Contribution
LIT-842 asked whether ambiguous phrases are contextual. It found corpus data always signalling, and it secured no contextual case. This paper turns to the signalling itself. It proves that CbD’s measure of signalling, Δ, is the least amount of direct causal influence of context that any of Jones’s canonical models of the data must contain. It then measures Δ, and how it splits between verb and noun, across 90 rank-2 noun–verb systems. It reports that the split, not the total, tracks whether a verb is homonymous or polysemous.
Key insight
Signalling is not noise to be subtracted before the interesting question. It is a measurable direct influence of context, and in a binary system it has an exact causal reading. Δ counts how often, at minimum, a hidden state that fixes all other influences would still have to give the word a different interpretation in the two contexts. On that reading, “how much does the partner word steer this word’s interpretation” has a number, and the number can be compared across kinds of word.
Assumptions
- Binary contents. Each word is restricted to two interpretations, labelled ±1, though some have more (§4). Proposition 1 needs the binary case: for ±1 variables, |⟨R⟩ − ⟨R′⟩| = 2|P(R = +1) − P(R′ = +1)|.
- Rank-2 cyclic systems. Two contents (verb, noun) and two contexts (verb–object, subject–verb). Each content occurs in exactly two contexts. The same verb and the same noun are treated as the same content in both grammatical roles.
- A context is the phrase, its grammatical structure, and “any extra information available, such as the corpus” (§4).
- Probabilities are relative frequencies of interpretations that the authors assigned by hand to BNC and ukWaC occurrences. No annotation protocol or agreement is reported. Each combination had to occur at least once in both orders. Systems whose uncertainty on Δ was “bigger than the range of possible values” were dropped; how err(Δ) was computed is not stated.
- Jones’s canonical models. Context C and a latent background Λ jointly determine each content F_q. Every system has such a model (Jones 2019, quoted).
- Classification by psycholinguistic lists. Verbs come from Pickering and Frisson (2001b) and Shutova (2010), nouns from Rayner and Duffy (1986) and Tanenhaus et al. (1979). Homonymous means “several meanings” and polysemous means “several senses”, following those sources.
Key results
- Eq. 1. Δ = Σ_i |⟨R_i^{j_i}⟩ − ⟨R_i^{j′_i}⟩|, and Δ = 0 iff the system is non-signalling.
- Eq. 3. The direct influence of switching context c to c′ on content q in a canonical model is Δ_{c,c′}(F_q) = P(Λ ∈ {λ : F_q(λ, c) ≠ F_q(λ, c′)}).
- Lemma 1 (via Dzhafarov and Kujala 2016, Theorem 3.3). Over couplings S, max P(S_q^c = S_q^{c′} = o) = min(P[R_q^c = o], P[R_q^{c′} = o]), and the maximum is attained.
- Proposition 2 (quoted from Jones 2019, Proposition 8.4). Canonical models and couplings correspond, with Δ_{c,c′}(F_q) = P(S_q^c ≠ S_q^{c′}). So the minimal direct influence is the minimal disagreement over couplings (Corollary 1).
- Proposition 1. For a binary cyclic system, Δ = 2 Σ_q Δ*(F_q), where Δ*(F_q) = 1 − Σ_{v∈{±1}} min(P[R_q^c = v], P[R_q^{c′} = v]) (Eq. 5). The proof is three lines given the above. Eq. 13 drops the absolute values that Eq. 12 carries. This is a slip, and the result is unaffected. In the worked example, Eq. 16 prints min(3/4, 3/4) where the data of Fig. 2 give min(3/4, 3/5); the stated value 3/20 is correct.
- Fig. 5a: Δ by class (mean ± standard error). Meanings-verb with meanings-noun 1.24 ± 0.29 (n = 14); meanings-verb with senses-noun 1.50 ± 0.43 (n = 4); senses-verb with meanings-noun 1.36 ± 0.15 (n = 61); senses-verb with senses-noun 1.38 ± 0.36 (n = 11); overall 1.35 ± 0.12 (n = 90). The class sizes are recounted here from the appendix; the paper does not print them. All the values reproduce from the appendix with the population standard deviation (ddof = 0).
- Fig. 5b: standard deviation of Δ. Meanings-verbs 1.04, senses-verbs 1.20, meanings-nouns 1.18, senses-nouns 1.11. All reproduce. The paper reads the larger values as more variable behaviour and cautions that the dataset is small. No test is given.
- §4: verb share 2Δ_v/Δ, over systems with Δ > 0. Meanings-verbs 0.70 (n = 14, from cast 4, admit 3, file 3, saw 2, plug 1 and tap 1), senses-verbs 0.48 (n = 55). Nouns: meanings-nouns 0.53 (n = 57), senses-nouns 0.47 (n = 12), both about 50%. The paper says the meanings-verb share is “strictly higher … with more than 95% confidence”, names no test, and shows 66% intervals in Fig. 6. Recomputed here: Welch t = 2.00, df ≈ 20, one-sided p ≈ 0.030, two-sided p ≈ 0.059; Mann–Whitney U = 511.5 of 770. Footnote 6 says setting the Δ = 0 systems to 50% changes nothing qualitative. Recomputed: 0.65 (n = 18) against 0.48 (n = 72).
- Footnote 4. A rank-2 model with Δ > 2 cannot be CbD-contextual. By the cyclic criterion of LIT-777, s_odd ≤ 2 in rank 2 and contextuality needs s_odd > Δ, so Δ ≥ 2 already suffices. 34 of the 90 systems are ruled out on Δ alone.
- §5. It recalls LIT-842’s contextual systems, adopt/boxer (1/30) and throw/pitcher (7/30); the extraction garbles them. Their rows here are Δ = 1.87 and 0.87, matching NOTE-645’s recomputation (56/30 and 13/15). “No contextual instances of rank-4 models have been found yet.”
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | In a binary cyclic system, Δ is twice the summed minimal direct influences over canonical models | strong (proof), given Jones’s Prop. 8.4 | Proposition 1, Lemma 1, Proposition 2 (quoted) |
| C2 | Mean Δ does not distinguish the four ambiguity classes | moderate | Fig. 5a; reproduced from Appendix A |
| C3 | Homonymous verbs carry a larger share of direct influence than polysemous verbs | weak | 14 against 55 systems, six homonymous verbs, unnamed test; p ≈ 0.06 two-sided on recomputation |
| C4 | The noun’s share does not depend on homonymy or polysemy | weak | Fig. 6b; a null on 12 senses-noun systems |
| C5 | Verbs on average have a larger direct influence than nouns (§1, §4) | not supported | the appendix gives an overall verb share of 0.52 ± 0.04 |
| C6 | The verb result agrees with Pickering and Frisson’s eye-tracking finding | informal argument | §4 discussion; no behavioural data here |
| C7 | Combinations with a polysemous verb or a homonymous noun vary more | weak | Fig. 5b standard deviations, untested |
Concepts
- content / context. A content is the interpretation of a word. A context is a phrase with its grammatical structure and its corpus.
- direct influence (Δ_{c,c′}(F_q)): in a canonical model, the probability over the background that switching context changes the content’s value. Minimal direct influence (Δ*): its least value over all canonical models compatible with the data.
- degree of signalling (Δ): the summed absolute differences of a content’s expectations across its contexts. The paper keeps the word “signalling” for what CbD calls inconsistent connectedness.
- meanings / senses: unrelated interpretations (homonymy, as in spring) versus related ones (polysemy, as in newspaper as object or as content). “Interpretation” covers both.
- verb share: 2Δ_v/Δ, the fraction of Δ due to the verb. By Proposition 1, verb and noun shares sum to 1.
Connections
The framework is CbD (LIT-777) and the analysis of LIT-264. Its systems are modelled on question-order effects (LIT-834). The causal side is Jones (2019), which builds on Cavalcanti (2018) and Pearl; neither is held here. It continues LIT-842. That paper used a few hand-chosen phrases to test contextuality. This one uses 90 systems built systematically from psycholinguistic word lists, and it measures signalling. The two share adopt/boxer and throw/pitcher, and their numbers agree. NOTE-645 summarises this paper from LIT-842’s side. This reading confirms its figures, and adds that the 95% claim is one-sided at best and that the overall verb-dominance claim does not survive the appendix. The psycholinguistic findings it appeals to (Frazier and Rayner 1990, Pickering and Frisson 2001b) are cited, not tested.
Bearing on the record
- It produces THEORY-177: in binary systems the CbD signalling measure is the minimal causal direct influence of context. The record’s CbD readings (LIT-777, NOTE-600) hold the maximal coupling, so they hold the total-variation half. They do not hold its identification with direct influence in a causal model.
- CLAIM-125 (transport between empirical models is open for signalling data). This paper does not address transport between scenarios, so it does not settle the claim’s second part. It bears on it in two ways. It shows that natural-language meaning-selection data are heavily signalling: mean Δ is 1.35 out of a possible 4, and about half of these rank-2 systems are signalling enough (Δ ≥ 2) to be noncontextual automatically. So the signalling case is the typical case for language, not an edge case. It also gives a quantity a transport could be asked to carry or bound: per-content minimal direct influence, which is a total-variation distance with a causal reading. Whether simulations between scenarios can increase it is not asked.
- CLAIM-037 (formal contextuality needs CbD when marginals shift). Consistent and reinforced: 69 of the 90 systems are signalling, and 18 of the 21 with Δ = 0 have err(Δ) of 1 or more, so even those are not evidence of non-signalling.
- THEORY-013. No bearing on its verdict. No system is tested for contextuality, and the two contextual ones are LIT-842’s, already weighed in NOTE-645.
- THEORY-168 (CbD verdicts depend on representation). It is illustrated in passing. Restricting each word to two interpretations is a dichotomisation choice, and the paper says the ±1 labelling does not affect Δ or contextuality. Which two interpretations are kept is not discussed.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Small and degenerate counts. Counts per system are not reported. 51 of 90 systems have Δ exactly 0, 2 or 4, and 18 have err(Δ) = 1.41. Both patterns suggest one or a few occurrences per context. Eight of the 14 homonymous-verb shares are exactly 0, 0.5 or 1.
- Non-independent units. Systems share verbs and nouns: six homonymous verbs supply the 14 systems behind the 70%. The test, if there is one, treats them as independent.
- Unnamed test, one-sided at best. “More than 95% confidence” holds one-sided under a Welch test on the appendix and fails two-sided. There is no correction for the several comparisons made (verbs, nouns, four classes, standard deviations).
- Annotation. The authors labelled the interpretations by hand, with no agreement measure. The corpora are British.
- A context includes the corpus, so pooling BNC and ukWaC occurrences is a modelling choice the paper does not examine.
- Psycholinguistic interpretation is by analogy. Δ is a corpus statistic, not a processing measure.
- No contextuality results, despite the title.
- Version. Read from an author copy whose provenance was not confirmed. The version of record was not compared.
Open questions
- Whether the homonymous/polysemous verb difference survives with reported counts, a mixed model with verb and noun as random effects, and more homonymous verbs.
- Whether minimal direct influence, as a per-content total-variation distance, is monotone under the simulations of THEORY-174. If it were, that would be a first step on CLAIM-125’s signalling part.
- Whether a human-judgement version (proposed in §5) gives the same Δ levels as corpus frequencies.
- Whether Proposition 1 has a counterpart for non-binary contents, where Δ as defined is not available but Δ* (Eq. 5) is (footnote 3).