In Chapter 2, Reichenbach's notion of common cause and the related Common Cause Principle was formulated explicitly in terms of classical, Kolmogorovian probability measure spaces. The Definition 2.4 of common cause followed Reichenbach's original definition closely, and insisting on the quite obvious methodological principle that probabilistic concepts and statements (in particular claims about random events being correlated or probabilistically independent) are only meaningful within the context of a fixed probability measure space in terms of which some segment of reality is modeled, we specified the notions of common cause incomplete and common cause complete probability theories: a theory was defined to be common cause complete if it contains a proper common cause of every correlation it predicts, common cause incomplete otherwise. (There is a strong version of common cause completeness as well: a probability space was defined to be strongly common cause closed if it contains common causes of all admissible types – Definition 4.4.) These explicitly defined notions form the basis on which one can start assessing the status of the Common Cause Principle in the spirit of empirical philosophy. Suppose we have an empirically confirmed scientific theory T describing some segment of reality using (possibly among other mathematical structures) a probability measure space ( X, S, p ).

Cambridge University Press eBooksMay 16, 20130 citations

Completion of the Causal Completability Problem

Michał Marczyk, Leszek Wroński

We give a few results concerning the notions of causal completability and causal closedness of classical probability spaces (Hofer-Szabó et al. [1999]; Gyenis and Rédei [2004]). We prove that (i) any classical probability space has a causally closed extension; (ii) any finite classical probability space with positive rational probabilities on the atoms of the event algebra can be extended to a causally up-to-three-closed finite space; and (iii) any classical probability space can be extended to a space in which all correlations between events that are logically independent modulo measure zero event have a countably infinite common-cause system. Collectively, these results show that it is surprisingly easy to find Reichenbach-style ‘explanations' for correlations, underlining doubts as to whether this approach can yield a philosophically relevant account of causality. 1 Introduction2 Basic Definitions and Results in the Literature3 Causal Completability the Easy Way: ‘Splitting the Atom’4 Causal Completability of Classical Probability Spaces: The General Case5 Infinite Statistical Common-Cause Systems for Arbitrary Pairs6 Conclusion Appendix A

The British Journal for the Philosophy of ScienceMar 23, 201417 citations

Introduction and overview

Gábor Hofer‐Szabó, Miklós Rédei, László E. Szabó

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.

Cambridge University Press eBooksMay 16, 20130 citations

2 Screening Off and Explanation: Formal Properties

Leszek Wroński

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.

Aug 19, 20140 citations

Reichenbach's Common Cause Principle and EPR correlations

Gábor Hofer‐Szabó, Miklós Rédei, László E. Szabó

Einstein-Podolsky-Rosen (EPR) correlations Explaining the correlations predicted by quantum mechanics in the case of joint quantum systems of the EPR type has proved to be a major challenge for the Common Cause Principle. The EPR correlations have been confirmed by a large number of sophisticated experiments carried out since the early 1980s, and, since quantum theory does not give us any hints as to where to look for possible common causes that could explain the correlations in question, the suspicion arose that common causes of EPR correlations might not exist at all. If this were indeed the case, then the EPR correlations would constitute strong empirical evidence against Reichenbach's Common Cause Principle. But how could this be possible in view of the results presented in the previous chapters? As we have seen (Chapter 3), every common cause incomplete probability space is common cause extendible (even strongly common cause extendible with respect to any finite set of correlations; Proposition 3.9); therefore every correlation can, in principle, be explained by a (possibly hidden) common cause, including the (finite number of) EPR correlations. Could the EPR correlations (not) be explained by referring to such hidden common causes? To answer this question, one has to realize first that it is not quite obvious how to link the Common Cause Principle to quantum mechanics because Reichenbach's notion of common cause was defined in terms of classical probability theory (Chapter 2), not in terms of quantum probability theory.

Cambridge University Press eBooksMay 16, 20130 citations