Ehtibar N. Dzhafarov, Ru Zhang, Janne Kujala (2016), Philosophical Transactions of the Royal Society A 374(2058):20150099 (arXiv v1 28 April 2015) — ARXIV-1504.07422, DOI-10.1098/rsta.2015.0099
Key takeaways
- The distinction. Each measurement is indexed by what it measures and by its context, and measurements in different contexts have no joint distribution. A system is noncontextual if some coupling of all contexts makes each object’s measurements coincide as often as their own marginals allow (m = m′). If an answer’s distribution merely depends on the context, that is “inconsistent connectedness”, and the authors call reading it as contextuality “a trivial sense of contextuality”.
- The criterion (Eq. 7, proved in the cited papers, not here): a binary cyclic system of rank n is noncontextual iff ΔC = s₁(⟨V₁W₂⟩, …, ⟨VₙW₁⟩) − (n − 2) − Σ|⟨Vᵢ⟩ − ⟨Wᵢ⟩| ≤ 0. With consistent marginals it is Leggett–Garg (n = 3), CHSH (n = 4) and KCBS (n = 5), all violated by some quantum systems.
- Question order (§3, S1). The QQ equality ⟨V₁W₂⟩ = ⟨V₂W₁⟩ makes ΔC ≤ 0 automatic. Of 73 pairs from Wang et al.’s collection, 67 have ΔC < 0. The six positive values are all small (≤ 0.063), and in five of them the QQ equality cannot be rejected. Over the 72 pairs other than Rose–Jackson, the pooled χ² test of the QQ equality gives p ≈ 0.35 (72 df). Rose–Jackson violates QQ (p < 10⁻⁷) and is still noncontextual, so QQ is sufficient for noncontextuality, not necessary.
- The other data (§§4–7). Schröder staircases: ΔC = −1.233. Aerts, Gabora and Sozzo’s animal–sound choices: ΔC = −3.357; the published CHSH “violation” assumed equal marginals, which the data do not have. Bruza et al.’s 23 primed word combinations: ΔC from −2.882 to −0.418 (rechecked here from S2). Psychophysical matching: no positive ΔC in any of between 3,024 and 11,663,568 dichotomisations per design.
- The conclusion is a working hypothesis: behavioural and social systems are noncontextual. The authors add that this does not make quantum formalisms inapplicable, since the quantum question-order model is what predicts noncontextuality here.
Standing in the record
Filed on 2026-09-27 while bringing the record’s contextuality threads
together at the owner’s request (see the curation entry of that date),
and filed Deferred because nobody had yet read it closely.
On 2026-10-09 the owner asked for it again, from the bibliography of the
manuscript what-survives-translation. It was already held, so it was
read rather than duplicated: NOTE-235, upgraded from Skimmed to Read. It
is the source of THEORY-013 and one of the sources of THEORY-166. The
two question-order papers it re-analyses are now held and read:
LIT-834 (the QQ model) and LIT-795 (the 72-study test). The
same criterion applied in physics is in LIT-263.
What a further reading should check:
- The proof of Eq. 7 (Kujala, Dzhafarov & Larsson 2015, PRL 115, 150401; Kujala & Dzhafarov, arXiv:1503.02181), which this paper cites and does not give.
- Later Contextuality-by-Default work (e.g. Cervantes & Dzhafarov 2018, human choice experiments) is reported to find contextuality in behaviour. It is not held or read here.