Contextuality and Dichotomizations of Random Variables

• Published in: Foundations of physics 2022-02, Vol.52 (1), Article 13
• Main Authors: Kujala, Janne V., Dzhafarov, Ehtibar N.
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Classical Mechanics; Continuity (mathematics); History and Philosophical Foundations of Physics; Philosophy of Science; Physics; Physics and Astronomy; Quantum Physics; Random variables; Real numbers; Relativity Theory; Set theory; Statistical Physics and Dynamical Systems; Topology
• ISSN: 0015-9018; 1572-9516
• DOI: 10.1007/s10701-021-00527-9

The Contextuality-by-Default approach to determining and measuring the (non)contextuality of a system of random variables requires that every random variable in the system be represented by an equivalent set of dichotomous random variables. In this paper we present general principles that justify the use of dichotomizations and determine their choice. The main idea in choosing dichotomizations is that if the set of possible values of a random variable is endowed with a pre-topology (V-space), then the allowable dichotomizations split the space of possible values into two linked subsets (“linkedness” being a weak form of pre-topological connectedness). We primarily focus on two types of random variables most often encountered in practice: categorical and real-valued ones (including continuous random variables, greatly underrepresented in the contextuality literature). A categorical variable (one with a finite number of unordered values) is represented by all of its possible dichotomizations. If the values of a random variable are real numbers, then they are dichotomized by intervals above and below a variable cut point.