The final work of a distinguished physicist, this remarkable volume examines the emotive significance of time, the time order of mechanics, the time direction of thermodynamics and microstatistics, the time direction of macrostatistics, and the time of quantum physics. Coherent discussions include accounts of analytic methods of scientific philosophy in the investigation of probability, quantum mechanics, the theory of relativity, and causality. "[Reichenbach’s] best by a good deal."—Physics Today. 1971 ed.

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.

We give a few results concerning the notions of causal completability and causal closedness of classical probability spaces . We prove that any classical probability space has a causally closed extension; 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 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 ConclusionAppendix A. (shrink)

If $\mathcal{A}$ (V) is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V 1 and V 2 are spacelike separated spacetime regions, then the system ( $\mathcal{A}$ (V 1 ), $\mathcal{A}$ (V 2 ), φ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A∈ $\mathcal{A}$ (V 1 ), B∈ $\mathcal{A}$ (V 2 ) correlated in the normal state φ there exists a projection C (...) belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V 1 and V 2 and disjoint from both V 1 and V 2 , a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system ( $\mathcal{A}$ (V 1 ), $\mathcal{A}$ (V 2 ), φ) with a locally normal and locally faithful state φ and suitable bounded V 1 and V 2 satisfies the Weak Reichenbach's Common Cause Principle. (shrink)

A partition $\{C_i\}_{i\in I}$ of a Boolean algebra $\cS$ in a probability measure space $(\cS,p)$ is called a Reichenbachian common cause system for the correlated pair $A,B$ of events in $\cS$ if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set $I$ is called the size of the common cause system. It is shown that given any correlation in $(\cS,p)$, and given any finite size $n>2$, the probability space (...) $(\cS,p)$ can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size $n$ for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of \cS$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated. (shrink)

The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be (...) common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach's Common Cause Principle. (shrink)

In this paper we give a positive answer to a problem posed by Hofer-Szabó and Rédei (Int. J. Theor. Phys. 43:1819–1826, 2004) regarding the existence of infinite Reichenbachian common cause systems (RCCSs). An example of a countably infinite RCCS is presented. It is also determined that no RCCSs of greater cardinality exist.

My Ph.D. dissertation written under the supervision of Prof. Tomasz Placek at the Institute of Philosophy of the Jagiellonian University in Kraków. In one of its most basic and informal shapes, the principle of the common cause states that any surprising correlation between two factors which are believed not to directly influence one another is due to their common cause. Here we will be concerned with a version od this idea which possesses a purely probabilistic formulation. It was introduced, in (...) the form of a general principle, by Hans Reichenbach in his posthumously published book "The Direction of Time". The central notion of the principle in Reichenbach's formulation, and of the current essay, is that of screening off: two correlated events are screened off by a third event if conditioning on the third event makes them probabilistically independent. Reichenbach's principle marks also the beginning of a new field of philosophy: namely, that of ``probabilistic causality''. For the most part, the current essay can be seen as an effort at checking how far one can go with the purely statistical notions revolving around Reichenbach's idea of common cause. In short, the answer is ``surprisingly far''; in some classes of probability spaces all correlations between ``interesting'' events possess explanations of such sort. However, this fact lends itself to opposing interpretations; more on that in the conclusion. Chapters 6 and 7 contain mathematical results concerning these issues. The screening-off condition requires an equality of a probabilistic nature to hold; chapter 8 is a short discussion of slightly weakened versions of the condition, which hold if the sides of the above mentioned equality differ to a small degree. In chapter 2, after some mathematical preliminaries, we study the various formulations of the principle which might be said to stem from the original idea of Reichenbach. We also examine a few of the most salient counterarguments, which undermine at least some of the formulations. Chapter 3 is of a formal nature, dealing with various probabilistic notions which can be thought of as generalizations of Reichenbach's concept of common cause. The next chapter concerns the relationship between the idea of common causal explanation and the Bell inequalities. In chapter 5 we briefly present the form of Reichenbach's principle which can be found in the field of representing causal structures by means of directed acyclic graphs. (shrink)

A partition $\{C_i\}_{i\in I}$ of a Boolean algebra Ω in a probability measure space (Ω, p) is called a Reichenbachian common cause system for the correlation between a pair A,B of events in Ω if any two elements in the partition behave like a Reichenbachian common cause and its complement; the cardinality of the index set I is called the size of the common cause system. It is shown that given any non-strict correlation in (Ω, p), and given any finite (...) natural number n > 2, the probability space (Ω,p) can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. (shrink)

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 Reichenbach's definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle. (shrink)