Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Beyond objects and subsets <s> The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By taking into account the dependence of (classical) probabilities on contexts (i.e. complexes of physical conditions), one can reproduce all distinct features of quantum probabilities such as the interference of probabilities and the violation of Bells inequality. Moreover, by starting with a formula for the interference of probabilities (which generalizes the well known classical formula of total probability), one can construct the representation of contextual probabilities by complex probability amplitudes or, in the abstract formalism, by normalized vectors of the complex Hilbert space or its hyperbolic generalization. Thus the Hilbert space representation of probabilities can be naturally derived from classical probabilistic assumptions. An important chapter of the book critically reviews known no-go theorems: the impossibility to establish a finer description of micro-phenomena than provided by quantum mechanics; and, in particular, the commonly accepted consequences of Bells theorem (including quantum non-locality). Also, possible applications of the contextual probabilistic model and its quantum-like representation in complex Hilbert spaces in other fields (e.g. in cognitive science and psychology) are discussed. <s> BIB001 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Beyond objects and subsets <s> Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach (“contextuality-by-default”) is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two measures essentially coincide. <s> BIB002 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Beyond objects and subsets <s> Contextuality means non-existence of a joint distribution for random variables recorded under mutually incompatible conditions, subject to certain constraints imposed on how the identity of these variables may change across these conditions. In simple quantum systems contextuality is indicated by violations of Bell-type or Leggett-Garg-type inequalities. These inequalities, however, are predicated on the assumption of no-signaling, defined as invariance of the distributions of measurement results with respect to other (e.g., earlier in time) measurements' settings. Signaling makes the inequalities inapplicable: a non-signaling system with any degree of contextuality, however high, loses any relation to this concept as soon as it exhibits any degree of signaling, however small. This is unsatisfactory. We describe a principled way of defining and measuring contextuality in arbitrary systems with random outputs, whether signaling is absent or present. <s> BIB003 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Beyond objects and subsets <s> We present a proof for a conjecture previously formulated by Dzhafarov, Kujala, and Larsson (Foundations of Physics, in press, arXiv:1411.2244). The conjecture specifies a measure for the degree of contextuality and a criterion (necessary and sufficient condition) for contextuality in a broad class of quantum systems. This class includes Leggett-Garg, EPR/Bell, and Klyachko-Can-Binicioglu-Shumovsky type systems as special cases. In a system of this class certain physical properties $q_{1},...,q_{n}$ are measured in pairs $(q_{i},q_{j})$; every property enters in precisely two such pairs; and each measurement outcome is a binary random variable. Denoting the measurement outcomes for a property $q_{i}$ in the two pairs it enters by $V_{i}$ and $W_{i}$, the pair of measurement outcomes for $(q_{i},q_{j})$ is $(V_{i},W_{j})$. Contextuality is defined as follows: one computes the minimal possible value $\Delta_{0}$ for the sum of $\Pr[V_{i}\not=W_{i}]$ (over $i=1,...,n$) that is allowed by the individual distributions of $V_{i}$ and $W_{i}$; one computes the minimal possible value $\Delta_{\min}$ for the sum of $\Pr[V_{i}\not=W_{i}]$ across all possible couplings of (i.e., joint distributions imposed on) the entire set of random variables $V_{1},W_{1},...,V_{n},W_{n}$ in the system; and the system is considered contextual if $\Delta_{\min}>\Delta_{0}$ (otherwise $\Delta_{\min}=\Delta_{0}$). This definition has its justification in the general approach dubbed Contextuality-by-Default, and it allows for measurement errors and signaling among the measured properties. The conjecture proved in this paper specifies the value of $\Delta_{\min}-\Delta_{0}$ in terms of the distributions of the measurement outcomes $(V_{i},W_{j})$. <s> BIB004
Defining a context as a subset of objects measured together   is less general than defining it by conditions under which certain objects are measured BIB002  BIB003 . For instance, by the first of these definitions {q 1 , q 2 } for a given pair of objects is a single context, while the second definition allows one to speak of the same pair of objects q 1 , q 2 forming several different contexts. Thus, if q 1 , q 2 are two tests, they can be given in one order or the other, (q 1 , q 2 ) or (q 1 , q 2 ). In fact, in all our previous discussion of cyclic systems we used the notation for ordered pairs, (q i , q i⊕1 ) rather than {q i , q i⊕1 }. This is inconsequential for cyclic systems of rank n ≥ 3. For n = 2, however, the difference between (q 1 , q 2 ) and (q 2 , q 1 ) is critical if n = 2 is to be a nontrivial system (with the distributions of the two bunches not identical). It can be shown that the system can be nontrivial, and n = 2 is a legitimate value for BIB001 and BIB004 . Being formal and mathematically rigorous here makes things simpler. A context is merely a label (say, superscript) at a random variable with the convention that identically superscripted variables are "bunched together," i.e., they are jointly distributed. An object is merely another label (in our notation, a subscript) that makes all the elements of a bunch different and indicates which elements from different bunches should be put together to form a a connection. So if there are six random variables grouped into three distinct bunches R , we can (but do not have to) interpret this as three different contexts involving the same two objects. From mathematical (and perhaps also philosophical) point of view, measurements grouped into bunches and connections are more fundamental than objects being measured within contexts.