Random variables representing measurements, broadly understood to include any responses to any inputs, form a system in which each of them is uniquely identified by its content (that which it measures) and its context (the conditions under which it is recorded). Two random variables are jointly distributed if and only if they share a context. In a canonical representation of a system, all random variables are binary, and every content-sharing pair of random variables has a unique maximal coupling (the joint distribution imposed on them so that they coincide with maximal possible probability). The system is contextual if these maximal couplings are incompatible with the joint distributions of the context-sharing random variables. We propose to represent any system of measurements in a canonical form and to consider the system contextual if and only if its canonical representation is contextual. As an illustration, we establish a criterion for contextuality of the canonical system consisting of all dichotomizations of a single pair of content-sharing categorical random variables.

We present unambiguous experimental evidence for (quantum-like) probabilistic contextuality in psychology. All previous attempts to find contextuality in a psychological experiment were unsuccessful because of the gross violations of marginal selectivity in behavioral data, making the traditional mathematical tests developed in quantum mechanics inapplicable. In our crowdsourcing experiment respondents were making two simple choices: of one of two characters in a story (The Snow Queen by Hans Christian Andersen), and of one of two characteristics, such as Kind and Evil, so that the character and the characteristic chosen matched the story line. The formal structure of the experiment imitated that of the Einstein-Podolsky-Rosen paradigm in the Bohm-Bell version. Marginal selectivity was violated, indicating that the two choices were directly influencing each other, but the application of a mathematical test developed in the Contextuality-by-Default theory, extending the traditional quantum-mechanical test, indicated a strong presence of contextuality proper, not reducible to direct influences.

In quantum physics there are well-known situations when measurements of the same property in different contexts (under different conditions) have the same probability distribution, but cannot be represented by one and the same random variable. Such systems of random variables are called contextual. More generally, true contextuality is observed when different contexts force measurements of the same property (in psychology, responses to the same question) to be more dissimilar random variables than warranted by the difference of their distributions. The difference in distributions is itself a form of context-dependence, but of another nature: it is attributable to direct causal influences exerted by contexts upon the random variables. The Contextuality-by-Default (CbD) theory allows one to separate true contextuality from direct influences in the overall context-dependence. The CbD analysis of numerous previous attempts to demonstrate contextuality in human judgments shows that all context-dependence in them can be accounted for by direct influences, with no true contextuality present. However, contextual systems in human behavior can be found. In this paper we present a series of crowdsourcing experiments that exhibit true contextuality in simple decision making. The design of these experiments is an elaboration of one introduced in the "Snow Queen" experiment (Decision 5, 193-204, 2018), where contextuality was for the first time demonstrated unequivocally.

Dzhafarovet al.(Dzhafarovet al.2016Phil. Trans. R. Soc. A374, 20150099. (doi:10.1098/rsta.2015.0099)) reviewed several behavioural datasets imitating the formal design of the quantum-mechanical contextuality experiments. The conclusion was that none of these datasets exhibited contextuality if understood in the generalized sense proposed by Dzhafarovet al.(2015Found. Phys.7, 762-782. (doi:10.1007/s10701-015-9882-9)), while the traditional definition of contextuality does not apply to these data because they violate the condition of consistent connectedness (also known as marginal selectivity, no-signalling condition, no-disturbance principle, etc.). In this paper, we clarify the relationship between (in)consistent connectedness and (non)contextuality, as well as between the traditional and extended definitions of (non)contextuality, using as an example the Clauser-Horn-Shimony-Holt inequalities originally designed for detecting contextuality in entangled particles.

Highlights Most of the patients with critical COVID-19 admitted to the 10 ICUs in Mexico were men over 57 years of age with hypertension and diabetes, and 6% were health-care workers. Patients with hypertension and diabetes had significantly decreased survival, but neither of these comorbidities were an independent factor associated with mortality Patients with critical COVID-19 who died in the hospital exhibit significantly higher C-reactive protein concentrations than survivors in our study