Common cause completability of non classical probability spaces

Abstract

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