It follows from the last two definitions that, for a well-fitting $\mathsf{C}$, a consistently connected system is $\mathsf{C}$-noncontextual if and only if it is noncontextual in the traditional sense (i.e., in the sense of Definition 2). In other words, any extension of the notion of contextuality using a well-fitting $\mathsf{C}$ properly reduces to the traditional notion when confined to consistently connected systems. This is not, obviously, sufficient to consider the extension of contextuality by means of a well-constructed $\mathsf{C}$. There may be other desiderata for a well-constructed notion of contextuality, and a specific choice of $\mathsf{C}$ may not satisfy them. The question we pose now is as follows:

• 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?

Note that we impose no constraints on what $\mathsf{X}$ may entail, except for its being related to contextuality. It may, e.g., for some relation B between systems, have the form “if system ${\mathcal{R}}_1$ is (non)contextual, then any system ${\mathcal{R}}_2$ related to ${\mathcal{R}}_1$ by B is (non)contextual” [24].

To answer the question Q*, we need the following result.

Theorem 1.For any well-fitting $\mathsf{C}$ and system $\mathcal{R}$, there is a consistently connected system ${\mathcal{R}}^{‡}$ that canonically models it (Definition 1), such that $\mathcal{R}$ is $\mathsf{C}$-contextual (Definition 4) if and only if ${\mathcal{R}}^{‡}$ is contextual in the traditional sense (Definition 2).

Proof. Let ${\mathcal{R}}^{‡}$ be a canonically modeling system for $\mathcal{R}$, with One can check that ${\mathcal{R}}^{‡}$ is consistently connected: every connection ${\mathcal{R}}_{\left(q,c\right)}$ of ${\mathcal{R}}^{‡}$ consists of precisely two variables, $R_{\left(q,c\right)}^{\left(·,c\right)}$ and $R_{\left(q,c\right)}^{\left(q,·\right)}$, where $R_{\left(q,c\right)}^{\left(·,c\right)}\stackrel{d}{=}{R}_{\left(q,c\right)}^{\left(q,·\right)}$. Indeed, $R_{\left(q,c\right)}^{\left(·,c\right)}\stackrel{d}{=}{R}_q^c$, because $R^{\left(·,c\right)}\stackrel{d}{=}{R}^c$ in any canonically modeling system and $R_{\left(q,c\right)}^{\left(q,·\right)}\stackrel{d}{=}{R}_q^c$ because we know from (*) that $R_{\left(q,c\right)}^{\left(q,·\right)}\stackrel{d}{=}{S}_q^c$, where $S_q^c\in \mathsf{C}\left[{\mathcal{R}}_q\right]$.The system ${\mathcal{R}}^{‡}$ thus constructed is referred to as a consistification of $\mathcal{R}$. We can now define the consistification $S^{‡}$ of a coupling S of a system in precisely the same way as for the system itself, except that (*) is replaced with the straightforward with the obvious correspondence between the different indexations within the two random vectors. Clearly, $S^{‡}$ is a coupling of ${\mathcal{R}}^{‡}$.Assume now that $\mathcal{R}$ is noncontextual. This means that it has a coupling S such that (a) $S^c\stackrel{d}{=}{R}^c$ for every $c\in C$ and (b) $S_q=\mathsf{C}\left[{\mathcal{R}}_q\right]$ for every $q\in Q$. Then, in the coupling $S^{‡}$ of system ${\mathcal{R}}^{‡}$, we have (a’) $S^{\left(·,c\right)}\stackrel{d}{=}{R}^{\left(·,c\right)}$ for every $\left(·,c\right)\in C^{‡}$, and (b’) $S^{\left(q,·\right)}=\mathsf{C}\left[{\mathcal{R}}_q\right]$ for every $\left(q,·\right)\in C^{‡}$. Moreover, since both $S_{\left(q,c\right)}^{\left(·,c\right)}$ and $S_{\left(q,c\right)}^{\left(q,·\right)}$ equal $S_q^c$, we have (c’) $S_{\left(q,c\right)}^{\left(·,c\right)}=S_{\left(q,c\right)}^{\left(q,·\right)}$. However, (a’)-(b’)-(c’) mean that ${\mathcal{R}}^{‡}$ is noncontextual in the traditional sense. The implication here is easily seen to be reversible, and we conclude that $\mathcal{R}$ is noncontextual if and only if so is ${\mathcal{R}}^{‡}$. □

$${\mathcal{R}}^{\left(q,·\right)}=\left\{R_{\left(q,c\right)}^{\left(q,·\right)}:\left(q,c\right){\prec }^{‡}\left(q,·\right)\right\}\stackrel{d}{=}\mathsf{C}\left[\left\{R_q^c:q\prec c\right\}\right]=\mathsf{C}\left[{\mathcal{R}}_q\right].\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\ast \right)$$

$$S^{\left(q,·\right)}=S_q,$$

In our example (4), $\mathcal{B}$ is a consistification of $\mathcal{A}$ if we specify the rule for the auxiliary bunches as follows: $R_{\left(q,c\right)}^{\left(q,·\right)}\stackrel{d}{=}{R}_{\left(q,c\right)}^{\left(·,c\right)}$, and the distribution of $R^{\left(q,·\right)}$ is the same as that of $\mathsf{C}\left[{\mathcal{R}}_q\right]$. If $\mathsf{C}$ is chosen as in CbD, the consistification of the system $\mathcal{R}$ in (2) is the system below (omitting, for simplicity, the parentheses and commas in $R_{\left(q,c\right)}^{\left(·,c\right)}$ and $R_{\left(q,c\right)}^{\left(q,·\right)}$):

(11)

where all variables are assumed to be dichotomous, and in each of the auxiliary bunches, the variables are pairwise equal with maximal possible probability.

For the purposes of contextuality analysis, ${\mathcal{R}}^{‡}$ can be viewed as a mere reformulation of $\mathcal{R}$, a different form of the same substance. We express this fact by saying that $\mathcal{R}$ and ${\mathcal{R}}^{‡}$ are contextually equivalent. (In Refs. [24,26], contextual equivalence is defined more narrowly, requiring also the numerical coincidence of certain measures of contextuality, such as contextual fraction [32]. In this paper, however, the level of abstraction is higher, and we only consider the notion of contextuality rather than its quantifications.)

Consider now a theory of (generally, extended) contextuality $\mathsf{T}=\mathsf{T}\left(\mathsf{C},\mathfrak{R}\right)$. In accordance with Theorem 1, we can form the class ${\mathfrak{R}}^{‡}$ of the consistifications of the elements of $\mathfrak{R}$ in a bijective correspondence with $\mathfrak{R}$. By extension of the term, we can say that $\mathsf{T}$ and ${\mathsf{T}}^{\prime }=\mathsf{T}\left({\mathsf{C}}_0,{\mathfrak{R}}^{‡}\right)$ are contextually equivalent. ${\mathsf{C}}_0$ here denotes the statement “the connection $R_q$ has an identity coupling” that underlies the traditional notion of contextuality, because by definition, it can be viewed as a special case of any well-fitting statement $\mathsf{C}$. We have now everything we need to demonstrate our main conclusion. Let there be a set of requirements $\mathsf{X}$ of the notion of contextuality that are satisfied by all consistently connected systems (using the traditional contextuality) and contravened by some inconsistently connected ones, using some version of $\mathsf{C}$-contextuality. Let $\mathsf{T}$ include some of the inconsistently connected systems contravening $\mathsf{X}$. Clearly then, requirements $\mathsf{X}$ contradict theory $\mathsf{T}$, but they are satisfied by the contextually equivalent theory ${\mathsf{T}}^{\prime }=\mathsf{T}\left({\mathsf{C}}_0,{\mathfrak{R}}^{‡}\right)$. Therefore, $\mathsf{X}$ is not a set of substantive requirements. We can summarize this as a formal theorem.

Theorem 2.For any well-fitting $\mathsf{C}$, there can be no set of substantive requirements $\mathsf{X}$ of the notion of contextuality that are satisfied by all consistently connected systems (using the traditional contextuality) and contravened by some inconsistently connected ones, using $\mathsf{C}$-contextuality.

Of course, a set of requirements $\mathsf{X}$ satisfied by ${\mathsf{T}}^{\prime }$ but not $\mathsf{T}$ can be readily formulated. The theorem says, however, that all it can do is lead one to prefer one of two equivalent representations of contextuality, without affecting the substance of the notion.

Note also that in the theorem just formulated, we assume no relationship between the set of requirements $\mathsf{X}$ and the bijective correspondence relating $\mathfrak{R}$ to ${\mathfrak{R}}^{‡}$. In particular, let $\mathsf{X}$ have the form “if system ${\mathcal{R}}_1$ is contextual, then any system ${\mathcal{R}}_2$ related to ${\mathcal{R}}_1$ by relation B is contextual.” It is not necessary then, although not excluded either, that ${\mathcal{R}}_2^{‡}$ is also related to ${\mathcal{R}}_1^{‡}$ by relation B. All that is stated in the theorem above is that if one wishes to use this $\mathsf{X}$ as a substantive principle in testing competing theories, then the failure of a theory to satisfy it cannot be selectively attributed to the fact that its $\mathfrak{R}$ contains inconsistently connected systems.