Causal contextuality and contextuality-by-default are different concepts

Abstract

In the framework of operational theories and ontological models, contextuality is defined as a causal (and hence probabilistic) dependence of the distribution of measurement outcomes in a given ontic state on simultaneous measurements. I will refer to this concept as causal contextuality and compare it with contextuality-by-default (CbD-contextuality) defined by Dzhafarov, Kujala and colleagues. The two concepts are different. First, causal contextuality is a property of the ontological model while CbD-contextuality is a feature of the operational theory. Second, signaling operational theories can be CbD-noncontextual but they admit only causally contextual ontological models. To highlight the difference between the two concepts, I construct a simple toy model which is causally contextual but CbD-noncontextual.