# Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Basic Notions

**Definition**

**1.**

**(canonical contents)**- ${Q}^{\u2020}=\left\{\left(q,c\right):q\prec c\right\}$,
**(canonical contexts)**- ${C}^{\u2020}=\left\{\left(\xb7,c\right):c\in C\right\}\bigsqcup \left\{\left(q,\xb7\right):q\in Q\right\}$,
**(canonical relation)**- $\left(q,c\right){\prec}^{\u2020}\left(\xb7,c\right)\u27fa\left(q,c\right)\in {Q}^{\u2020}$, and $\left(q,c\right){\prec}^{\u2020}\left(q,\xb7\right)\u27fa\left(q,c\right)\in {Q}^{\u2020},$
**(main bunches)**- ${R}^{\left(\xb7,c\right)}=\left\{{R}_{\left(q,c\right)}^{\left(\xb7,c\right)}:\left(q,c\right){\prec}^{\u2020}\left(\xb7,c\right)\right\}\stackrel{d}{=}\left\{{R}_{q}^{c}:q\prec c\right\}={R}^{c}$
**(auxiliary bunches)**- ${R}^{\left(q,\xb7\right)}=\left\{{R}_{\left(q,c\right)}^{\left(q,\xb7\right)}:\left(q,c\right){\prec}^{\u2020}\left(q,\xb7\right)\right\}$ is uniquely determined by the distributions of the corresponding variables in ${\mathcal{R}}_{\left(q,\xb7\right)}=\left\{{R}_{\left(q,c\right)}^{\left(\xb7,c\right)}:\left(q,c\right){\prec}^{\u2020}\left(q,\xb7\right)\right\}$.

- 1.
- The formats of $\mathcal{R}$ and ${\mathcal{R}}^{\u2020}$ are reconstructible from each other, and so are the bunches of the two systems. Moreover, ${\mathcal{R}}^{\u2020}$ faithfully replicates the bunches of $\mathcal{R}$. This allows one to say that $\mathcal{R}$ and ${\mathcal{R}}^{\u2020}$ describe the same empirical or theoretical situation.
- 2.
- One might wonder why we need the auxiliary contexts at all, and they are indeed unnecessary if all one wants is a system modeling another system, e.g.,However, we will see the utility of the auxiliary contexts when we introduce consistifications and contextual equivalence, in Section 4.
- 3.
- The contents in the modeling system are “contextualized”. For instance, system $\mathcal{A}$ in (4) may be describing an experiment in which two questions, $q=1$ and $q=2$, are asked in two orders, $c=1$ indicating “1 then 2” and $c=2$ indicating “2 then 1” [30,31]. In this case, in the modeling system, the content $q=\left(1,2\right)$ should be interpreted as “question 1 asked second”, and $q=\left(1,1\right)$ should be interpreted as “question 1 asked first”. We return to the issue of interpretation in Section 5.1.
- 4.
- The indexation of the variables in a canonical model is clearly redundant, and it can be simplified. It is more important, however, to maintain the general logic of indexing the variables by their contents and contexts.

## 3. Traditional and Extended Contextuality

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 4. Equivalence and Impossibility Theorems

**Q***:- Is it possible to formulate a set of such desiderata/requirements $\mathsf{X}$ for the notion of contextuality that, for some choice of $\mathsf{C}$, (1) $\mathsf{X}$ is satisfied for any consistently connected system, but (2) $\mathsf{X}$ is not satisfied for some inconsistently connected systems?

**Q***, we need the following result.

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

## 5. Miscellaneous Remarks

#### 5.1. Interpretation of Contents and Contexts

#### 5.2. Hidden Variable Models

#### 5.3. The Existence and Uniqueness Constraint

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Dzhafarov, E.N.; Kujala, J.V. Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies. *Entropy* **2023**, *25*, 581.
https://doi.org/10.3390/e25040581

**AMA Style**

Dzhafarov EN, Kujala JV. Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies. *Entropy*. 2023; 25(4):581.
https://doi.org/10.3390/e25040581

**Chicago/Turabian Style**

Dzhafarov, Ehtibar N., and Janne V. Kujala. 2023. "Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies" *Entropy* 25, no. 4: 581.
https://doi.org/10.3390/e25040581