The classical interpretation of probability together with the principle of indifference is formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called labelling invariance is defined in the category of Haar probability spaces; it is shown that labelling invariance is violated, and Bertrand’s paradox is interpreted as the proof of violation of labelling invariance. It is shown that Bangu’s attempt to block the emergence of Bertrand’s paradox by requiring the re-labelling (...) of random events to preserve randomness cannot succeed non-trivially. A non-trivial strategy to preserve labelling invariance is identified, and it is argued that, under the interpretation of Bertrand’s paradox suggested in the paper, the paradox does not undermine either the principle of indifference or the classical interpretation and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena. It is shown in particular that violation of labelling invariance does not entail that labelling of random events affects the probabilities of random events. It also is argued, however, that the content of the principle of indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful. (shrink)

By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that (...) the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935). (shrink)

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal (...) companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. (shrink)

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. (shrink)

We argue that, contrary to some analyses in the philosophy of science literature, ergodic theory falls short in explaining the success of classical equilibrium statistical mechanics. Our claim is based on the observations that dynamical systems for which statistical mechanics works are most likely not ergodic, and that ergodicity is both too strong and too weak a condition for the required explanation: one needs only ergodic-like behaviour for the finite set of observables that matter, but the behaviour must ensure that (...) the approach to equilibrium for these observables is on the appropriate time-scale. (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)

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 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. (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)

John von Neumann (1903-1957) was undoubtedly one of the scientific geniuses of the 20th century. The main fields to which he contributed include various disciplines of pure and applied mathematics, mathematical and theoretical physics, logic, theoretical computer science, and computer architecture. Von Neumann was also actively involved in politics and science management and he had a major impact on US government decisions during, and especially after, the Second World War. There exist several popular books on his personality and various collections (...) focusing on his achievements in mathematics, computer science, and economy. Strangely enough, to date no detailed appraisal of his seminal contributions to the mathematical foundations of quantum physics has appeared. Von Neumann's theory of measurement and his critique of hidden variables became the touchstone of most debates in the foundations of quantum mechanics. Today, his name also figures most prominently in the mathematically rigorous branches of contemporary quantum mechanics of large systems and quantum field theory. And finally - as one of his last lectures, published in this volume for the first time, shows - he considered the relation of quantum logic and quantum mechanical probability as his most important problem for the second half of the twentieth century. The present volume embraces both historical and systematic analyses of his methodology of mathematical physics, and of the various aspects of his work in the foundations of quantum physics, such as theory of measurement, quantum logic, and quantum mechanical entropy. The volume is rounded off by previously unpublished letters and lectures documenting von Neumann's thinking about quantum theory after his 1932 Mathematical Foundations of Quantum Mechanics. The general part of the Yearbook contains papers emerging from the Institute's annual lecture series and reviews of important publications of philosophy of science and its history. (shrink)

Reichenbach's principles of a probabilistic common cause of probabilistic correlations is formulated in terms of relativistic quantum field theory, and the problem is raised whether correlations in relativistic quantum field theory between events represented by projections in local observable algebrasA(V1) andA(V2) pertaining to spacelike separated spacetime regions V1 and V2 can be explained by finding a probabilistic common cause of the correlation in Reichenbach's sense. While this problem remains open, it is shown that if all superluminal correlations predicted by the (...) vacuum state between events inA(V1) andA(V2) have a genuinely probabilistic common cause, then the local algebrasA(V1) andA(V2) must be statistically independent in the sense of C*-independence. (shrink)

We prove new results on common cause closedness of quantum probability spaces, where by a quantum probability space is meant the projection lattice of a non-commutative von Neumann algebra together with a countably additive probability measure on the lattice. Common cause closedness is the feature that for every correlation between a pair of commuting projections there exists in the lattice a third projection commuting with both of the correlated projections and which is a Reichenbachian common cause of the correlation. The (...) main result we prove is that a quantum probability space is common cause closed if and only if it has at most one measure theoretic atom. This result improves earlier ones published in [1]. The result is discussed from the perspective of status of the Common Cause Principle. Open problems on common cause closedness of general probability spaces (L, ϕ) are formulated, where L is an orthomodular bounded lattice and ϕ is a probability measure on L. (shrink)

The Bayes Blind Spot of a Bayesian Agent is, by definition, the set of probability measures on a Boolean σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-algebra that are absolutely continuous with respect to the background probability measure of a Bayesian Agent on the algebra and which the Bayesian Agent cannot learn by a single conditionalization no matter what evidence he has about the elements in the Boolean σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (...) $$\end{document}-algebra. It is shown that if the Boolean algebra is finite, then the Bayes Blind Spot is a very large set: it has the same cardinality as the set of all probability measures ; it has the same measure as the measure of the set of all probability measures ; and is a “fat” set in topological sense in the set of all probability measures taken with its natural topology. Features of the Bayes Blind Spot are determined from the perspective of repeated Bayesian learning when the Boolean algebra is finite. Open problems about the Bayes Blind Spot are formulated in probability spaces with infinite Boolean σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-algebras. The results are discussed from the perspective of Bayesianism. (shrink)

Because of the complex interdependence of physics and mathematics their relation is not free of tensions. The paper looks at how the tension has been perceived and articulated by some physicists, mathematicians and mathematical physicists. Some sources of the tension are identified and it is claimed that the tension is both natural and fruitful for both physics and mathematics. An attempt is made to explain why mathematical precision is typically not welcome in physics.

Reichenbach’s Common Cause Principle is the claim that if two events are correlated, then either there is a causal connection between the correlated events that is responsible for the correlation or there is a third event, a so called common cause, which brings about the correlation. The paper reviews some results concerning Reichenbach’s notion of common cause, results that are directly relevant to the problem of how one can falsify Reichenbach’s Common Cause Principle. Special emphasis will be put on the (...) question of whether EPR-type correlations can have an explanation in terms of Reichenbachian common causes. Most of the results to be recalled indicate that falsifying Reichenbach’s Common Cause Principle is much more tricky than one may have thought and that, contrary to some claims in the literature, there is no conclusive proof yet that the EPR correlations predicted by ordinary, non-relativistic quantum mechanics cannot have a common cause; furthermore, recent results about possible Reichenbachian common causes of correlations predicted by quantum field states between spacelike separated local observable algebras in algebraic quantum field theory strongly indicate that there may very well exist Reichenbachian common causes of superluminal correlations predicted by quantum field theory. (shrink)

The idea of quantum logic first appears explicitly in the short Section 5 of Chapter III. in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics [31]; however, the real birthplace of quantum logic is commonly identified with the 1936 seminal paper co-authored by G. Birkhoff and J. von Neumann [5]. The aim of this review is to recall the main idea of the Birkhoff-von Neumann concept1 of quantum logic as this was put forward in the 1936 paper. (...) The review is motivated partly by two facts related to quantum logic: one, peculiar, is that the 1936 von Neumann concept is an almost totally neglected2 topic in the enormous quantum logic literature [17]; the other, not very well-known, is that von Neumann was never completely satisfied with how he had worked out quantum logic. (shrink)

Relativistic locality is interpreted in this paper as a web of conditions expressing the compatibility of a physical theory with the underlying causal structure of spacetime. Four components of this web are distinguished: spatiotemporal locality, along with three distinct notions of causal locality, dubbed CL-Independence, CL-Dependence, and CL-Dynamic. These four conditions can be regimented using concepts from the categorical approach to quantum field theory initiated by Brunetti, Fredenhagen, and Verch (2003). A covariant functor representing a general quantum field theory is (...) defined to be causally local if it satisfies the three CL conditions. Any such theory is viewed as fully compliant with relativistic locality. We survey current results indicating the extent to which an algebraic quantum field theory satisfying the Haag–Kastler axioms is causally local. (shrink)

We distinguish two sub-types of each of the two causality principles formulated in connection with the Common Cause Principle in Henson (2005) and raise and investigate the problem of logical relations among the resulting four causality principles. Based in part on the analysis of the status of these four principles in algebraic quantum field theory we will argue that the four causal principles are non-equivalent.

This paper has two parts, a historical and a systematic. In the historical part it is argued that the two major axiomatic approaches to relativistic quantum field theory, the Wightman and Haag-Kastler axiomatizations, are realizations of the program of axiomatization of physical theories announced by Hilbert in his 6th of the 23 problems discussed in his famous 1900 Paris lecture on open problems in mathematics, if axiomatizing physical theories is interpreted in a soft and opportunistic sense suggested in 1927 by (...) Hilbert, Nordheim, and von Neumann. To show this, Section 2 recalls first some of Hilbert's views about the axiomatic approach to physical theories as these were formulated in his 6th problem. It will be .. (shrink)

We investigate the general properties of general Bayesian learning, where “general Bayesian learning” means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If (...) a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes’ rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization. (shrink)

The projection latticesP(ℳ1),P(ℳ2) of two von Neumann subalgebras ℳ1, ℳ2 of the von Neumann algebra ℳ are defined to be logically independent if A ∧ B≠0 for any 0≠AεP(ℳ1), 0≠BP(ℳ2). After motivating this notion in independence, it is shown thatP(ℳ1),P(ℳ2) are logically independent if ℳ1 is a subfactor in a finite factor ℳ andP(ℳ1),P(ℳ2 commute. Also, logical independence is related to the statistical independence conditions called C*-independence W*-independence, and strict locality. Logical independence ofP(ℳ1,P(ℳ2 turns out to be equivalent to the (...) C*-independence of (ℳ1,ℳ2) for mutually commuting ℳ1,ℳ2 and it is shown that if (ℳ1,ℳ2) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP(ℳ1,P(ℳ2 are logically independent in the following cases: ℳ is a finite-dimensional full-matrix algebra and ℳ1,ℳ2 are C*-independent; (ℳ1,ℳ2) is a W*-independent pair; ℳ1,ℳ2 have the property of strict locality. (shrink)

Recently, new types of independence of a pair of C *- or W *-subalgebras (1,2) of a C *- or W *-algebra have been introduced: operational C *- and W *-independence (Rédei and Summers, http://arxiv.org/abs/0810.5294, 2008) and operational C *- and W *-separability (Rédei and Valente, How local are local operations in local quantum field theory? 2009). In this paper it is shown that operational C *-independence is equivalent to operational C *-separability and that operational W *-independence is equivalent to (...) operational W *-separability. Specific further sub-types of both operational C *- and W *-separability and operational C *- and W *-independence are defined and the problem of characterization of the logical interdependencies of the independence notions is raised. (shrink)

Weak and strong consistency of the Abstract Principal Principle are defined in terms of classical probability measure spaces. It is proved that the Abstract Principal Principle is both weakly and strongly consistent. The Abstract Principal Principle is strengthened by adding a stability requirement to it. Weak and strong consistency of the resulting Stable Abstract Principal Principle are defined. It is shown that the Stable Abstract Principal Principle is weakly consistent. Strong consistency of the Stable Abstract Principal principle remains an open (...) question. (shrink)

It is shown that the Kolmogorovian Censorship Hypothesis, according to which quantum probabilities are interpretable as conditional probabilities in a classical probability measure space, holds not only for Hilbert space quantum mechanics but for general quantum probability theories based on the theory of von Neumann algebras.

Based partly on proving that algebraic relativistic quantum field theory (ARQFT) is a stochastic Einstein local (SEL) theory in the sense of SEL which was introduced by Hellman (1982b) and which is adapted in this paper to ARQFT, the recently proved maximal and typical violation of Bell's inequalities in ARQFT (Summers and Werner 1987a-c) is interpreted in this paper as showing that Bell's inequalities are, in a sense, irrelevant for the problem of Einstein local stochastic hidden variables, especially if this (...) problem is raised in connection with ARQFT. This leads to the question of how to formulate the problem of local hidden variables in ARQFT. By giving a precise definition of hidden-variable theory within the operator algebraic framework of quantum mechanics, it will be argued that the aim of hidden-variable investigations is to determine those classes of quantum theories whose elements represent a statistical content that cannot be reduced in a given way. In some particular way to be stated, a proposition will be stated which distinguishes quantum field theories whose statistical content cannot be reduced without violating some relativistic locality principle. (shrink)

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal (...) companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. (shrink)

The classical interpretation of probability together with the principle of indifference is formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called labelling invariance is defined in the category of Haar probability spaces; it is shown that labelling invariance is violated, and Bertrand’s paradox is interpreted as the proof of violation of labelling invariance. It is shown that Bangu’s attempt to block the emergence of Bertrand’s paradox by requiring the re-labelling (...) of random events to preserve randomness cannot succeed non-trivially. A non-trivial strategy to preserve labelling invariance is identified, and it is argued that, under the interpretation of Bertrand’s paradox suggested in the paper, the paradox does not undermine either the principle of indifference or the classical interpretation and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena. It is shown in particular that violation of labelling invariance does not entail that labelling of random events affects the probabilities of random events. It also is argued, however, that the content of the principle of indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful. 1 The Main Claims2 The Elementary Classical Interpretation of Probability3 The General Classical Interpretation of Probability in Terms of Haar Measures4 Labelling Invariance and Labelling Irrelevance5 General Bertrand’s Paradox6 Attempts to Save Labelling Invariance7 Comments on the Classical Interpretation of Probability. (shrink)

We distinguish two sub-types of each of the two causality principles formulated in connection with the Common Cause Principle in Henson and raise and investigate the problem of logical relations among the resulting four causality principles. Based in part on the analysis of the status of these four principles in algebraic quantum field theory we will argue that the four causal principles are non- equivalent.

Reichenbach’s Common Cause Principle is the claim that if two events are correlated, then either there is a causal connection between the correlated events that is responsible for the correlation or there is a third event, a so called (Reichenbachian) common cause, which brings about the correlation. The paper reviews some results concerning Reichenbach’s notion of common cause, results that are directly relevant to the problem of how one can falsify Reichenbach’s Common Cause Principle. Special emphasis will be put on (...) the question of whether EPR-type correlations can have an explanation in terms of Reichenbachian common causes. Most of the results to be recalled indicate that falsifying Reichenbach’s Common Cause Principle is much more tricky than one may have thought and that, contrary to some claims in the literature, there is no conclusive proof yet that the EPR correlations predicted by ordinary, non-relativistic quantum mechanics cannot have a common cause; furthermore, recent results about possible Reichenbachian common causes of correlations predicted by quantum field states between spacelike separated local observable algebras in algebraic quantum field theory strongly indicate that there may very well exist Reichenbachian common causes of superluminal correlations predicted by quantum field theory. (shrink)

Abstract Based on recalling two characteristic features of Bayesian statistical inference in commutative probability theory, a stability property of the inference is pointed out, and it is argued that that stability of the Bayesian statistical inference is an essential property which must be preserved under generalization of Bayesian inference to the non?commutative case. Mathematical no?go theorems are recalled then which show that, in general, the stability can not be preserved in non?commutative context. Two possible interpretations of the impossibility of generalization (...) of Bayesian statistical inference to the non?commutative case are offered, none of which seems to be completely satisfying. (shrink)

We investigate the general properties of general Bayesian learning, where ``general Bayesian learning'' means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If (...) a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes' rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization. (shrink)

It is argued that in his critique of standard nonrelativistic quantum mechanics Einstein formulated three requirements as necessary for a physical theory to be compatible with the field-theorectical paradigm, and it is shown that local, relativistic, algebraic quantum field theory typically satisfies those criteria-although, there are still open questions concerning the status of operational separability of quantum systems localized in space like separated space-time regions. It is concluded that local algebraic quantum field theory can be viewed as a research program (...) that Einstein suggested informally in 1948 and that was realized only later in mathematical physics. (shrink)

The Common Cause Principle, stating that correlations are either consequences of a direct causal link between the correlated events or are due to a common cause, is assessed from the perspective of its viability and it is argued that at present we do not have strictly empirical evidence that could be interpreted as disconfirming the principle. In particular it is not known whether spacelike correlations predicted by quantum field theory can be explained by properly localized common causes, and EPR correlations (...) are known to be unexplainable only by common causes that are required to satisfy conditions that are empirically not tested. (shrink)

The paper takes thePrincipal Principle to be a norm demanding that subjective degrees of belief of a Bayesian agent be equal to the objective probabilities once the agent has conditionalized his subjective degrees of beliefs on the values of the objective probabilities, where the objective probabilities can be not only chances but any other quantities determined objectively. Weak and strong consistency of the Abstract Principal Principle are defined in terms of classical probability measure spaces. It is proved that the Abstract (...) Principal Principle is weakly consistent and that it is strongly consistent in the category of probability measure spaces where the Boolean algebra representing the objective random events is finite. It is argued that it is desirable to strengthen the Abstract Principal Principle by adding a stability requirement to it. Weak and strong consistency of the resulting Stable Abstract Principal Principle are defined, and the strong consistency of the Abstract Principal Principle is interpreted as necessary for a non-omniscient Bayesian agent to be able to have rational degrees of belief in all epistemic situations. It is shown that the Stable Abstract Principal Principle is weakly consistent, but the strong consistency of the Stable Abstract Principal principle remains an open question. We conclude that we do not yet have proof that Bayesian agents can have rational degrees of belief in every epistemic situation. (shrink)

A probability space is common cause closed if it contains a Reichenbachian common cause of every correlation in it and common cause incomplete otherwise. It is shown that a probability space is common cause incomplete if and only if it contains more than one atom and that every space is common cause completable. The implications of these results for Reichenbach's Common Cause Principle are discussed, and it is argued that the principle is only falsifiable if conditions on the common cause (...) are imposed that go beyond the requirements formulated by Reichenbach in the definition of common cause. (shrink)

The role of measure theoretic atomicity in common cause closedness of general probability theories with non-distributive event structures is raised and investigated. It is shown that if a general probability space is non-atomic then it is common cause closed. Conditions are found that entail that a general probability space containing two atoms is not common cause closed but it is common cause closed if it contains only one atom. The results are discussed from the perspective of the Common Cause Principle.

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)

This paper reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory. In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw Ulam, this paper clarifies von Neumann's contribution to discovering the relationship between spatial isomorphism and spectral isomorphism. The main message of the paper is that von Neumann's argument described in his letter to Ulam is the very first proof that (...) spatial isomorphism and spectral isomorphism are not equivalent because spectral isomorphism is weaker than spatial isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical systems with mixed spectra need not be spatially isomorphic. (shrink)

Given two unital C*-algebrasA, ℬ and their state spacesE A , Eℬ respectively, (A,E A ) is said to have (ℬ, Eℬ) as a hidden theory via a linear, positive, unit-preserving map L: ℬ →A if, for all ϕ εE A , L*ϕ can be decomposed in Eℬ into states with pointwise strictly less dispersion than that of ϕ. Conditions onA and L are found that exclude (A,E A ) from having a hidden theory via L. It is shown in (...) particular that, ifA is simple, then no (ℬ, Eℬ) can be a hidden theory of (A,E A ) via a Jordan homomorphism; it is proved furthermore that, ifA is a UHF algebra, it cannot be embedded into a larger C*-algebra ℬ such that (ℬ, Eℬ) is a hidden theory of (A,E A ) via a conditional expectation from ℬ ontoA. (shrink)

Butterfield's (1992a,b,c) claim of the equivalence of absence of Lewisian probabilistic counterfactual causality (LC) to Hellman's stochastic Einstein locality (SEL) is questioned. Butterfield's assumption on which the proof of his claim is based would suffice to prove that SEL implies absence of LC also for appropriately given versions of these notions in algebraic quantum field theory, but the assumption is not an admissible one. The conclusion must be that the relation of SEL and absence of LC is open, and that (...) they may be independent. (shrink)