Ehtibar N. Dzhafarov

Ehtibar N. Dzhafarov, Victor H. Cervantes

We introduce a new notion, that of a contextuality profile of a system. Rather than characterizing a system's contextuality by a single number, its overall degree of contextuality, we show how it can be characterized by a curve relating degree of contextuality to level at which the system is considered,\begin{array}{c|c|c|c|c|c|c|c} \textnormal{level} & 1 & \cdots & n-1 & n>1 & n+1 & \cdots & N\\ \hline \textnormal{degree} & 0 & \cdots & 0 & d_{n}>0 & d_{n+1}\geq d_{n} & \cdots & d_{N}\geq d_{N-1} \end{array},where N is the maximum number of variables per system's context. A system is represented at level n if one only considers the joint distributions with k\leq n variables, ignoring higher-order joint distributions. We show that the level-wise contextuality analysis can be used in conjunction with any well-constructed measure of contextuality. We present a method of concatenated systems to explore contextuality profiles systematically, and we apply it to the contextuality profiles for three major measures of contextuality proposed in the literature.

Damir D. Dzhafarov, Ehtibar N. Dzhafarov

We view sorites in terms of stimuli acting upon a system and evoking this system's responses. Supervenience of responses on stimuli implies that they either lack tolerance (i.e., they change in every vicinity of some of the stimuli), or stimuli are not always connectable by finite chains of stimuli in which successive members are `very similar'. If supervenience does not hold, the properties of tolerance and connectedness cannot be formulated and therefore soritical sequences cannot be constructed. We hypothesize that supervenience in empirical systems (such as people answering questions) is fundamentally probabilistic. The supervenience of probabilities of responses on stimuli is stable, in the sense that `higher-order' probability distributions can always be reduced to `ordinary' ones. In making rules about which stimuli ought to correspond to which responses, the main characterization of choices in soritical situations is their arbitrariness. We argue that arbitrariness poses no problems for classical logic.

Ehtibar N. Dzhafarov, Janne V. Kujala

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.