The notion of (non) contextuality pertains to sets of properties measured one subset (context) at a time. We extend this notion to include so-called inconsistently connected systems, in which the measurements of a given property in different contexts may have different distributions, due to contextual biases in experimental design or physical interactions (signaling): a system of measurements has a maximally noncontextual description if they can be imposed a joint distribution on in which the measurements of any one property in different contexts are equal to each other with the maximal probability allowed by their different distributions. We derive necessary and sufficient conditions for the existence of such a description in a broad class of systems including Klyachko-Can-Binicioglu-Shumvosky-type (KCBS), EPR-Bell-type, and Leggett-Garg-type systems. Because these conditions allow for inconsistent connectedness, they are applicable to real experiments. We illustrate this by analyzing an experiment by Lapkiewicz and colleagues aimed at testing contextuality in a KCBS-type system.

Funding Agencies|NSF [SES-1155956]; AFOSR [FA9550-14-1-0318]; A. von Humboldt Foundation; FQXi through the Silicon Valley Community Foundation

[Correction Notice: An Erratum for this article was reported in Vol 121(1) of Psychological Review (see record 2014-03591-005). The link to supplemental material was missing. All versions of this article have been corrected.] Much current research on speeded choice utilizes models in which the response is triggered by a stochastic process crossing a deterministic threshold. This article focuses on 2 such model classes, 1 based on continuous-time diffusion and the other on linear ballistic accumulation (LBA). Both models assume random variability in growth rates and in other model components across trials. We show that if the form of this variability is unconstrained, the models can exactly match any possible pattern of response probabilities and response time distributions. Thus, the explanatory or predictive content of these models is determined not by their structural assumptions but, rather, by distributional assumptions (e.g., Gaussian distributions) that are traditionally regarded as implementation details. Selective influence assumptions (i.e., which experimental manipulations affect which model parameters) are shown to have no restrictive effect, except for the theoretically questionable assumption that speed-accuracy instructions do not affect growth rates. The 2nd contribution of this article concerns translation of falsifiable models between universal modeling languages. Specifically, we translate the predictions of the diffusion and LBA models (with their parametric and selective influence assumptions intact) into the Grice modeling framework, in which accumulation processes are deterministic and thresholds are random variables. The Grice framework is also known to reproduce any possible pattern of response probabilities and times, and hence it can be used as a common language for comparing models. It is found that only a few simple properties of empirical data are necessary predictions of the diffusion and LBA models.

This paper provides a systematic yet accessible presentation of the Contextuality-by-Default theory. The consideration is confined to finite systems of categorical random variables, which allows us to focus on the basics of the theory without using full-scale measure-theoretic language. Contextuality-by-Default is a theory of random variables identified by their contents and their contexts, so that two variables have a joint distribution if and only if they share a context. Intuitively, the content of a random variable is the entity the random variable measures or responds to, while the context is formed by the conditions under which these measurements or responses are obtained. A system of random variables consists of stochastically unrelated "bunches," each of which is a set of jointly distributed random variables sharing a context. The variables that have the same content in different contexts form "connections" between the bunches. A probabilistic coupling of this system is a set of random variables obtained by imposing a joint distribution on the stochastically unrelated bunches. A system is considered noncontextual or contextual according to whether it can or cannot be coupled so that the joint distributions imposed on its connections possess a certain property (in the present version of the theory, "maximality"). We present a criterion of contextuality for a special class of systems of random variables, called cyclic systems. We also introduce a general measure of contextuality that makes use of (quasi-)couplings whose distributions may involve negative numbers or numbers greater than 1 in place of probabilities.

There has been a strong recent interest in applying quantum theory (QT) outside physics, including in cognitive science. We analyze the applicability of QT to two basic properties in opinion polling. The first property (response replicability) is that, for a large class of questions, a response to a given question is expected to be repeated if the question is posed again, irrespective of whether another question is asked and answered in between. The second property (question order effect) is that the response probabilities frequently depend on the order in which the questions are asked. Whenever these two properties occur together, it poses a problem for QT. The conventional QT with Hermitian operators can handle response replicability, but only in the way incompatible with the question order effect. In the generalization of QT known as theory of positive-operator-valued measures (POVMs), in order to account for response replicability, the POVMs involved must be conventional operators. Although these problems are not unique to QT and also challenge conventional cognitive theories, they stand out as important unresolved problems for the application of QT to cognition. Either some new principles are needed to determine the bounds of applicability of QT to cognition, or quantum formalisms more general than POVMs are needed.

selective influence, conditional independence, marginal selectivity, processing architectures, Thurstonian modeling,