On Universality of Classical Probability with Contextually Labeled Random Variables

TL;DR

This paper demonstrates that classical probability theory can handle empirical phenomena when random variables are labeled contextually, clarifying misconceptions and linking to quantum contextuality concepts.

Contribution

It shows that contextual labeling of random variables resolves apparent contradictions in classical probability and allows analysis of contextuality in empirical systems.

Findings

Contextual labeling prevents misconceptions about probability theory.

Certain empirical systems are noncontextual when properly labeled.

The double-slit experiment and related systems are shown to be noncontextual.

Abstract

One can often encounter claims that classical (Kolmogorovian) probability theory cannot handle, or even is contradicted by, certain empirical findings or substantive theories. This note joins several previous attempts to explain that these claims are unjustified, illustrating this on the issues of (non)existence of joint distributions, probabilities of ordered events, and additivity of probabilities. The specific focus of this note is on showing that the mistakes underlying these claims can be precluded by labeling all random variables involved contextually. Moreover, contextual labeling also enables a valuable additional way of analyzing probabilistic aspects of empirical situations: determining whether the random variables involved form a contextual system, in the sense generalized from quantum mechanics. Thus, to the extent the Wang-Busemeyer QQ equality for the question order effect holds, the system describing them is noncontextual. The double-slit experiment and its behavioral analogues also turn out to form a noncontextual system, having the same probabilistic format (cyclic system of rank 4) as the one describing spins of two entangled electrons.