## Abstract

Most behavioural and social experiments aimed at revealing contextuality are confined to cyclic systems with binary outcomes. In quantum physics, this broad class of systems includes as special cases Klyachko–Can–Binicioglu–Shumovsky-type, Einstein–Podolsky–Rosen–Bell-type and Suppes–Zanotti–Leggett–Garg-type systems. The theory of contextuality known as contextuality-by-default allows one to define and measure contextuality in all such systems, even if there are context-dependent errors in measurements, or if something in the contexts directly interacts with the measurements. This makes the theory especially suitable for behavioural and social systems, where direct interactions of ‘everything with everything’ are ubiquitous. For cyclic systems with binary outcomes, the theory provides necessary and sufficient conditions for non-contextuality, and these conditions are known to be breached in certain quantum systems. We review several behavioural and social datasets (from polls of public opinion to visual illusions to conjoint choices to word combinations to psychophysical matching), and none of these data provides any evidence for contextuality. Our working hypothesis is that this may be a broadly applicable rule: behavioural and social systems are non-contextual, i.e. all ‘contextual effects’ in them result from the ubiquitous dependence of response distributions on the elements of contexts other than the ones to which the response is presumably or normatively directed.

## 1. Introduction

Although the word is widely used in linguistics, psychology and philosophy, the notion of contextuality as it is used in this paper comes from quantum mechanics, where in turn it came from logic [1]. The reason for the prominence of this notion in quantum theory is that classical-mechanical systems are not contextual while some quantum-mechanical systems are. Contextuality is sometimes even presented as one of the ‘paradoxes’ of quantum mechanics. In psychology, as it turns out, a certain variety of (non-)contextuality has been prominent too, but it is known under a different name: selectiveness of influences, or lack thereof (for details, see [2,3]).

The term ‘contextuality’ refers to properties of systems of random variables each of which can be viewed (sometimes artificially) as a measurement of some ‘object’ in some *context*. For instance, an object *q* may be a question, and the context may be defined by what other question *q*′ it is asked in combination with. Then the answer to this question is a random variable that can be interpreted as the measurement of *q* in the context (*q*,*q*′). If the same question *q* is then asked in combination with some other question *q*′′, then the measurement is a different random variable, . More generally, context in which *q* is measured is defined by the conditions *c* under which the measurement is made, yielding random variable . This notation (or one of numerous variants thereof) is called *contextual notation* for random variables: it codifies the idea that the identity of a measurement is defined both by what is measured and by the conditions under which it is measured [4–11].

Within each context, the measurements are made ‘together’, because of which they have an empirically defined *joint distribution*. Thus, in context (*q*,*q*′), we have two jointly distributed random variables and . We call the set of all random variables jointly recorded in a given context a *bunch* (of random variables, or of measurements). Two different bunches have no joint distribution, because there is no empirically defined way of coupling the values of one bunch with those of another. We say that they are *stochastically unrelated*. Thus, in
1.1
any component of *R*^{(q,q′)} is stochastically unrelated to any component of *R*^{(q,q′′)}, including and .

This work is based on the theory of contextuality dubbed contextuality-by-default (CbD) [9–17] (for precursors of this theory, see [6,7,11]). On a very general level, its main idea is that
*a system of different, stochastically unrelated bunches of random variables can be characterized by considering all possible ways in which they can be coupled under well-chosen constraint*s *imposed, for each object, on the relationship between the measurements of this object in different contexts*.

To *couple* different bunches simply means to impose a joint distribution on them. In the example above, this means finding four jointly distributed random variables (*A*,*B*,*X*,*Y*) such that, in reference to (1.1),
1.2
∼ standing for ‘is distributed as’. The quadruple (*A*,*B*,*X*,*Y*) is then called a *coupling* for the bunches *R*^{(q,q′)} and *R*^{(q,q′′)}. The ‘well-chosen constraints’ is a key notion in the formulation above. In our example, these constraints should apply to *A* and *X*, the coupling counterparts of and measuring (answering) the same question *q* in two different contexts.

Intuitively, ‘non-contextuality’ means ‘independence of context’, and because of this it is tempting to say that the system of two bunches in (1.1) is non-contextual if we can consider and as ‘one and the same’ random variable, *R*_{q}. This may appear simple, but in fact it is logically impossible: since and are stochastically unrelated, they cannot be ‘the same’. A random variable cannot be stochastically unrelated to itself. The precise meaning here comes from considering couplings (*A*,*B*,*X*,*Y*) for the two bunches. Clearly, in every such coupling and . We can say that the measurement of *q* in the system is context-independent if among all possible couplings (*A*,*B*,*X*,*Y*) there is at least one in which . In this particular example, due to its simplicity (only three random variables involved in two contexts), it can be shown that such a coupling does exist, provided . In a more complex system, such a coupling may not exist even if the system is *consistently connected*: which means that in this system the measurements of one and the same ‘object’ always have the same distribution.

The traditional approaches to contextuality were confined to consistent connectedness, but this condition is too restrictive in quantum physics [15,16,18] and virtually inapplicable in social and behavioural sciences: almost always, a response to question (or stimulus) *q* will depend on the context in which it is asked, which may translate into and having different distributions. There is nothing wrong in calling any such a case contextual, and this is done by many (see §§??sec3 and ??sec6). It is, however, more informative to separate inconsistent connectedness from contextuality, and this is what is done in the CbD theory. We use the term *inconsistently connected* for the systems that are not necessarily consistently connected (but may be so, as a special or limit case).

The logic of the CbD approach is as follows. We first consider separately the random variables measuring the same object in different contexts, in our example and . We call this set of random variables the *connection* (for the measured object, in our case *q*). Among all possible couplings (*A*′,*X*′) for the connection , i.e. among all jointly distributed (*A*′,*X*′) such that and , we find the minimal value *m*′ of . Then we look at the entire system of the bunches, in our case (1.1), and among all possible couplings (*A*,*B*,*X*,*Y*) for this system, we find the minimal value *m* for . It should be clear that *m*′ cannot exceed *m*, because in every coupling (*A*,*B*,*X*,*Y*) for (1.1), the part (*A*,*X*) forms a coupling for the connection . But they can be equal, *m*=*m*′, and then we say that the system is non-contextual. If *m*>*m*′, the system is contextual. Again, due to its simplicity, the system consisting of the two bunches (1.1) cannot be contextual, but this may very well be the case in more complex systems.

As an example of the latter, consider a system with two bunches
1.3
in which there are only two ‘objects’ *q*,*q*′, and the two contexts differ in the order in which these objects are measured. We have two connections here,
1.4
Let us assume the measurements are binary, with values +1 and −1 (e.g. corresponding to answers Yes and No), and let us further assume that all four random variables are ‘fair coins’, with equal probabilities of +1 and −1. Then the distributions of the bunches *R*^{(q,q′)} and *R*^{(q′,q)} in (1.3) are uniquely defined by the product expected values
1.5
It is easy to see that, across all possible couplings (*A*′,*X*′) for , the minimum value *m*′_{1} of is 0, and the same is true for the minimum value *m*′_{2} of across all possible couplings (*B*′,*Y* ′) for . However, it follows from the general theory that across all possible couplings (*A*,*B*,*X*,*Y*) for the entire system (1.3), the values *m*_{1} of and *m*_{2} of cannot be both zero unless
1.6
The latter need not be the case: it may, for example, very well be that (perfect correlation) and (perfect anti-correlation). In this case *m*_{1}+*m*_{2}≥1, whence either *m*_{1}>*m*′_{1}=0 or *m*_{2}>*m*′_{2}=0, indicating that the system is contextual.

As we show in this paper, the general rule for a broad spectrum of behavioural and social systems of measurements seems to be that *they are all non-contextual in the sense of CbD*.

## 2. Cyclic systems of arbitrary rank

In this section and throughout the rest of the paper, we assume that all our measurements are binary random variables, with values ±1.

We apply the logic of the CbD theory to systems in which all objects are measured in pairs so that each object belongs to precisely two pairs. We call such systems *cyclic*, because we can enumerate the objects in such a system *q*_{1},…,*q*_{n} and arrange them in a cycle
2.1

in which any two successive objects form a context. The number *n* is referred to as the *rank* of the system. Our last example in the previous section is a cyclic system of rank 2, the smallest possible.

In accordance with our notation, each object *q*_{i} in a cyclic system is measured by two random variables: and , where the operations ⊕ and ⊖ are cyclic addition and subtraction (so that *n*⊕1=1 and 1⊖1=*n*). Since there are no other random variables involved, we can simplify notation: we will denote , measuring the first object in the context, by *V* _{i}, and , measuring the second object in the context, by *W*_{i}. As a result, each bunch in a cyclic system has the form (*V* _{i},*W*_{i⊕1}); for example, the bunch of measurements for (*q*_{1},*q*_{2}) is (*V* _{1},*W*_{2}), for (*q*_{n},*q*_{1}) the bunch is (*V* _{n},*W*_{1}), etc.

Now we can represent a cyclic system of measurements in the form of a *V* –*W* cycle:
2.2

where solid lines indicate bunches (joint measurements) and point lines indicate connections (measurements of the object in different contexts).

It is proved in [12,15,16] that such a system is non-contextual if and only if its bunches satisfy the following inequality:
2.3
where 〈⋅〉 denotes expected value, and the *s*_{1}-part is the maximum of all linear combinations ±〈*V* _{1}*W*_{2}〉±…±〈*V* _{n−1}*W*_{n}〉±〈*V* _{n}*W*_{1}〉 with the proviso that the number of minuses is odd. Note that the criterion is written entirely in terms of the expectations of *V* _{i}, *W*_{i} and of the products *V* _{i},*W*_{i⊕1} (*i*=1,…,*n*). This means that the information about a cyclic system we need can be presented in the form of the diagram
2.4

We will use such diagrams to discuss experimental data in the subsequent sections.

This criterion of non-contextuality is generally breached by quantum-mechanical systems. Thus, for consistently connected systems, for *n*=3, the inequality reduces to Suppes–Zanotti–Leggett–Garg inequality [19,20], for *n*=4 it acquires the form of the Clauser–Horn–Shimony–Holt inequalities for the Einstein–Podolsky–Rosen–Bell paradigm [21–23], and for *n*=5 (with an additional constraint) it becomes what is known as Klyachko–Can–Binicioglu–Shumovsky inequality [24]. All of them are predicted by quantum theory and supported by experiments to be violated by some quantum-mechanical systems. For *n*=3, using the criterion (2.3), violations are also predicted for inconsistently connected systems [18]; and for *n*=5, violations of (2.3) were demonstrated experimentally [25] (as analysed in [16]).

By contrast, we find no violations of (2.3) in all known to us behavioural and social experiments aimed at revealing contextuality: Δ*C* never exceeds zero. In the subsequent sections, we demonstrate this ‘failure to fail’ the non-contextuality criterion for several experimental studies, for cyclic systems of rank 2, 3 and 4.

## 3. Question order effect (cyclic systems of rank 2)

Wang *et al*. [26] considered 73 polls in which two questions, *A* and *B* (playing the role of ‘objects’ *q*_{1},*q*_{2} being measured), were asked in two possible orders, and (forming two contexts). The possible answers to each question, random variables
3.1
were binary: +1 (Yes) or −1 (No). For instance, in the Gallup poll results used in [27], one pair of questions was (paraphrasing)
*A*: Do you think many white people dislike black people?*B*: Do you think many black people dislike white people?

with the resulting estimates of joint and marginal probabilities

We translate ‘Yes to *A*’ into *V* _{1}=1 in and into *W*_{1}=1 in ; correspondingly, ‘Yes to *B*’ translates into *W*_{2}=1 in and into *V* _{2}=1 in . Using the notation (2.4), we deal here with the system
To make sure the calculations are clear, for any ±1 random variables *X*,*Y* ,
The non-contextuality criterion (2.3) for cyclic systems of rank 2 specializes to the form
3.2
For the values in the diagram above, Δ*C*=−0.406, so there is no evidence that the system is contextual.

Reference [26] contains analysis of 73 such pairs of questions, including 66 taken from PEW polls (with *N* ranging from 125 to 927), four taken from Gallup polls reported by Moore [27] (with *N* about 500), and three pairs of questions with *N* ranging from 106 to 305. (The data were kindly provided to us by Wang *et al*. [26]; our computations based on these data are shown in electronic supplementary material, S1.)

The analysis is simplified if we accept the empirical regularity discovered by Wang & Busemeyer [28] and convincingly corroborated in [26]: using our notation, the discovery is that for the vast majority of question pairs, 3.3 while 3.4

The last inequality is what is traditionally called the question order effect [27], and (3.3) is dubbed by Wang and Busemeyer the *quantum question* (QQ) equality. Wang & Busemeyer [28] theoretically justify the QQ equality by positing that the process of answering two successive questions can be modelled by orthogonally projecting a state vector *ψ* twice in succession in a Hilbert space. Denoting the projectors corresponding to response Yes to the questions *A* and *B* by *P* and *Q*, respectively, we have *P*^{2}=*P*, *Q*^{2}=*Q*. The orthogonal projectors corresponding to response No to the same two questions are then *I*−*P* and *I*−*Q*, with *I* denoting the identity operator. We have, for the question order ,
and it is readily shown that
As *P* and *Q* enter in this expression symmetrically, the expression is precisely the same for
The empirical QQ effect now follows from the assumption that the operators *P*,*Q* do not vary across respondents (being determined by the questions alone), whereas the mixture of the initial states *ψ* has the same distribution in any two large groups of respondents. At the same time, the question order effect follows from the fact that ∥*QPψ*∥^{2} is not generally the same as ∥*PQψ*∥^{2}.

The QQ equality trivially implies (3.2), i.e. lack of contextuality. Therefore, to the extent that the QQ equality can be viewed as an empirical law (and [26] demonstrates this convincingly for 72 out of 73 question pairs), the criterion of non-contextuality should be satisfied for any 〈*V* _{1}〉,〈*W*_{1}〉,〈*V* _{2}〉,〈*W*_{2}〉. We can confirm and complement the statistical analysis presented in [26] of the 72 questions by pointing out that the overall *χ*^{2} test of the equality (3.3) over all of them yields *p*>0.35, d.f.=72.

The singled out pair of questions that violates the QQ equality is taken from the Gallup poll study reported in [27]: paraphrasing,
*A*: Should Pete Rose be admitted to the baseball hall of fame?*B*: Should shoeless Joe Jackson be admitted to the baseball hall of fame?

References [26,28] provide an explanation for why the double-projection model should not apply to this particular pair of questions, but we need not be concerned with it. The diagram of the results for this pair is
and it is readily seen to violate the equality 〈*V* _{1}*W*_{2}〉=〈*V* _{2}*W*_{1}〉 (*p*<10^{−7}, *χ*^{2} test with d.f.=1). At the same time, the diagram yields Δ*C*=−0.422, no evidence of contextuality. This example serves as a good demonstration for the fact that while the QQ equality is a sufficient condition for lack of contextuality, it is by no means necessary.

Considering the question pairs one by one, all but six Δ*C* values out of 73 are negative. In five of these six cases, the QQ equality |〈*V* _{1}*W*_{2}〉−〈*V* _{2}*W*_{1}〉|=0 cannot be rejected with *p*-values ranging from 0.06 to 0.47. Therefore, (3.2) cannot be rejected either. In the remaining case, *p*-value for the QQ equality is 0.008, and Δ*C*=0.063. While this case is suspicious, we do not think it warrants a special investigation: using conventional significance values, say, 0.01, for 73 similar cases, we get the probability of at least one rejection inflated to 0.52.

Note that in the literature cited, including [26,28], the term ‘contextual effect’ is used to designate the question order effect (3.4). This meaning of contextuality corresponds to what we call here inconsistent connectedness (or violations of marginal selectivity), and it should not be confused with the meaning of contextuality as defined in §§??sec1 and ??sec2 and indicated by the sign of Δ*C*.

## 4. Schröder's staircase illusion (a cyclic system of rank 3)

Asano *et al.* [29] studied a cyclic system of rank 3, using as ‘objects’ *q*_{1},*q*_{2},*q*_{3} Schröder's staircases tilted at three different angles, *θ*=40°,45°,50°, as shown in figure 1. In fact, these three angles formed the middle part of a set of 11 angles ranging from 0° to 90° and presented either in the descending order (context *c*_{1}), or in the ascending order (context *c*_{2}), or else in a random order (context *c*_{3}). Each context involved a separate set of about 50 participants, and each participant in response to each of 11 angles had to indicate whether she/he sees the surface A in front of B (+1) or B in front of A (−1). From these 11 responses, in each context, the authors selected two. In context *c*_{1}, the selected responses were those to *θ*=40°,45°, so, formally, *c*_{1} can be identified with (*q*_{1},*q*_{2}); in contexts *c*_{2} and *c*_{3} the selected responses were those to *θ*=45°,50° and to *θ*=50°,40°, respectively, making *c*_{2}=(*q*_{2},*q*_{3}) and *c*_{3}=(*q*_{3},*q*_{1}). It is irrelevant to the logic of the analysis that each context in fact contained all three tilts *q*_{1},*q*_{2},*q*_{3}, as well as eight other tilts. (Reference [29] includes a variety of other combinations of three objects and three contexts extracted from the experiment in question. The dataset for the combination described here was kindly made available to us by Asano *et al*. [29].)

The results of the experiment are shown in the diagram of expected values below: The criterion of non-contextuality for a rank 3 cyclic system has the form 4.1

where
The calculation shows Δ*C*=−1.233, no evidence for contextuality.

Search for contextuality is the specific goal of Asano *et al*. [29], but the meaning of the concept there is different from ours: there, it means violations of the Suppes–Zanotti–Leggett–Garg inequality (which is the consistently connected case of (4.1)), irrespective of whether these violations are due to inconsistent connectedness or due to contextuality in our sense.

## 5. Conjoint choices: animals and sounds they make (a cyclic system of rank 4)

Aerts *et al.* [30] present results of an experiment in which each of 81 participants had to choose between two animals and between two animal sounds, under four conditions *c*_{1},*c*_{2},*c*_{3},*c*_{4} (contexts), as shown below:

The ‘objects’ to be measured here are the choices offered:
Each of the four contexts corresponds to a pair of these objects,
and the choices made are binary measurements (random variables)
The table of the results above translates into the diagram of expected values
The non-contextuality criterion for rank 4 has the form
5.1
where
The value computed from the data is Δ*C*=−3.357, providing no evidence for contextuality.

Reference [30] reports that contextuality in this dataset is present because 5.2 i.e. the data violate the classical CHSH inequalities [22,23]. As pointed out in [31], the CHSH inequalities are predicated on the assumption of consistent connectedness (marginal selectivity). Without this assumption, they cannot be derived as a necessary or sufficient condition of non-contextuality, and this assumption is clearly violated in the data. Aerts [32] has developed a theory which allows for inconsistent connectedness, but it is unclear to us how this justifies the use of CHSH inequalties in [30].

## 6. Word combinations and priming (cyclic systems of rank 4)

Bruza *et al.* [33] studied ambiguous two-word combinations, such as ‘apple chip’. One can understand this word combination to refer to an edible chip made of apples or to an apple computer component. It is even possible to imagine such meanings as a piece chipped off of an apple computer, or a computer component made of apples. In the experiments referred to the participants were asked to explain how they understood the first and the second word in a combination: one meaning of each word (e.g. the fruit meaning for ‘apple’, the edible product meaning for ‘chip’, etc.) can be taken for +1, any other meaning being classified as −1. The meanings were determined by asking the participants to explain how they understood the words. For each two-word combination, the experimenters used one of four pairs of priming words presumably affecting the meanings. For the ‘apple chip’ combination, the priming words could be
forming four contexts
The order in which we list the words in a context is chosen to create a cycle: (*q*_{1},*q*_{2}),(*q*_{2},*q*_{3}), etc. Although this is not intuitive, formally, the measured ‘objects’ here are the priming words *q*_{1},*q*_{2},*q*_{3},*q*_{4}, while the measurements are binary random variables indicating in what meaning (±1) the participant understood ‘apple’ and ‘chip’. In (*V* _{1},*W*_{2}) and (*V* _{3},*W*_{4}), the *V* 's are meanings of ‘apple’ and *W*'s the meanings of ‘chip’; in (*V* _{2},*W*_{3}) and (*V* _{4},*W*_{1}), it is vice versa. (This is no more than a notational convention, purely for the purposes of using the cyclic indexation.)

Reference [33] presents data on 23 word combinations preceded by priming words (each combination in each context being shown to each of 61–65 participants). In all 23 cases, the computed values of Δ*C* are negative, ranging from −2.882 to −0.418 (for the ‘apple chip’ example the value is −1.640). We conclude, once again, that there is no evidence in favour of contextuality. (Bruza *et al*. [33] kindly provided to us the word pairs and priming words, with the computed values of *s*_{1} and equivalents of |〈*V* _{i}〉−〈*W*_{i}〉| (*i*=1,…,4), for all 23 word combinations; they are presented, with permission, in electronic supplementary material, S2, with the computation of Δ*C* added.)

The aim of Bruza *et al.* [33] was not to study contextuality. Rather they were interested in the property called *compositionality*, defined, in our terms, as consistent connectedness together with lack of contextuality. Violations of this condition therefore amount to either inconsistent connectedness or, if connectedness is consistent, to contextuality in our sense.

## 7. Psychophysical matching (cyclic systems of rank 4)

All experiments discussed so far use participants as replicants: the estimate of in a given context is the proportion of participants who responded (*v*,*w*), *v*=±1, *w*=±1. In the question order effect and Schröder's staircase illusion studies, different groups of people participated in different contexts, whereas the conjoint choices and word combinations studies employed repeated measures design: each participant made one choice in each of the four contexts.

In our laboratory, we searched for possible contextual effects in a large series of psychophysical experiments where each of very few (usually, three) participants was repeatedly ‘measuring’ the same four ‘objects’ in the same four contexts. In each of the seven experiments, the number of replications per participant was 1000–2000, evenly divided between different contexts.

The logic of an experiment was as follows. The participant was shown two stimuli, target one (*T*) and adjustable one (*A*), both completely specified by two parameters. In each trial, the values *α* and *β* of these parameters (real numbers) in the target stimulus *T*(*α*,*β*) are fixed at one of several values, each pair of values determining a context; in the adjustable stimulus, the two parameters can be simultaneously or (in some experiments) successively changed by the participant rotating a trackball. At the end of each trial, the participant reaches some values *X* and *Y* of these parameters that she/he judges to make *A*(*X*,*Y*) match (i.e. look the same as) *T*(*α*,*β*). In most experiments, *α* and *β* vary on several levels each (or even quasi-continuously within certain ranges), and we always choose four specific values or subranges of their values: *q*_{1},*q*_{3} for *α* and *q*_{2},*q*_{4} for *β*. They form four contexts that can be cyclically arranged as
and for each of them we get empirical distributions of *X* and *Y* : (*X*_{12},*Y* _{12}) for context (*q*_{1},*q*_{2}), (*X*_{41},*Y* _{41}) for context (*q*_{4},*q*_{1}), etc. In this notation, of the two objects *q*_{i},*q*_{j}, the random variable *X*_{ij} ‘measures’ the *q* with an odd index (1 or 3), whether *i* or *j*; analogously, *Y* _{ij} ‘measures’ the *q* with the even index.

The values of *X* and *Y* are then dichotomized in the following way: we choose a value *x*_{i} and a value *y*_{j} (*i*=1,3, *j*=2,4) and define
7.1
and
7.2
The values of (*x*_{1},*x*_{3},*y*_{2},*y*_{4}) can be chosen in a variety of ways, and for each choice, we apply to the obtained *V* and *W* variables the criterion (5.1).

As an example, in one of the experiments, the stimuli *T* and *A* were two dots in two circles, like the ones shown in figure 2*a*, with a dot's position within a circle described in polar coordinates (*α* and *X* denoting distance from the centre in pixels, *β* and *Y* denoting angle in degrees measured counterclockwise from the horizontal rightward radius-vector). We extract from this experiment a 2×2 subdesign as shown in figure 3. Then we choose a value of *x*_{1} as any integer (in pixels) between and , we choose *y*_{2} as any integer (in degrees) between and , and analogously for *x*_{3} and *y*_{4}. This yields 25×23×21×79 quadruples of (*x*_{1},*x*_{3},*y*_{2},*y*_{4}), and the corresponding number of cyclic systems of binary random variables (*V* _{1},*W*_{2},*V* _{2},*W*_{3},*V* _{3},*W*_{4},*V* _{4},*W*_{1}). Consider, for example, one such choice: (*x*_{1},*x*_{3},*y*_{2},*y*_{4})=(72 px,67 px,60°,23°). The diagram of this system is

and the value of Δ*C*=−2.137, no evidence of contextuality. In fact, negative values of Δ*C* are obtained for all 25×23×21×79 dichotomizations. Clearly, different dichotomizations of the same random variables are not stochastically independent, but there is no mathematical reason for Δ*C* to be of the same sign in all of them.

In electronic supplementary material, S3, we describe in detail how the dichotomizations were made, their number ranging from 3024 to 11 663 568 per 2×2 (sub)design in each experiment for each participant. The outcome is: not a single case with positive Δ*C* observed.

## 8. Conclusion

The empirical data analysed above suggest that the non-contextuality boundaries, which are generally breached in quantum physics, are not breached by behavioural and social systems. This may seem a disappointing conclusion for some. With the realization that quantum formalisms may be used to construct models in various areas outside physics [34–37], the expectation was created that human behaviour should exhibit contextuality, perhaps even on a greater scale than allowed by quantum theory. However, if the no-contextuality conclusion of this paper is proved to be a rule for a very broad class of behavioural and social systems, it is rather fortunate for behavioural and social sciences. Non-contextuality means more constrained behaviour, and constraints allow one to make predictions. The power of quantum mechanics is not in that quantum systems breach the classical-mechanical bounds of non-contextuality, but in the theory that imposes other, equally strict constraints. Presence of contextuality, in the absence of a general theory like quantum mechanics, translates into unpredictability.

It must be noted that absence of contextuality in behavioural and social systems does not mean that quantum formalisms are not applicable to them. A good argument for why this conclusion would be groundless is provided by the question order effect discussed in §??sec3: it is precisely the applicability of a quantum-mechanical model in the question order effect analysis [26,28] that allows one to predict the lack of contextuality in this paradigm.

When discussing contextuality, one should be aware of the likelihood of purely terminological confusions. It is clear that in the behavioural and social systems, a context generally influences the measurement of an object within it. For instance, the distribution of answers to a question depends on a question asked and answered before it. One could call this contextuality, and many do. This is, however, a trivial sense of contextuality, on a par with the fact that the distribution of answers to a question depends on what this question is. One should not be surprised that other factors (such as temperature in the laboratory or questions asked and answered previously) can influence this distribution too. We call this inconsistent connectedness, and we offer a theory that distinguishes this ubiquitous feature from contextuality in a different, one could argue more interesting meaning.

## Data accessibility

The computations discussed in §§3 and 6 are presented in electronic supplementary material, S1 and S2, respectively. The original datasets are available from Wang *et al*. [26] and Bruza *et al*. [33]. Details of the experiments discussed in §7 are presented in electronic supplementary material, S3; the datasets are available as ‘Contextuality in Psychophysical Matching’, http://dx.doi.org/10.7910/DVN/OJZKKP, Harvard Dataverse, V1.

## Authors' contributions

All authors contributed equally to writing the paper. The theory was developed primarily by E.N.D. and J.K. The experiments in §7 were conducted and analysed primarily by R.Z.

## Competing interests

We declare we have no competing interests.

## Funding

This research has been supported by NSF grant no. SES-1155956, AFOSR grant no. FA9550-14-1-0318 and A. von Humboldt Foundation.

## Acknowledgements

We are grateful to Wang *et al*. [26], Asano *et al*. [29] and Bruza *et al*. [33] for kindly providing datasets for our analysis. We have benefited from discussions with Jan- Åke Larsson and Victor H. Cervantes (who pointed out a mistake in an earlier version of the paper).

## Footnotes

One contribution of 14 to a theme issue ‘Quantum probability and the mathematical modelling of decision making’.

- Accepted August 25, 2015.

- © 2015 The Author(s)