Ehtibar N. Dzhafarov, Víctor H. Cervantes and Janne V. Kujala (2017), Philosophical Transactions of the Royal Society A 375(2106), 20160389 — ARXIV-1703.01252
Key takeaways
- Canonical representation (Section 3). Contextuality is decided on a system in which (A) every variable is binary and, after multimaximality, (B) connections are handled pairwise. A k-valued variable is replaced by its k one-value “detector” splits, plus the splits of any coarsening one cares about. Joint variables are formed by joining when within-context dependencies must count (Example 3.1). Which transformations to include “reflects what aspects of the empirical situation one is interested in”. The canonical system is unique given that choice.
- General CbD (Section 2). Noncontextuality is relative to a chosen set T of connection couplings (Definition 2.1). The degree of contextuality is min‖X‖ − 1 over quasi-couplings agreeing with T (Theorems 2.4–2.5, from LIT-777).
- One pair, all splits (Section 4). For R_1^1, R_1^2 with values 1…k and masses p_i, q_i, keep all 2^{k−1} − 1 splits. Then only the 1- and 2-splits matter (Theorems 4.1, 4.3). A maximally connected coupling of the 1-2 system is unique if it exists, and is supported on the diagonal plus one row or one column (Theorem 4.4). It exists if and only if p_i > q_i for at most one i, or p_i < q_i for at most one i (Corollary 4.5). The whole system is noncontextual exactly then (Theorem 4.6: nominal dominance).
- Consequences the authors draw (Section 5). Nominal dominance is “a stringent necessary condition for noncontextuality, likely to be violated in many empirical systems”. Until then, almost all dichotomous behavioural systems examined had been noncontextual. Canonical representations of multiple-choice responses “offer new possibilities”. For continuous densities, all splits make the pair contextual unless f = g. Restricting splits to intervals or cuts of an ordered scale is left as future work.
Standing in the record
Filed on 2026-10-09 from the manuscript bibliography of 2026-10-09 (work
what-survives-translation) at the owner’s request: a work the manuscript
considered and dropped from its final reference list. See the curation entry
of that day.
Read on 2026-10-09 (NOTE-640). It is the source of THEORY-168, which states that CbD’s verdict depends on the chosen representation. It bears on THEORY-013’s scope: that theory’s data are binary, and this paper’s verdict for multi-valued responses can differ sharply.