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CLAIM-037: Formal contextuality is a failure of global extension, distinct from ordinary context dependence, and needs Contextuality-by-Default when marginals shift with context

4 min read

Properties13
StatusProposed
Version4
Rolethesis
Warrantinterpretive
Tagscontextuality, philosophy-of-language
Linepragmatic-transport, formal-apparatus
Workswhat-survives-translation
Rests onCLAIM-374 — On a consistently connected system, Contextuality-by-Default and the sheaf criterion agree: a maximally connected coupling exists exactly when a global law has the contexts' laws as its marginals
GroundsTHEORY-012 — An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality, LIT-016 — The sheaf-theoretic structure of non-locality and contextuality, LIT-777 — Context–content systems of random variables: The Contextuality-by-Default theory, LIT-831 — Contextuality-by-Default 2.0: Systems with Binary Random Variables, LIT-842 — On the Quantum-like Contextuality of Ambiguous Phrases, LIT-844 — Quantum-Like Contextuality in Large Language Models, THEORY-014 — In both formalisms of contextuality, classicality is the existence of a nonnegative version of a linear representation that always exists over the reals, so negativity is universal and only its unavoidability is nonclassical, THEORY-020 — Every Boolean algebra is the clopen algebra of its Stone space of prime ideals, and a finite one is the power set of its atoms, so a single Boolean context always has two-valued valuations and the Kochen–Specker obstruction lies in how contexts overlap
ComplementsCLAIM-041 — Context-sensitive interpretation, order effects and noncommuting framing operations do not by themselves establish nonclassical probability or quantum contextuality, CLAIM-044 — Ambiguity and contextuality are different phenomena: an ambiguous utterance admits several compatible global assignments, a contextual one admits none, CLAIM-131 — A compatible family of sections of the event sheaf always glues, so where a communicative object lacks a global realization the obstruction lies in its supports or distributions, not in the event sheaf, CLAIM-141 — Causal-loop consistency and sheaf contextuality are both local-to-global problems, but passing to distributions affects them oppositely: a loop with no deterministic solution has a distributional fixed point, while a model with no global section in its supports has no global distribution either, CLAIM-143 — Local descriptions that disagree are not thereby incompatible: whether there is an obstruction can be assessed only relative to a specified class of admissible correspondences between their observations, independently validated and not admitted because they remove the disagreement, CLAIM-310 — Sorting schemes over objects with determinate properties never conflict as descriptions, since the product of any sortings refines them all; they conflict as arrangements, since a fixed number of bins realizes one partition when each item is stored once, and which costs least depends on what is searched for and on what stands out within a bin
Illustrated byCASE-035 — Solidarity, condemnation and humour: the parity obstruction, CASE-037 — The parity triangle with uniform weights: a no-signalling, strongly contextual model beside a family of point sections that glues, CASE-095 — Two builders sorting one collection: the brick that one bins as red and two-wide and the other as a curved connector has a bin that respects both, and what the builders cannot share is an arrangement, not a description
Cited as “concludes” byARG-022 — The 2016 Contextuality-by-Default paper grounds the criterion the manuscript's general observational covers need
Rendered intodocs/arguments-by-conclusion.md

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The claim

An observational scenario is a cover of observables by jointly accessible contexts; an empirical model is noncontextual when a global law has the contexts’ laws as marginals (Abramsky and Brandenburger, LIT-016). The manuscript insists that this is not every context effect: asking two questions together may shift one’s marginal, and then Contextuality-by-Default (Dzhafarov and Kujala, LIT-777) is the right tool. Order effects (AB ≠ BA) occur in classical Markov systems and establish no contextuality (§9).

What it does not say

It does not claim pragmatic observables are contextual; the parity example (solidarity, condemnation, humour) is offered as an obstruction pattern, not a finding about judgements. NOTE-600’s reader adds that for consistently connected systems the two criteria coincide, and that this is the reader’s inference, not the paper’s.

Language data are signalling

In Wang et al. (LIT-842), every model estimated from the corpora is signalling. The sheaf criterion could not be applied to any of them, and only Contextuality-by-Default could. That is direct support for this claim.

Abramsky, Barbosa and Searle (LIT-843) offer a second route. In their approach, signalling from a declared causal past becomes part of the classical model.

The BERT study, again

In Lo et al. (LIT-844) every instance signals unless all its probabilities are ½. The two criteria that allow for signalling then diverge widely: 71.1% of instances are contextual by Contextuality-by-Default, 0.148% by the sheaf criterion. That supports this claim.

Note of 2026-10-12: evidence now linked

Two THEORYs now ground this claim:

  • THEORY-014: in Contextuality-by-Default a signed maximally connected quasi-coupling always exists, signalling or not, and noncontextuality is the existence of a nonnegative one. This sharpens “global extension” here to mean nonnegative extension.
  • THEORY-020: the Kochen–Specker failure cannot sit in any one context, only in the agreement where contexts overlap. It is the deterministic case of THEORY-012, which already grounds this claim.

Note of 2026-10-19: split under ADR-043

That the two criteria coincide on consistently connected systems is now CLAIM-374. It is proved there, for connections of any size, and it is Active. The defeat condition’s second disjunct, a consistently connected system on which they disagree, now belongs to that claim, so it is removed here. This claim keeps the thesis about systems whose marginals shift. It is supported by LIT-842 and LIT-844, and contested by LIT-843’s causal-past route, which folds signalling into the classical model instead. Per CLAIM-069, the version of Contextuality-by-Default it should name is the one with multimaximal couplings, for observables in more than two contexts. The thesis is interpretive until the record compares the two routes on a signalling data set, or concedes “needs” to “can handle”.

Note of 2026-10-29: its source argument is Rejected

ARG-022, which held that the 2016 Contextuality-by-Default paper (LIT-777) grounds the criterion the manuscript’s general observational covers need, is Rejected: THEORY-168’s note of this date works a binary system with a content in four contexts on which the 2016 criterion and CbD 2.0 disagree. For contents in more than two contexts, the ground this claim’s “needs Contextuality-by-Default” should cite is CbD 2.0, Dzhafarov and Kujala’s multimaximal criterion (LIT-831, read in NOTE-643), with THEORY-722 and THEORY-168 for how the verdict depends on the representation. LIT-777 still grounds it for cyclic and consistently connected systems, where the two versions agree. Moving the ground is left to the next pass on this claim.

Note of 2026-10-29: what holds it back

“Needs Contextuality-by-Default” is contested by two sheaf routes the record holds that allow for signalling: LIT-843’s causal past and LIT-844’s signalling-corrected test. LIT-844’s data show the two verdicts diverging (0.148% against 71.1%) without showing which is right. It goes Active on a comparison of a sheaf route with CbD 2.0 on one signalling data set.

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