A Qualified Kolmogorovian Account of Probabilistic Contextuality

TL;DR

This paper develops a mathematical framework to analyze all possible contextuality patterns in systems where outputs depend on inputs, using couplings of unrelated outputs and constraints like Bell inequalities.

Contribution

It introduces a comprehensive language for characterizing contextuality through couplings and influence patterns, extending the understanding of probabilistic contextuality.

Findings

Provides a method to determine all contextuality patterns

Characterizes systems using influence relations and constraints

Connects contextuality with couplings and inequalities

Abstract

We describe a mathematical language for determining all possible patterns of contextuality in the dependence of stochastic outputs of a system on its deterministic inputs. The central notion is that of all possible couplings for stochastically unrelated outputs indexed by mutually incompatible values of inputs. A system is characterized by a pattern of which outputs can be "directly influenced" by which inputs (a primitive relation, hypothetical or normative), and by certain constraints imposed on the outputs (such as Bell-type inequalities or their quantum analogues). The set of couplings compatible with these constraints represents a form of contextuality in the dependence of outputs on inputs with respect to the declared pattern of direct influences.