Gábor Hofer-Szabó

This paper is the philosopher-friendly version of our more technical work. It aims to give a clear-cut definition of Bell's notion of local causality. Having provided a framework, called local physical theory, which integrates probabilistic and spatiotemporal concepts, we formulate the notion of local causality and relate it to other locality and causality concepts. Then we compare Bell's local causality with Reichenbach's Common Cause Principle and relate both to the Bell inequalities. We find a nice parallelism: both local causality and the Common Cause Principle are more general notions than captured by the Bell inequalities. Namely, Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.

The aim of this paper is to give a sharp definition of Bell's notion of local causality. To this end, first we unfold a framework, called local physical theory, integrating probabilistic and spatiotemporal concepts. Formulating local causality within this framework and classifying local physical theories by whether they obey local primitive causality---a property rendering the dynamics of the theory causal, we then investigate what is needed for a local physical theory, with or without local primitive causality, to be locally causal. Finally, comparing Bell's local causality with the Common Cause Principles and relating both to the Bell inequalities we find a nice parallelism: Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.

This paper is a further consideration of Hemmo and Shenker’s ideas about the proper conceptual characterization of macrostates in statistical mechanics. We provide two formulations of how macrostates come about as elements of certain partitions of the system’s phase space imposed on by the interaction between the system and an observer, and we show that these two formulations are mathematically equivalent. We also reflect on conceptual issues regarding the relationship of macrostates to distinguishability, thermodynamic regularity, observer dependence, and the general phenomenon of measurement.

In a recent paper, Cabbolet argues that the PBR theorem is nonreal since in the ensemble interpretation of quantum mechanics the entangled measurement used in the derivation of the PBR theorem is nonexisting. However, Cabbolet (1) does not provide any argument for the nonexistence of entangled measurements beyond the incompatibility of the existence of entangled measurements and the existence of $$\psi$$ -epistemic models which we already know from the PBR theorem; and (2) he does not show why it is more reasonable to abandon entangled measurements instead of $$\psi$$ -epistemic models. Hence, the PBR theorem remains intact.

This book collects research papers on the philosophical foundations of probability, causality, spacetime and quantum theory. The papers are related to talks presented in six subsequent workshops organized by The Budapest-Kraków Research Group on Probability, Causality and Determinism. Coverage consists of three parts. Part I focuses on the notion of probability from a general philosophical and formal epistemological perspective. Part II applies probabilistic considerations to address causal questions in the foundations of quantum mechanics. Part III investigates the question of indeterminism in spacetime theories. It also explores some related questions, such as decidability and observation. The contributing authors are all philosophers of science with a strong background in mathematics or physics. They believe that paying attention to the finer formal details often helps avoiding pitfalls that exacerbate the philosophical problems that are in the center of focus of contemporary research. The papers presented here help make explicit the mathematical-structural assumptions that underlie key philosophical argumentations. This formally rigorous and conceptually precise approach will appeal to researchers and philosophers as well as mathematicians and statisticians.

It will be shown that the Peres–Mermin square admits value-definite noncontextual hidden-variable models if the observables associated with the operators can be measured only sequentially but not simultaneously. Namely, sequential measurements allow for noncontextual models in which hidden states update between consecutive measurements. Two recent experiments realizing the Peres–Mermin square by sequential measurements will also be analyzed along with other hidden-variable models accounting for these experiments.

​This book summarizes the results of research the authors have pursued in the past years on the problem of implementing Bell's notion of local causality in local physical theories and relating it to other important concepts and principles in the foundations of physics such as the Common Cause Principle, Bell's inequalities, the EPR scenario, and various other locality and causality concepts. The book is intended for philosophers of science with an interest in the formal background of sciences, philosophers of physics and physicists working in foundation of physics.

It is a well known fact that a common common causal explanation of the EPR scenario which consists in providing a local, non-conspiratorial common common cause system for a set of EPR correlations is excluded by various Bell inequalities. But what if we replace the assumption of a common common cause system by the requirement that each correlation of the set has a local, non-conspiratorial separate common cause system? In the paper we show that this move does not yield a solution by providing a general recipe how to derive any Bell(δ) inequality—that is an inequality differing from some Bell inequality in a term of order of δ—from the assumption that an appropriate set of almost perfect anticorrelations has a separate common causal explanation

Standard derivations of the Bell inequalities assume a common common cause system that is a common screener-off for all correlations and some additional assumptions concerning locality and no-conspiracy. In a recent paper (Grasshoff et al., 2005) Bell inequalities have been derived via separate common causes assuming perfect correlations between the events. In the paper it will be shown that the assumptions of this separate-common-cause-type derivation of the Bell inequalities in the case of perfect correlations can be reduced to the assumptions of common-common-cause-system-type derivation. However, in the case of non-perfect correlations a non-reducible separate-common-cause-type derivation of some Bell-like inequalities can be given. The violation of these Bell-like inequalities proves Szabó's (2000) conjecture concerning the non-existence of a local, non-conspiratorial, separate-common-cause-model for a delta δ-neighborhood of perfect EPR correlations.

Standard derivations of the Bell inequalities assume a common-commoncause-system that is a common screener-off for all correlations and some additional assumptions concerning locality and no-conspiracy. In a recent paper Graßhoff et al., "The British Journal for the Philosophy of Science", 56, 663–680 ) Bell inequalities have been derived via separate common causes assuming perfect correlations between the events. In the paper it will be shown that the assumptions of this separate-common-cause-type derivation of the Bell inequalities in the case of perfect correlations can be reduced to the assumptions of a common-common-cause-system-type derivation. However, in the case of non-perfect correlations a non-reducible separate-common-cause-type derivation of some Bell-like inequalities can be given. The violation of these Bell-like inequalities proves Szabo's ) conjecture concerning the non-existence of a local, non-conspiratorial, separate-common-cause-model for a δ-neighborhood of perfect EPR correlations.

In the paper it will be shown that Reichenbach’s Weak Common Cause Principle is not valid in algebraic quantum field theory with locally finite degrees of freedom in general. Namely, for any pair of projections A, B supported in spacelike separated double cones ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ , respectively, a correlating state can be given for which there is no nontrivial common cause (system) located in the union of the backward light cones of ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ and commuting with the both A and B. Since noncommuting common cause solutions are presented in these states the abandonment of commutativity can modulate this result: noncommutative Common Cause Principles might survive in these models

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

The aim of the paper is to relate Bell’s notion of local causality to the Causal Markov Condition. To this end, first a framework, called local physical theory, will be introduced integrating spatiotemporal and probabilistic entities and the notions of local causality and Markovity will be defined. Then, illustrated in a simple stochastic model, it will be shown how a discrete local physical theory transforms into a Bayesian network and how the Causal Markov Condition arises as a special case of Bell’s local causality and Markovity

I will argue that the Peres-Mermin square does not necessarily rule out a value-definite (deterministic) noncontextual hidden variable model if the operators are not given a physical interpretation satisfying the following two requirements: (i) each operator is uniquely realized by a single physical measurement; (ii) commuting operators are realized by simultaneous measurements. To underpin this claim, I will construct three hidden variable models for three different physical realizations of the Peres-Mermin square: one violating (i), another violating (ii), and a third one violating both (i) and (ii).

In the paper the relation between the standard probabilistic characterization of the common cause and Bell's notion of local causality will be investigated. It will be shown that the probabilistic common cause follows from local causality if one accepts, as Bell did, two assumptions concerning the common cause: first, the common cause is localized in the intersection of the past of the correlating events; second, it provides a complete specification of the `beables' of this intersection. However, neither assumptions are a priori requirements. In the paper the logical role of these assumptions will be studied and it will be shown that only the second assumption is necessary for the derivation of the probabilistic common cause from local causality.

This essay has two main claims about EPR’s Reality Criterion. First, we claim that the application of the Reality Criterion makes an essential difference between the EPR argument and Einstein’s later arguments against quantum mechanics. We show that while the EPR argument, making use of the Reality Criterion, does derive that certain interpretations of quantum mechanics are incomplete, Einstein’s later arguments, making no use of the Reality Criterion, do not prove incompleteness, but rather point to the inadequacy of the Copenhagen interpretation. We take this fact as an indication that the Reality Criterion is a crucial, indispensable component of the incompleteness argument(s). The second claim is more substantive. We argue that the Reality Criterion is a special case of the Common Cause Principle. Finally, we relate this proposal to Tim Maudlin’s recent assertion that the Reality Criterion is an analytic truth.

This paper aims to implement Bell’s notion of local causality into a framework, called local physical theory, which is general enough to integrate both probabilistic and spatiotemporal concepts and also classical and quantum theories. Bell’s original idea of local causality will then arise as the classical case of our definition. First, we investigate what is needed for a local physical theory to be locally causal. Then we compare local causality with Reichenbach’s common cause principle and relate both to the Bell inequalities. We find a nice parallelism: both local causality and the common cause principle are more general notions than captured by the Bell inequalities. Namely, Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively

In the paper we compare three principles accounting for correlations, namely Reichenbach's Common Cause Principle, Bell's Local Causality Principle, and Einstein's Reality Criterion and relate them to the Bell inequalities. We show that there are two routes connecting the principles to the Bell inequalities. In case of Reichenbach's Common Cause Principle and Bell's Local Causality Principle one assumes a non-conspiratorial joint common cause for a set of correlations. In case of Einstein's Reality Criterion one assumes strongly non-conspiratorial separate common causes for a set of perfect correlations. Strongly non-conspiratorial separate common causes for perfect correlations, however, form a non-conspiratorial joint common cause. Hence the two routes leading the Bell inequalities meet.

Our approach aims at accounting for causal claims in terms of how the physical states of the underlying dynamical system evolve with time. Causal claims assert connections between two sets of physicals states—their truth depends on whether the two sets in question are genuinely connected by time evolution such that physical states from one set evolve with time into the states of the other set. We demonstrate the virtues of our approach by showing how it is able to account for typical causes, causally relevant factors, being ‘the’ cause, and cases of overdetermination and causation by absences.

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.

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 common cause. The principle is of central importance in the philosophy of science, especially in causal explanation, causal modeling and in the foundations of quantum physics. Written for philosophers of science, physicists and statisticians, this book contributes to the debate over the validity of the common cause principle, by proving results that bring to the surface the nature of explanation by common causes. It provides a technical and mathematically rigorous examination of the notion of common cause, providing an analysis not only in terms of classical probability measure spaces, which is typical in the available literature, but in quantum probability theory as well. The authors provide numerous open problems to further the debate and encourage future research in this field.