Characterizing Common Cause Closed Probability Spaces

Gyenis, Zalán; Rédei, Miklós

doi:10.1086/660302

Cited by 15 publications

(14 citation statements)

References 27 publications

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“…The converse is not true, a classical probability space can be not purely nonatomic and still common cause closed; in fact, common cause closedness of classical probability spaces can be characterized completely: a classical probability space is common cause closed if and only if it has at most one measure theoretic atom [5].…”

Section: Definition 24mentioning

confidence: 99%

“…This result was strengthened by proving that classical probability spaces are common cause completable with respect to any set of correlations [5], i.e. every classical probability space is common cause completable.…”

Section: Definition 24mentioning

confidence: 99%

“…every classical probability space is common cause completable. In fact, it was showed in [24] that every common cause incomplete classical probability space has an extension that is common cause closed (also see [5] and [10][ Proposition 4.19]). This settled the problem of common cause completability of classical probability spaces.…”

Section: Definition 24mentioning

confidence: 99%

“…For classical spaces it is enough to observe that the space (Q, ρ) is set-represented provided (L, ρ) is set-represented. A somewhat detailed form of the construction in this case can be found in [5].…”

Section: The Extension In the Classical And Quantum Casementioning

confidence: 99%

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Common cause completability of non classical probability spaces

Gyenis¹,

Rédei²

2016

Belgr Philosop Ann

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We prove that under some technical assumptions on a general, non-classical probability space, the probability space is extendible into a larger probability space that is common cause closed in the sense of containing a common cause of every correlation between elements in the space. It is argued that the philosophical significance of this common cause completability result is that it allows the defence of the Common Cause Principle against certain attempts of falsification. Some open problems concerning possible strengthening of the common cause completability result are formulated. Main resultIn this paper we prove a new result on the problem of common cause completability of non-classical probability spaces. A non-classical (also called: general) probability space is a pair (L, φ), where L is an orthocomplemented, orthomodular, non-distributive lattice and φ : L → [0, 1] is a countably additive probability measure. Taking L to be a distributive lattice (Boolean algebra), one recovers classical probability theory; taking L to be the projection lattice of a von Neumann algebra, one obtains quantum probability theory. A general probability space (L, φ) is called common cause completable if it can be embedded into a larger general probability space which is common cause complete (closed) in the sense of containing a common cause of every correlation in it. Our main result (Proposition 4.2.3) states that under some technical conditions on the lattice L a general probability space (L, φ) is common cause completable. This result utilizes earlier results on common cause closedness of general probability theories ( The main conceptual-philosophical significance of the common cause extendability result proved in this paper is that it allows one to deflect the arrow of falsification directed against the Common Cause Principle for a very large and abstract class of probability theories. This will be discussed in section 5 in the more general context of how one can assess the status of the Common Cause Principle, viewed as a general metaphysical claim about the causal structure of the world. Further sections of the paper are organized as follows: Section 2 fixes some notation and recalls some facts from lattice theory and general probability spaces needed to formulate the main result. Section 3

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Section: Definition 24mentioning

confidence: 99%

Section: Definition 24mentioning

confidence: 99%

Section: Definition 24mentioning

confidence: 99%

Section: The Extension In the Classical And Quantum Casementioning

confidence: 99%

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Common cause completability of non classical probability spaces

Gyenis¹,

Rédei²

2016

Belgr Philosop Ann

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“…Gyenis and Rédei [2] gave a characterization of common cause closedness of classical probability measure spaces. In the proof of that characterization the following result of Johnson [12] played an important role.…”

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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Characterizing Common Cause Closed Probability Spaces

Cited by 15 publications

References 27 publications

Common cause completability of non classical probability spaces

Common cause completability of non classical probability spaces

Characterizing common cause closedness of quantum probability theories

Assessing the Status of the Common Cause Principle

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