Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Contextuality and quantum mechanics (QM) <s> Forty years after the advent of quantum mechanics the problem of hidden variables, that is, the possibility of imbedding quantum theory into a classical theory, remains a controversial and obscure subject. Whereas to most physicists the possibility of a classical reinterpretation of quantum mechanics remains remote and perhaps irrelevant to current problems, a minority have kept the issue alive throughout this period. (See Freistadt [5] for a review of the problem and a comprehensive bibliography up to 1957.) As far as results are concerned there are on the one hand purported proofs of the non-existence of hidden variables, most notably von Neumann’s proof, and on the other, various attempts to introduce hidden variables such as de Broglie [4] and Bohm [1] and [2]. One of the difficulties in evaluating these contradictory results is that no exact mathematical criterion is given to enable one to judge the degree of success of these proposals. <s> BIB001 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Contextuality and quantum mechanics (QM) <s> 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. <s> BIB002 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Contextuality and quantum mechanics (QM) <s> We present general principles underlying analysis of the dependence of random variables (outputs) on deterministic conditions (inputs). Random outputs recorded under mutually exclusive input values are labeled by these values and considered stochastically unrelated, possessing no joint distribution. An input that does not directly influence an output creates a context for the latter. Any constraint imposed on the dependence of random outputs on inputs can be characterized by considering all possible couplings (joint distributions) imposed on stochastically unrelated outputs. The target application of these principles is a quantum mechanical system of entangled particles, with directions of spin measurements chosen for each particle being inputs and the spins recorded outputs. The sphere of applicability, however, spans systems across physical, biological, and behavioral sciences. <s> BIB003 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Contextuality and quantum mechanics (QM) <s> Contextual situations are those in which seemingly 'the same' random variable changes its identity depending on the conditions under which it is recorded. Such a change of identity is observed whenever the assumption that the variable is one and the same under different conditions leads to contradictions when one considers its joint distribution with other random variables (this is the essence of all Bell-type theorems). In our Contextuality-by-Default approach, instead of asking why or how the conditions force 'one and the same' random variable to change 'its' identity, any two random variables recorded under different conditions are considered different 'automatically.' They are never the same, nor are they jointly distributed, but one can always impose on them a joint distribution (probabilistic coupling). The special situations when there is a coupling in which these random variables are equal with probability 1 are considered noncontextual. Contextuality means that such couplings do not exist. We argue that the determination of the identity of random variables by conditions under which they are recorded is not a causal relationship and cannot violate laws of physics. <s> BIB004
The notion of (non)contextuality has its origins in logic  , but since the publication of Ref. BIB001 it has been widely considered a QM notion. QM indeed provides the only known to us theoretically justified examples of contextual systems. (Non)contextuality per se, however, is a purely probabilisitic concept, squarely within the classical, Kolmogorovian probability theory (that includes the notion of stochastic unrelatedness and that of couplings) BIB002 BIB004 BIB003 . When contextuality is present in a QM system, QM is relevant to answering the question of exactly how the noncontextuality conditions in the system are violated, but it is not relevant to the question of what these conditions are.