On Joint Distributions, Counterfactual Values, and Hidden Variables in Relation to Contextuality

Ehtibar Dzhafarov

Published 2018-09-03Version 1

Contextuality can be understood in three ways: (1) in terms of (non)existence of certain joint distributions involving measurements made in several mutually exclusive contexts, (2) in terms of relationship between factual measurements in a given context and counterfactual measurements that would be made if one used other contexts, and (3) in terms of (non)existence of "hidden variables" that map into the outcomes of measurements by context-independent functions. It is believed by many that the three meanings are equivalent, but the issue remains controversial. In particular, Robert Griffiths recently claimed (Phys. Rev. A 96:032110, 2017) that the first and the second meanings are not equivalent. We show that if the first meaning is formulated within the framework of the Contextuality-by-Default (CbD) theory, then the traditional view can be upheld unambiguously. The language of probabilistic couplings, which is at the heart of CbD, is a mathematically rigorous way of speaking of counterfactual values, whereas the "hidden variables" mapped into measurement outcomes by appropriately defined functions is merely a variant of introducing counterfactual values.