Contribution
Behavioural “contextuality” had been claimed from order effects, priming and violations of CHSH-type inequalities computed as if each measurement had one distribution in every context. The paper applies a criterion that holds when distributions do vary with context, the Contextuality-by-Default cyclic-system criterion, to five kinds of behavioural data, two of them previously published as contextual. None is contextual. It also shows that the quantum question-order model predicts the absence of contextuality.
Key insight
Two things are called “context effects” and only one of them is contextuality. An answer’s distribution depending on what came before is inconsistent connectedness: ordinary, ubiquitous and, in the authors’ words, “a trivial sense of contextuality”. Contextuality is the further fact that no joint model can make each question’s answers across contexts as similar as their own distributions allow. Behavioural data have plenty of the first and, so far, none of the second.
Assumptions
- Binary measurements, ±1, arranged as a cyclic system: each object is measured in exactly two contexts, and successive objects form the contexts (§2).
- Bunches from different contexts are stochastically unrelated. A coupling imposes a joint distribution on them. Noncontextuality is the existence of a coupling in which each connection attains its own maximal coincidence probability.
- Participants as replicants in §§3–6: probabilities are proportions across people, between-subjects (§§3–4) or within-subjects (§§5–6). §7 uses repeated trials from very few participants.
- The criterion is quoted: Eq. 7 is proved in refs. [14, 16, 17].
Key results
- Eq. 7: noncontextual iff ΔC = s₁(⟨V₁W₂⟩, …, ⟨VₙW₁⟩) − (n − 2) − Σᵢ|⟨Vᵢ⟩ − ⟨Wᵢ⟩| ≤ 0, s₁ being the maximum over sign patterns with an odd number of minuses. For consistent marginals and n = 3, 4, 5 this is Suppes–Zanotti–Leggett–Garg, CHSH and (with a constraint) KCBS.
- Rank 2, question order (Eq. 10): ΔC = |⟨V₁W₂⟩ − ⟨V₂W₁⟩| − (|⟨V₁⟩ − ⟨W₁⟩| + |⟨V₂⟩ − ⟨W₂⟩|). The white–black poll gives ΔC = −0.406; I reproduced this from Wang & Busemeyer’s Table 1.
- The QQ equality from the model (§3): for projectors P, Q, (1 + ⟨V₁W₂⟩)/2 = ⟨(P Q P + (I − P)(I − Q)(I − P))ψ|ψ⟩, and P Q P + (I − P)(I − Q)(I − P) = I − (P + Q) + (P Q + Q P) is symmetric in P and Q, so ⟨V₁W₂⟩ = ⟨V₂W₁⟩ for any state. The empirical equality then needs the same mixture of states in the two groups. QQ ⇒ ΔC ≤ 0.
- S1: 73 pairs (66 Pew, four Gallup from Moore, three others). ΔC > 0 in six Pew pairs, all ≤ 0.063. In five the QQ equality survives (p 0.06–0.47); in the sixth p = 0.008 and ΔC = 0.063, discounted for multiplicity (P(at least one rejection at 0.01 in 73) ≈ 0.52). Pooled χ² for QQ over the 72 non-Rose–Jackson pairs: the text says p > 0.35; the S1 values sum to 76.04 on 72 df, p ≈ 0.350.
- Rose–Jackson violates QQ (p < 10⁻⁷) yet is noncontextual: QQ is sufficient, not necessary.
- §4, Schröder staircases (Asano et al.), rank 3: ΔC = −1.233.
- §5, animals and sounds (Aerts, Gabora & Sozzo), rank 4: ΔC = −3.357. The published CHSH violation (s₁ − 2 > 0) presupposes consistent connectedness, which the data violate.
- §6, Bruza et al.’s primed ambiguous word pairs, rank 4: all 23 ΔC negative, −2.882 to −0.418 (“apple chip” −1.640). I recomputed all 23 from S2’s s₁ and marginal differences; they agree. Bruza et al.’s “compositionality” is, in these terms, consistent connectedness plus noncontextuality.
- §7 and S3, psychophysical matching, seven experiments: between 3,024 and 11,663,568 dichotomisations per 2 × 2 design, and not one positive ΔC.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A binary cyclic system is noncontextual iff ΔC ≤ 0 | strong, but proved elsewhere | Eq. 7, refs. [14, 16, 17] |
| C2 | The QQ equality implies noncontextuality for question-order systems | strong (proof) | §3 |
| C3 | Question-order data (73 pairs) show no contextuality | strong for these data | §3, S1 |
| C4 | The CHSH violation reported for conjoint choices is an artefact of assuming equal marginals | strong | §5 |
| C5 | Primed word combinations and psychophysical matching show no contextuality | strong for these data | §§6–7, S2, S3 |
| C6 | Behavioural and social systems are noncontextual in general | weak: a working hypothesis from five paradigms | §8 |
Concepts
- bunch: the measurements recorded jointly in one context.
- connection: the measurements of one object across contexts.
- consistent connectedness: every connection’s measurements have the same distribution (marginal selectivity). Inconsistent connectedness is its failure, and the paper’s name for most “context effects”.
- contextuality (CbD): no coupling of the bunches attains, on every connection, that connection’s own maximal coincidence probability.
- cyclic system of rank n: n objects, n contexts, each object in two consecutive contexts.
Connections
The question-order data and model are Wang & Busemeyer’s (LIT-834) and Wang, Solloway, Shiffrin & Busemeyer’s (LIT-795); the data were supplied by the authors. The general theory is Contextuality-by-Default (LIT-777 in this record, read 2026-10-09). The same criterion applied to physics is reviewed in LIT-263.
Bearing on the record
- THEORY-013 is sourced here, and the full reading supports what it states. Two refinements. First, its “73 poll question-order pairs” includes three non-poll studies (S1 rows 70–72; two are laboratory experiments). Second, “apart from one poll pair the authors discount for multiple comparisons” is right: that pair is a Pew pair with ΔC = 0.063. Its first promote_when condition, a close reading of the proof of the criterion, is not met by this reading: the proof is not in this paper.
- THEORY-166 takes from this paper the re-derivation of the QQ equality and its implication of noncontextuality.
- Corrections to the skim of 2026-09-27. The skim said “72 of 73 pairs fit the QQ equality”. The paper says so, but by its own S1 flags the QQ equality is rejected at 0.05 for two pairs besides Rose–Jackson (a Pew pair at p = 0.008 and one of the three non-Gallup, non-Pew studies at χ² = 5.48, p ≈ 0.02). That is about what 72 tests produce by chance, and the pooled test holds.
- An error in §3, not affecting the conclusion. The Rose–Jackson diagram swaps ⟨W₁⟩ and ⟨V₂⟩, and the two product expectations, against the variable definitions of Eq. 9. Recomputed from Wang & Busemeyer’s Table 1: ⟨V₁⟩ = .324, ⟨W₂⟩ = −.289, ⟨V₂⟩ = −.035, ⟨W₁⟩ = .078, ⟨V₁W₂⟩ = .316, ⟨V₂W₁⟩ = .619, so ΔC = −0.197, not −0.422. S1’s row for the same pair lists different marginals again and ΔC = −0.851. All three are negative, so the pair is noncontextual on any reading.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- The criterion’s proof is cited, not given.
- Five paradigms, mostly re-analyses of others’ data; the general conclusion is offered as a working hypothesis.
- §§3–4 compare different groups of people across contexts, so the “random variables” are population proportions, not one system measured repeatedly.
- Only cyclic systems with binary outcomes are treated.
Open questions
- Is there any behavioural paradigm with ΔC robustly above zero? Later Contextuality-by-Default work (e.g. Cervantes & Dzhafarov 2018) is reported to find one; not held or read here.
- Where does the Rose–Jackson row of S1 come from? It matches neither the §3 diagram nor the published proportions.
Note of 2026-10-10: a documented case
CASE-053, found by a search the owner asked for, reads this paper’s null beside the later tests of contextuality on elicited human judgements. It supplies a scope for the “Key insight” sentence “Behavioural data have plenty of the first and, so far, none of the second”. That “none” is out of date for behavioural data in general: Cervantes and Dzhafarov (2018, doi:10.1037/dec0000095) report CbD-contextuality in an instructed choice task, with a 99.99% bootstrap interval excluding zero, and Basieva et al. (2019) in five of six more, though Yearsley and Halliwell (2019, arXiv:1905.12570) dispute that direct influences are excluded. This paper’s open question about Cervantes and Dzhafarov is answered: yes, as reported, and disputed. The sentence should be read as “none in pragmatic or meaning data adequately tested”. On that scope it still holds: Wang’s thesis (2024) finds no Bell-type contextual model in human meaning judgements, and Lo, Sadrzadeh and Mansfield’s Winograd schemas (2023), the one pragmatic test, violate by 0.192 ± 0.176. The case does not change this reading’s results, which concern the paper’s own data sets, and it does not show that pragmatic judgements are noncontextual: no adequately powered pragmatic test has been run. CLAIM-201’s concession rests on this sentence and survives the rescoping.