Reichenbach’s Common Cause in an Atomless and Complete Orthomodular Lattice

Kitajima, Yuichiro

doi:10.1007/s10773-007-9475-2

Cited by 8 publications

(4 citation statements)

References 12 publications

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“…ind . Kitajima (2008) extended Gyenis and Rédei's result to a special class of non-classical spaces: these in which the OML, on which the measure is de ned, is atomless and complete (has suprema of all its subsets). Kitajima proves that in such a case the nonclassical probability space is atomless, too, and that all such spaces contain SCCs for each pair of logically independent, correlated events.…”

Section: Causal Closedness Of Atomless Spacesmentioning

confidence: 87%

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Reichenbach’s Paradise Constructing the Realm of Probabilistic Common “Causes”

Wroński¹

2014

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The Principle of the Common Cause and the Causal Markov Condition 63. DAGs-an Introduction 64. The Causal Markov Condition 66. Conclusions 68 Causal Closedness 70. A Preliminary Formal Remark 70. Causal (up-ton -)closedness 70. . Introduction 70. . Preliminary De nitions 72. . Summary of the Results of this Chapter 74. Proofs 74. . Some Useful Parameters 74. . Proof of Theorem 3 75. . Proof of Lemmas 10-12 84. The "Proper" / "Improper" Common Cause Distinction and the Relations of Logical Independence 86. Other Independence Relations 87. A Slight Generalization 88. . Examples 90. Application for Constructing Bayesian Networks-a Negative Opinion 91. The Existence of Deductive Explanantes 93. Conclusions and Problems 94. Causal Closedness of Atomless Spaces 95 Causal Completability 96. Causal Completability the Easy Way-"Splitting the Atom" 97. Causal Completability of Classical Probability Spaces-the General Case 99. Some Remarks on Causal Completability of Non-classical Probability Spaces 102. The Impact of the Results 104 Conclusion 105 Bibliography 108 Index 112

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Section: Causal Closedness Of Atomless Spacesmentioning

confidence: 87%

“…Fact 19 (Kitajima (2008)). If in a non-classical probability space L, P L is an atomless and complete OML then L, P is causally closed w.r.t.…”

Section: Causal Closedness Of Atomless Spacesmentioning

confidence: 99%

Reichenbach’s Paradise Constructing the Realm of Probabilistic Common “Causes”

Wroński¹

2014

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“…Making use of Q-decomposability, Gyenis and Rédei [1] strengthened Proposition 3.9 in [13] in the following way:…”

Section: Definition 4 ([1] P 445)mentioning

confidence: 99%

“…Making use of Q-decomposability, Gyenis and Rédei [6] strengthened Proposition 3.9 in [17] in the following way: Proposition 5 ([6] Proposition 3.10). Let L be a σ-complete orthomodular lattice and let φ be a faithful σ-additive probability measure on L. If there is at most one φ-atom Q in L, and φ is Q-decomposable, then (L, φ) is common cause closed.…”

Section: A Sufficient Condition For Common Cause Closednessmentioning

confidence: 99%

Characterizing common cause closedness of quantum probability theories

Kitajima

Rédei

2015

Studies in History and Philosophy of Science Part B: Studies in

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This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it.Characterizing common cause closedness of quantum probability theories AbstractWe 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.

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Assessing the Status of the Common Cause Principle

Rédei

2014

New Directions in the Philosophy of Science

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Reichenbach’s Common Cause in an Atomless and Complete Orthomodular Lattice

Cited by 8 publications

References 12 publications

Reichenbach’s Paradise Constructing the Realm of Probabilistic Common “Causes”

Reichenbach’s Paradise Constructing the Realm of Probabilistic Common “Causes”

Characterizing common cause closedness of quantum probability theories

Assessing the Status of the Common Cause Principle

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