Volume 112, February 2023, 102726Author links open overlay panelAbstract

Contextuality (or lack thereof) is a property of systems of random variables. Among the measures of the degree of contextuality, two have played important roles. One of them, Contextual Fraction (CNTF) was proposed within the framework of the sheaf-theoretic approach to contextuality, and extended to arbitrary systems in the Contextuality-by-Default approach. The other, denoted CNT2, was proposed as one of the measures within the Contextuality-by-Default approach. In this note, I prove that CNTF=2CNT2 within a class of systems, called cyclic, that have played a prominent role in contextuality research.

Section snippetsContextuality-by-default

This section introduces the definitions and results from the Contextuality-by-Default theory (CbD; Dzhafarov, 2019, Dzhafarov and Kujala, 2016, Dzhafarov and Kujala, 2017a, Dzhafarov, Kujala and Cervantes, 2016, Dzhafarov et al., 2020a, Kujala and Dzhafarov, 2019) that are needed for the proof. Throughout the text, I shall use the data from Cervantes and Dzhafarov (2018) to illustrate the defined objects and computations. Their data come from an online experiment in which participants were

Main theorem

Theorem 1

If Rn is a cyclic system of rank n, then CNTF=2CNT2.

Proof

Let Rn be a cyclic system of rank n. Note that if Rn is noncontextual, both measures CNT2 and CNTF equal zero and, trivially, CNTF=2CNT2. Thus, we will assume that Rn is contextual.

Without loss of generality, assume that the system Rn is consistently connected. If not, take the consistification Rn‡ of Rn. Consider the vector ϕ(b∗) and the corresponding sets ℂb, Db, Rb, and Eb. Theorem 15 of Dzhafarov et al. (2020a) states that CNT2=14‖ϕ(b∗),Eb‖

Discussion

The proof presented in this paper improves the understanding of the relationship between the different approaches to studying contextuality. Dzhafarov et al. (2020a) showed that in cyclic systems CNT2 is equal to another measure, CNT1, and in Kujala and Dzhafarov (2019), it was conjectured that these measures are proportional to a measure computed based on negative probabilities, CNT3. If this conjecture is found to be true, the current result shows that all these different approaches to the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Disclosures and acknowledgments

The author carried out most of this work as an Illinois Distinguished Postdoctoral Researcher at the University of Illinois at Urbana-Champaign.

The author is grateful to Sandra Camargo, Ehtibar Dzhafarov and an anonymous reviewer for feedback on earlier drafts, and to Ehtibar Dzhafarov and Giulio Camillo for fruitful conversations. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or

References (37)AsanoM. et al.Violation of contextual generalization of the Leggett-Garg inequality for recognition of ambiguous figures

Physica Scripta

(2014)

BasievaI. et al.True contextuality beats direct influences in human decision making

Journal of Experimental Psychology: General

(2019)

BellJ.S. On the Einstein-Podolsky-Rosen paradox

Physics

(1964)

BellJ.S. On the problem of hidden variables in quantum mechanics

Reviews of Modern Physics

(1966)

BusemeyerJ.R. et al.Quantum models of cognition and decision

(2012)

BusemeyerJ.R. et al.What is quantum cognition, and how is it applied to psychology?

Current Directions in Psychological Science

(2015)

CervantesV.H. et al.Exploration of contextuality in a psychophysical double-detection experimentCervantesV.H. et al.Snow queen is evil and beautiful: Experimental evidence for probabilistic contextuality in human choices

Decision

(2018)

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