Extending the ideas from (Hofer-Szabó and Rédei [2006]), we introduce the notion of causal up-to-n-closedness of probability spaces. A probability space is said to be causally up-to-n-closed with respect to a relation of independence R_ind iff for any pair of correlated events belonging to R_ind the space provides a common cause or a common cause system of size at most n. We prove that a finite classical probability space is causally up-to-3-closed w.r.t. the relation of logical independence iff its probability measure is constant on the set of atoms of non-0 probability. (The latter condition is a weakening of the notion of measure uniformity.) Other independence relations are also considered.
Extending the ideas from (Hofer-Szabó and Rédei [2006]), we introduce the notion of causal up-to-n-closedness of probability spaces. A probability space is said to be causally up-to-n-closed with respect to a relation of independence R_ind iff for any pair of correlated events belonging to R_ind the space provides a common cause or a common cause system of size at most n. We prove that a finite classical probability space is causally up-to-3-closed w.r.t. the relation of logical independence iff its probability measure is constant on the set of atoms of non-0 probability. (The latter condition is a weakening of the notion of measure uniformity.) Other independence relations are also considered.
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.
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 aim of this paper is to give a review of the known results and open problems concerning causal completeness (closedness) of classical and non-classical (quantum) probability spaces. Causal closedness of a probabilistic theory means that the theory is causally rich enough to be able to explain causally all the correlations it predicts. It is natural to ask whether probabilistic theories are causally closed if one assumes that Reichenbach’s Common Cause Principle holds: this principle states that if two events A and B are probabilistically correlated then either the correlation is due to a causal interaction between A and B, or, if A and B are causally independent, Rind(A,B), then there exists a third event C, a so-called common cause that explains the correlation by being related to A and B probabilistically in a specific way (see Definition 1). Causal closedness of a probabilistic theory is intended to express that the theory complies with the Common Cause Principle; accordingly, and more precisely, a probability theory is defined to be causally closed with respect to a causal independence relation
Causal closedness and common cause closedness Assuming that Reichenbach's Common Cause Principle is valid, one is led to the question of whether our probabilistic theories predicting probabilistic correlations can be causally rich enough to also contain the causes of all the correlations they predict. The aim of this chapter is to formulate precisely and investigate this question. According to the Common Cause Principle, causal richness of a theory T would manifest in T 's being causally closed (complete) in the sense of being capable of explaining the correlations by containing a common cause of every correlation between causally independent events A, B . This feature of a theory is formulated in the next two definitions of causal closedness. In both definitions ( X, S, p )is a probability space and R ind is a two-place causal independence relation that is assumed to have been defined between elements of S . We treat the relation R ind as a variable in the problem of causal closedness; hence, at this point we leave open what properties R ind should have to be acceptable as a causal independence relation – later we will return to the issue of how to specify it. Definition 4.1 The probability space ( X, S, p ) is called causally closed with respect to R ind , if for every correlation Corr p ( A, B ) ＞ 0 with A ϵ S and B ϵ S such that R ind ( A, B ) holds, there exists a common cause C of some type in S .
Common causes and common common causes Proposition 3.9 tells us that every common cause incomplete classical probability space can be strongly common cause extended with respect to any (finite) set of correlations; Proposition 4.19 states that every classical probability space is even common cause completable. Note that what these propositions say is not that for a set of correlations between ( A i , B i )( i = 1, 2 … n ) there exists a single, common common cause C in the extension (or completion) for the whole set of correlations; in fact, the common causes C i constructed explicitly in the proof of Proposition 3.9 are all different: C i ≠ C j ( i ≠ j ). This observation leads to the following question. Let ( A i , B i )( i = 1, … n )be a finite set of pairs of events in( X, S, p )that are correlated [ Corr p ( A i , B i ) ＞ 0 for every i ]. We say that C is a common common cause of these correlations if C is a Reichenbachian common cause of the correlated pair ( A i , B i )for every i . Does every set of correlations in a classical probability space have a common common cause? In view of the generality of this question one may surmise that the answer to it is “yes”; that is to say, one may conjecture that given any two correlations there can always exist a Reichenbachian common cause which is a common cause for both correlations, since, one may reason, we just have to refine our picture of the world by adding more and more events to the original event structure in a consistent manner, and finally we shall find a single common cause that explains both correlations.
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 .