Causal Completeness in General Probability Theories

Gyenis, Balázs; Rédei, Miklós

doi:10.1007/978-1-4020-9904-5_7

Cited by 6 publications

(3 citation statements)

References 22 publications

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“…The first is to switch to the more general context of random variables, as opposed to events. Initial steps in this direction have been taken in Gyenis & Rédei [2010]. The second would be to attempt to transfer the results to non-classical probability spaces.…”

Section: Resultsmentioning

confidence: 99%

Completion of the Causal Completability Problem

Marczyk

Wroński

2015

The British Journal for the Philosophy of Science

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Section: Resultsmentioning

confidence: 99%

Completion of the Causal Completability Problem

Marczyk

Wroński

2015

The British Journal for the Philosophy of Science

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“…Both common cause complete and common cause incomplete probability spaces exist: It was shown in [3]) (also see [4] and [10][Chapter 4]) that no classical probability space with a Boolean algebra of finite cardinality can be common cause complete and that a dense classical probability space is common cause closed. 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%

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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“…19 The last option would be to move the discussion to the more general context of random variables, as opposed to events. First steps in this direction have been provided by Gyenis and Rédei (2010).…”

Section: Conclusion and Problemsmentioning

confidence: 99%

A New Notion of Causal Closedness

Wroński

Marczyk

2013

Erkenn

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[745][746][747][748][749][750][751][752][753][754][755][756] 2006). In this paper we introduce a new and natural notion similar to causal closedness and prove a number of theorems which can be seen as extensions of earlier results from the literature. Most notably we prove that a finite probability space is causally closed in our sense iff its measure is uniform. We also present a generalisation of this result to a class of non-classical probability spaces.

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Causal Completeness in General Probability Theories

Cited by 6 publications

References 22 publications

Completion of the Causal Completability Problem

Completion of the Causal Completability Problem

Common cause completability of non classical probability spaces

A New Notion of Causal Closedness

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