Zheng Wang, Tyler Solloway, Richard M. Shiffrin, Jerome R. Busemeyer (2014), Proceedings of the National Academy of Sciences 111(26):9431–9436 — DOI-10.1073/pnas.1407756111
Key takeaways
- The regularity. In a table of context effects (the cell-by-cell difference between the two orders’ answer proportions), the two cells of each diagonal sum to approximately zero: q = [p(AyBy) + p(AnBn)] − [p(ByAy) + p(BnAn)] ≈ 0. The number of people who switch from yes–yes to no–no in one order is offset by those who switch the other way. Nothing in probability forces it: the set of possible context-effect tables is a three-dimensional pyramid, and the QQ equality picks out a triangular plane inside it.
- The data. There are 72 studies: 66 Pew Research Center surveys from 2001–2011, which are all the Pew surveys of that decade that varied the order of two questions; three Gallup polls from Moore (2002); Schuman, Presser and Ludwig’s 1981 abortion study; and two laboratory experiments. The Rose–Jackson poll is excluded because the model predicts it should fail. Paired context effects fall along the a priori line of slope −1 (r = −0.82; −0.73 without two extremes), and the ratio of q to the size of the order effect falls towards zero as order effects grow (17 studies with order effect > 0.10).
- The test that carries the weight is on the 66 Pew surveys alone, chosen without regard to their outcome: the distribution of χ² values for order effects departs from the null (p = 0.0004), and the distribution for q does not (p = 0.4625).
- The model and its condition. The prediction is the Lüders double projection of LIT-834, and it holds for mixed states, so it survives individual differences. It requires that the questions be asked back to back. New information between them applies different transformations, P_A U″ P_B versus P_B U′ P_A, and the equality is no longer expected, which is the Rose–Jackson case (q = 0.1514, χ²(1) = 28.57).
- What the authors concede. “It is possible to construct a model that is narrowly constrained to satisfy the QQ equality, but these constraints could also prevent the model from accounting for order effects”, with two examples in the supporting information, which was not read here. They ask for alternative accounts. They say the brain need not be quantum: quantum probability may describe reasoning even if neural processes are classical.
Standing in the record
Filed on 2026-10-09 at the owner’s request, from the bibliography of the
owner’s manuscript (work what-survives-translation) as it stood on
2026-10-08: a work the bibliography considered and the final reference
list dropped. See the curation entry of that day.
Read on 2026-10-09 (NOTE-604). It is the primary source of THEORY-166. The model it tests is LIT-834. Its data set, supplied by the authors, is re-analysed under Contextuality-by-Default in LIT-264, which finds the equality implies noncontextuality. The paper cites LIT-316 for proofs; that book is unread here.