In his constructive and well-informed commentary, Andrei Khrennikov acknowledges a privileged status of classical probability theory with respect to statistical analysis. He also sees advantages offered by the Contextuality-by-Default theory, notably, that it “demystifies quantum mechanics by highlighting the role of contextuality,” and that it can detect and measure contextuality in inconsistently connected systems. He argues, however, that classical probability theory may have difficulties in describing empirical phenomena if they are described entirely in terms of observable events. We disagree: contexts (...) in which random variables are recorded are as observable as the variables’ values. Khrennikov also argues that the Contextuality-by-Default theory suffers the problem of non-uniqueness of couplings. We disagree that this is a problem: couplings are all possible ways of imposing counterfactual joint distributions on random variables that de facto are not jointly distributed. The uniqueness of modeling experiments by means of quantum formalisms brought up by Khrennikov is achieved for the price of additional, substantive assumptions. This is consistent with our view of quantum theory as a special-purpose generator of classical probabilities. Khrennikov raises the issue of “mental signaling,” by which he means inconsistent connectedness in behavioral systems. Our position is that it is as inherent to behavioral systems as their stochasticity. (shrink)

The Contextuality-by-Default approach to determining and measuring the (non)contextuality of a system of random variables requires that every random variable in the system be represented by an equivalent set of dichotomous random variables. In this paper we present general principles that justify the use of dichotomizations and determine their choice. The main idea in choosing dichotomizations is that if the set of possible values of a random variable is endowed with a pre-topology (V-space), then the allowable dichotomizations split the space (...) of possible values into two linked subsets (“linkedness” being a weak form of pre-topological connectedness). We primarily focus on two types of random variables most often encountered in practice: categorical and real-valued ones (including continuous random variables, greatly underrepresented in the contextuality literature). A categorical variable (one with a finite number of unordered values) is represented by all of its possible dichotomizations. If the values of a random variable are real numbers, then they are dichotomized by intervals above and below a variable cut point. (shrink)