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TERM-042: communicative object, as a context-indexed system of empirical constraints over a cover

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Properties6
StatusActive
Tagscontextuality, philosophy-of-language
Linepragmatic-transport
Used byCLAIM-038 — A communicative object is a context-indexed system of empirical constraints over a cover of contexts, and these need not extend to one globally realizable interpretation, 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-133 — Communicative identity is relational continuity under admissible transformation: translation transports relationally organized observational structure, correspondence fixes what can be compared, admissibility which transformations count, and fidelity which communicatively consequential distinctions remain recoverable, CLAIM-247 — The bearer of communicative identity reversed, from an operational object defined on observed responses to a kind that can outrun observation, and no turn argues the change; both stances stand in the record
SupersedesTERM-002 — communicative object, as a compatible family of local sections
Cited as “uses” byARG-029 — The operational account, kept as a rule of method, supports a realist kind that can outrun observation as the bearer of communicative identity, CLAIM-042 — The levels at which communicative observables exist organize observables, not objects: a communicative object is identified jointly by constraints from observables at several levels, which, where those observables share one state space, is the intersection of the regions they pick out

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Definition

A129 §2: “A communicative observational object is a context-indexed system of empirically admissible local assignments or distributions, equipped with restriction relationships. Its global realizability is a property to be investigated, not part of its definition.”

A129 keeps three structures apart on an observational scenario (X, M, O):

  • the event sheaf ℰ, “All mathematically possible local assignments and their restrictions”;
  • the supports S_C ⊆ ℰ(C), “Which assignments are permitted in each context”;
  • the distributions e_C ∈ D(ℰ(C)), “How probability is allocated among local assignments”.

The object is the family of supports or distributions, one per context, with their restrictions. “Then a compatible family of local sections describes a realization of the observational object.” When the e_C agree on overlaps, the family is an empirical model in Abramsky and Brandenburger’s sense (THEORY-012, LIT-016). Whether they agree is part of what the data show. Whether the family has a global extension is the open question.

Why TERM-002 is superseded

The event presheaf E(U) = ∏_(x∈U) O_x is a sheaf, and the cover covers X. So any family of sections that agrees on overlaps glues uniquely to an assignment on X (CLAIM-131). TERM-002’s “compatible family of local sections … with no global assignment presupposed” therefore says nothing at the level of sections. A125 §2 found this: “In the ordinary event sheaf, a compatible family of local sections already glues to a global section.” The owner answered at U41, “this is good stuff, double click on this”, and A129 §2 made the correction. It is restated at A151 §2, A173, A178 §4.2, A203 §11 and A218 §6.

The obstruction lives one level up, and it has three grades (THEORY-012):

  • Possibilistic (logical). Some locally admissible section belongs to no compatible family of support sections. A compatible family of support sections still glues, because the supports form a subpresheaf of a sheaf.
  • Strong. No compatible family of support sections exists at all: no global assignment satisfies every support. This is what A129 calls the “possibilistic extension” question. It tests only the strongest grade.
  • Probabilistic. A compatible family of distributions has no nonnegative global distribution whose marginals are the e_C.

Only at the distribution level does a compatible family fail to glue. The manuscript §3 sentence quoted in TERM-002 had this right: the question arises “when empirical supports or distributions rule out the resulting global assignment”. The error was in A78’s prose and in TERM-002’s title.

What it is not

  • Not a compatible family of sections of ℰ, which is already a global assignment (CLAIM-131).
  • Not TERM-014’s observational equivalence class, which needs a shared space of situations. The two coincide when a global realization exists.
  • Not restricted to no-signalling data. The definition does not require the e_C to agree on overlaps. Language data signal (CLAIM-038, LIT-842), and for them Contextuality-by-Default applies (LIT-777). A129 does not say this. Every contextuality measure the record reads, the contextual fraction (LIT-265) among them, needs no-signalling.
  • Not one presheaf. A129 leaves the object as “assignments or distributions”. The support presheaf and the distribution presheaf D_R∘ℰ are different objects, and a claim stated in this term should say which it means.

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