@cite{geurts-pouscoulous-2009} — Embedded Implicatures?!?

@cite{geurts-pouscoulous-2009}

Geurts, B. & Pouscoulous, N. (2009). Embedded implicatures?!? Semantics & Pragmatics, 2(4), 1–34. https://doi.org/10.3765/sp.2.4

Two threads

The paper makes two interlocking arguments, both formalized here:

1. Empirical: four experiments show that mainstream conventionalist theories of scalar implicature (@cite{landman-1998}, @cite{levinson-2000}, @cite{recanati-2003}, @cite{chierchia-2004}, @cite{chierchia-2006}, @cite{fox-2007}, @cite{chierchia-fox-spector-2008}) — and even a minimal conventionalism that only claims local-SI readings exist — fail to predict the patterns observed. The embedded-implicature problem is traced back to @cite{cohen-1971}; @cite{landman-1998} is credited in the paper's footnote 2 with first noting the belief-report case.
2. Methodological: introspection systematically inflates SI rates (Worries §2 + Exp 2 + Exp 3); "subtle intuitions are best classified as ipso facto suspect" (page 4:18). The verification paradigm is the more reliable instrument.

Paper structure (taxonomy followed by section blocks below)

§	Content
§1	Experiments 1a and 1b — embedding-type variation (Table 2)
§2	Worries about the paradigm — three a-priori concerns
§3	Experiment 2 — paradigm bias on simple sentences
§4	Monotonicity — sentences (25)–(27) for Exp 3
§5	Experiment 3 — verification vs inference (Table 3)
§6	Minimal conventionalism — auxiliary-assumption argument
§7	Experiment 4 — ambiguity-detection task (Tables 4, 5)
§8	Meanwhile in the Gricean camp… — competence-based explanation

Empirical data captured

All numerical values in this file are taken directly from the paper's Tables 2–5 and §1.4 / §3.2 / §5.2 / §7.2. Pages are cited in the paper's article-prefixed format (4:N) used by S&P.

Statistical-test attribution

The paper uses different statistical tests at different points:

• Exp 1a–b overall: Cochran's Q-test (UNVERIFIED: Q = 49.750 for Exp 1a, Q = 21.68 for Exp 1b; both p < .001 — exact Q-values reproduced from the paper but not externally cross-checked)
• Exp 1a–b pairwise: McNemar's test (specific p-values per pair)
• Exp 2 inference vs verification: McNemar's test (n = 29, p < .01)
• Exp 2 order effects: Fisher's Exact test (the paper writes "Fischer's", presumably a typo; inference p = .45, verification p = 1)
• Exp 3 pairwise: McNemar's with Bonferroni correction
• Exp 4 critical vs ambiguous controls: UNVERIFIED Wilcoxon's Exact test (W = 208, n = 20, p < .0001) — n=20 reflects 2 exclusions from the 22 UCL participants

Linglib integration

The canonical some/all world model SomeAllWorld lives in Phenomena.ScalarImplicatures.Basic; this file uses it for the implicature-spine bridge. The §8 derivation routes through Implicature.Competence.processAlternative (Sauerland-style competence machinery). The think empirical condition is bridged to Embedded.Attitudes.AttitudeInterpretation.local_.

Subsequent literature (forward pointers)

The most-important post-publication replies and extensions, none yet formalized in linglib:

• Sauerland 2010 "Embedded implicatures and experimental constraints" (S&P 3:2) — paradigm-choice critique
• Clifton & Dube 2010 S&P reply
• Ippolito 2010 S&P reply
• Chemla & Spector 2011 J. of Semantics 28:359–400 — the canonical empirical successor; uses graded truth-value judgments and finds ~73% local-SI in universal contexts, directly contesting this paper's Exp 4 null result. The most important reply.
• Foppolo, Guasti & Chierchia 2012 — children's interpretation
• Cremers, Tieu & Romoli (2017–2018) — acquisition follow-ups

These belong in their own Studies/ files; this paper's claims are formalized as historical-state-of-knowledge claims, per linglib's chronological-dependency rule.

inductive Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.QuantifierContext : Type

Quantifier contexts tested in Experiments 3–4 (paper §4 sentences (25)–(27) and §5 setup).

• all : QuantifierContext
• moreThanOne : QuantifierContext
• exactlyTwo : QuantifierContext
• notAll : QuantifierContext
• notMoreThanOne : QuantifierContext

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instDecidableEqQuantifierContext : DecidableEq QuantifierContext

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instDecidableEqQuantifierContext x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprQuantifierContext : Repr QuantifierContext

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprQuantifierContext = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprQuantifierContext.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprQuantifierContext.repr : QuantifierContext → ℕ → Std.Format

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.quantifierMonotonicity : QuantifierContext → Core.NaturalLogic.ContextPolarity

Map a quantifier context to its monotonicity using the canonical Semantics.Entailment.Polarity.ContextPolarity enum.

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inductive Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.ConventionalistVariant : Type

Conventionalist variants distinguished by the paper. The paper itself draws the lexicalist vs syntax-based dichotomy (page 4:3, "Conventionalism comes in two main varieties: lexicalist and syntax-based") and crosses it with a mainstream vs minimal axis (page 4:24): mainstream variants commit to preferring the local SI in non-DE contexts; minimal variants only claim local-SI readings exist.

Footnote 3 places Levinson 2000 with the lexicalists ("his position is closer to that of Landman and Chierchia 2004 than it is to Horn's").

The file's three-way enum is the projection onto distinct prediction profiles: both mainstream variants predict preference (collapsed here into one prediction function); minimal makes no preference claim. The paper's underlying 2×2 (lexicalist/syntactic × mainstream/minimal) can be reconstructed from this projection plus the conventionalist literature.

• mainstreamLexicalist : ConventionalistVariant
• mainstreamSyntactic : ConventionalistVariant
• minimal : ConventionalistVariant

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instDecidableEqConventionalistVariant : DecidableEq ConventionalistVariant

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instDecidableEqConventionalistVariant x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprConventionalistVariant.repr : ConventionalistVariant → ℕ → Std.Format

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@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprConventionalistVariant : Repr ConventionalistVariant

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.predictsLocalSI : ConventionalistVariant → QuantifierContext → Bool

Does this variant predict a preference for local SIs in the given context? Mainstream variants predict preference in non-DE; minimal makes no preference claim.

Caveat: this is a flat Bool collapsing several distinctions the paper preserves at page 4:5–4:6 and 4:19. Specifically: strong defaultism (Levinson 2000) vs conditional preference (Landman 1998 / Chierchia 2004) vs mere possibility (Chierchia 2004 on non-monotonic). For non-monotonic exactly two, the paper notes Geurts 2009 says local SI is not default, Van Rooij & Schulz 2004 say opinions are divided, and Chierchia 2004 calls it "merely possible". The flat Bool above collapses these to "non-DE → true". Refine if a downstream consumer needs the finer-grained predictions.

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• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.predictsLocalSI Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.ConventionalistVariant.minimal x✝ = false

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.conventionalistPredictsLocalSI : QuantifierContext → Bool

Convenience: the prediction shared by both mainstream variants (paper's running "mainstream conventionalism" of §1).

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§1 Experiments 1a and 1b

inductive Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.EmbeddingType : Type

Embedding types tested in Experiments 1a and 1b (paper §1).

• simple : EmbeddingType
• think : EmbeddingType
• want : EmbeddingType
• must : EmbeddingType
• all : EmbeddingType

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instDecidableEqEmbeddingType : DecidableEq EmbeddingType

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instDecidableEqEmbeddingType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingType.repr : EmbeddingType → ℕ → Std.Format

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@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingType : Repr EmbeddingType

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingType = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingType.repr }

structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.EmbeddingResult : Type

One row from Exp 1a-b results (paper Table 2). Rates are integer percentages; the paper reports proportions (.93 = 93%).

• embedding : EmbeddingType
• localSIRate : ℕ

  Percentage of participants endorsing the local-SI inference

• n : ℕ

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingResult.repr : EmbeddingResult → ℕ → Std.Format

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@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingResult : Repr EmbeddingResult

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingResult = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprEmbeddingResult.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.lookupRate? (results : List EmbeddingResult) (e : EmbeddingType) : Option ℕ

Look up the local-SI rate for a given embedding type. Returns none if the embedding wasn't tested. Use this for queries where non-existence should be observable.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.lookupRate (results : List EmbeddingResult) (e : EmbeddingType) : ℕ

Local-SI rate for an embedding type known to be present in results, defaulting to 0 for untested embeddings. The 0-default is deliberate for use in derived aggregates (thinkAvgRate, complexConditionRates) where presence is ensured by the data tables; for general queries use lookupRate?.

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.lookupRate results e = (Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.lookupRate? results e).getD 0

Method (paper §1.1–1.3): Exp 1a tested some embedded under think, deontic must, and the universal quantifier all; Exp 1b under think and want. Inference-task questionnaires; same design as Figure 1 ("Emilie says: ‘Betty thinks that Fred heard some of the Verdi operas.' Would you infer from this that Betty thinks that Fred didn't hear all the Verdi operas?").

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp1aResults : List EmbeddingResult

Experiment 1a results (paper Table 2, n = 30, French-speaking students at the École Nationale des Arts Décoratifs in Paris).

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp1bResults : List EmbeddingResult

Experiment 1b results (paper Table 2, n = 31, same population).

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.thinkAvgRate : ℚ

Cross-experiment average for think (paper §1.4 page 4:9): "local SIs were relatively frequent with think (57.5% across the two experiments)." Derived directly from Table 2 via lookupRate.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.complexConditionRates : List ℕ

All embedded (i.e. non-simple) condition rates from Exp 1a-b.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.complexConditionsMean : ℚ

Mean rate across all complex (i.e. embedded) conditions in Exp 1a–b (paper §1.4 page 4:9): "complex conditions yielded local SIs at a much reduced mean rate of 35%." Derived from complexConditionRates: (50 + 27 + 3 + 65 + 32) / 5 = 35.4.

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theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.embedding_not_systematic : lookupRate exp1bResults EmbeddingType.simple - lookupRate exp1aResults EmbeddingType.must = 91

Paper's headline finding (page 4:9): SI rates vary from 3% (must, Exp 1a) to 94% (simple, Exp 1b) — a 91-point range. Anchored in the data tables rather than literal arithmetic. This contradicts the conventionalist claim that local SIs occur "systematically and freely in arbitrarily embedded positions."

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.think_above_35_others_below : thinkAvgRate = 575 / 10 ∧ lookupRate exp1aResults EmbeddingType.all < 35 ∧ lookupRate exp1aResults EmbeddingType.must < 35 ∧ lookupRate exp1bResults EmbeddingType.want < 35

Among embedded conditions, only think shows substantial local-SI endorsement (averaging 57.5% across both experiments). Other embeddings — all (27%), must (3%), want (32%) — fall below 35%, the mean of complex-condition rates that the paper carries forward as the disconfirmation baseline (page 4:9).

§1.4 Discussion: refuting the implausibility defense

Paper §1.4 (pages 4:9–4:13) anticipates the conventionalist response that low embedded-SI rates are due to implausibility of the strengthened readings, and refutes it with three objections:

• Objection #1 (page 4:10): the implausibility argument doesn't work for all and think — paraphrases (8a-b)/(9a-b) show that the strengthened reading isn't markedly less plausible than the literal one. Footnote 7 makes the structural observation: asymmetric entailment (a) → (b) → (b) is at least as plausible as (a), so the implausibility defense cannot ground out at "(a) is implausible relative to (b)" (this is mt from Lean core).
• Objection #2 (page 4:10–4:11): the argument might be restricted to must/want. Paper refutes with examples (12)–(14) — bona fide scalar implicatures that arise despite the strengthened reading being implausible.
• Objection #3 (page 4:12): the gradient prediction (must < all < want < think correlating with implausibility ordering) is empirically false. Paper offers (18a)/(18b)/(18c) as a triple where the three readings are roughly equally plausible, contradicting the predicted gradient.

structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.ImplausibilityCounterexample : Type

A paper §1.4 example of a bona fide scalar implicature that arises despite the strengthened reading being independently implausible (i.e., the implausibility defense, if correct, would predict the SI not to arise — but it does). Paper (12)–(14) and (15)–(17).

• utterance : String

  The utterance (verbatim from paper).

• siInference : String

  The SI that arises despite implausibility.

• implausibilityReason : String

  Why the SI's content is independently implausible.

• siArises : Bool

  Paper claim: the SI nevertheless arises.

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprImplausibilityCounterexample : Repr ImplausibilityCounterexample

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprImplausibilityCounterexample.repr : ImplausibilityCounterexample → ℕ → Std.Format

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.objection2_counterexamples : List ImplausibilityCounterexample

§1.4 Objection #2 counterexamples: SIs that arise despite their strengthened readings being implausible. Page 4:11, examples (12)–(14) plus the "Harry hopes" parallels (15)–(17). All five entries have siArises := true — the data IS the refutation; no separate theorem needed.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.objection3_data : List (String × ℕ)

§1.4 Objection #3: the implausibility defense predicts a gradient (must << all << want << think) of SI rates aligning with implausibility ordering. Paper page 4:12 example (18) gives three readings (18a-c) that, by the paper's intuitions, are roughly equally plausible — yet their empirical SI rates differ markedly (think 57.5%, all 27%, must 3%): a 54.5pp spread the implausibility defense cannot ground. The data itself is the refutation.

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§2 Worries about the paradigm

Paper §2 (pages 4:14–4:16) introduces three a-priori reasons to suspect the inference paradigm exaggerates SI rates. These motivate Exp 2.

• Worry #1 (page 4:14): the rate at which people spontaneously draw a conclusion ϕ from premises A is generally lower than the rate at which they endorse the corresponding argument "A, therefore ϕ" when asked explicitly (cf. @cite{evans-newstead-byrne-1993}).
• Worry #2 (page 4:15): asking whether (b) might be implied by (a) changes the context: it makes the question of whether the speaker believes (b) contextually relevant. The inference paradigm thereby creates the relevance condition that makes the SI more likely.
• Worry #3 (page 4:15): superficial similarity to valid syllogisms (@cite{chater-oaksford-1999}) may cause errors. SIs derived in the inference task may piggyback on this surface-similarity heuristic.

structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.StampsFollowUp : Type

Paper §1.4 page 4:13 informal follow-up: "we asked 31 Dutch philosophy students to decide if the following inference is valid: (19) Mary has to put some but not all of the stamps in a blue envelope. SO: She may not put all the stamps in the blue envelope. 27 (or 89%) of our students said that the inference was valid."

Function: shows that the logical form of the conventionalist inferences is not intrinsically difficult to evaluate — so high inference-task rates can't be dismissed as logical-difficulty artifacts, and low inference-task rates aren't blocked by logical form.

• n : ℕ
• endorsementCount : ℕ
• endorsementRate : ℕ

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprStampsFollowUp.repr : StampsFollowUp → ℕ → Std.Format

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@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprStampsFollowUp : Repr StampsFollowUp

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprStampsFollowUp = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprStampsFollowUp.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.maryStampsFollowUp : StampsFollowUp

The Mary-stamps follow-up data (page 4:13). 27/31 = 89% endorsement defuses the "logical-difficulty" alternative explanation for the low embedded-SI rates of Exp 1a-b.

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.maryStampsFollowUp = { n := 31, endorsementCount := 27, endorsementRate := 89 }

structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.ChaterOaksfordDatum : Type

@cite{chater-oaksford-1999}'s syllogism meta-analysis (paper §2 Worry #3, page 4:15): people endorse the valid All A → All C syllogism ~90% of the time across independent experiments, and endorse three superficially-similar invalid syllogisms ~63% on average. The 27pp valid-vs-invalid gap is the documented surface-similarity confound that motivates the paradigm-bias hypothesis.

• validRate : ℕ
• invalidRate : ℕ

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprChaterOaksfordDatum : Repr ChaterOaksfordDatum

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprChaterOaksfordDatum = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprChaterOaksfordDatum.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprChaterOaksfordDatum.repr : ChaterOaksfordDatum → ℕ → Std.Format

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.chaterOaksfordSyllogisms : ChaterOaksfordDatum

The Chater-Oaksford data cited at page 4:15.

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.chaterOaksfordSyllogisms = { validRate := 90, invalidRate := 63 }

§3 Experiment 2

Method (paper §3.1, page 4:16): 29 Dutch-speaking students at the University of Nijmegen, within-subjects: same Dutch equivalent of Some of the B's are in the box on the left (24) tested in both inference and verification tasks (counterbalanced order: 15 inference- first, 14 verification-first).

Stats (paper §3.2, page 4:17): no order effect (Fisher's Exact test, inference p = .45, verification p = 1). Inference rate (62%) vs verification rate (34%) significantly different (McNemar's test, n = 29, p < .01). Filler control accuracy 97% (paper specifies this is for verification fillers, ruling out positive response bias on the verification side).

structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.Exp2Data : Type

Aggregate Exp 2 data (paper §3.2).

• inferenceRate : ℕ
• verificationRate : ℕ
• verificationFillerAccuracy : ℕ
• n : ℕ

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp2Data : Repr Exp2Data

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp2Data = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp2Data.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp2Data.repr : Exp2Data → ℕ → Std.Format

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp2 : Exp2Data

Exp 2 results from paper §3.2. The 28pp inference-vs-verification gap (62% vs 34%) is the load-bearing inflation estimate the paper carries into Exp 3; the 97% verification-filler accuracy rules out a positive response bias on the verification side. The data IS the finding — no separate trivial-arithmetic theorems are needed.

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp2 = { inferenceRate := 62, verificationRate := 34, verificationFillerAccuracy := 97, n := 29 }

§4 Monotonicity setup

Paper §4 (pages 4:18–4:19) introduces the conventionalist predictions tested in Exp 3:

• (25) DE contexts (not all, not more than one): no preference for local SIs — all theories agree (and conventionalistPredictsLocalSI returns false here).
• (26) UE contexts (all, more than one): mainstream conventionalism predicts local SIs are preferred.
• (27) NM context (exactly two): mainstream theories diverge here, none of them flatly predicting local SIs:
  • @cite{geurts-2009}: the default construal of (26) and (27) is without local SIs.
  • @cite{vanrooij-schulz-2004}: opinions are divided, but local SIs can occur in UE contexts.
  • @cite{chierchia-2004}: local SIs are possible in (26a,b) and (27), but not argued to be preferred.

Exp3Row.verificationPred and inferencePred encode the bracketed predictions of Table 3 (page 4:22), which the paper attributes to the stronger version of mainstream conventionalism (i.e. local SIs are preferred in all non-DE environments). The flat conventionalistPredictsLocalSI collapses the "preferred"/"possible"/"divided"/"not default" distinctions the paper preserves; consult the docstring on predictsLocalSI before reusing.

§5 Experiment 3

Method (paper §5.1, page 4:20): 26 first-year humanities students at the University of Nijmegen, within-subjects: same five sentences (25)–(27) tested in both verification and inference tasks. Verification block first, then an unrelated experiment of about 20 minutes, then the inference block. Verification used six items (since exactly two had two verification conditions); inference used five.

Stats (paper §5.2, page 4:23): McNemar's with Bonferroni correction — all p < .005, not all p < .001, more than one p < .0005, not more than one p < .05, exactly two p < .005 (both conditions).

structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.Exp3Row : Type

One row of Exp 3 results (paper Table 3, page 4:22). The two exactly two verification trials correspond to the two situations described at page 4:21: one in which two squares are connected to some-but-not-all of the circles while one square is connected to all (mainstream conventionalism predicts "true"), and the converse (mainstream predicts "false"). The bracketed numbers in Table 3 are the stronger-version-of-mainstream-conventionalism trend predictions captured here as verificationPred / inferencePred.

• quantifier : QuantifierContext
• verificationTrueRate : ℕ

  % saying "true" in verification task.

• verificationPred : Bool

  Conventionalist predicted trend for verification ("should say true"?) per Table 3.

• inferenceRate : ℕ

  % endorsing local-SI inference task.

• inferencePred : Bool

  Conventionalist predicted trend for inference ("should endorse"?) per Table 3.

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp3Row.repr : Exp3Row → ℕ → Std.Format

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@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp3Row : Repr Exp3Row

Equations

• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp3Row = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp3Row.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp3Results : List Exp3Row

Experiment 3 results (paper Table 3, page 4:22, n = 26).

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theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.verification_no_local_SI_in_UE : ((List.filter (fun (r : Exp3Row) => quantifierMonotonicity r.quantifier == Core.NaturalLogic.ContextPolarity.upward) exp3Results).all fun (r : Exp3Row) => r.verificationTrueRate == 100) = true

Verification shows zero local SIs in UE contexts (100% "true", accepting the classical reading), directly contradicting mainstream conventionalism's prediction (Table 3 column predicts "false").

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.verification_tracks_classical_reading : (exp3Results.all fun (r : Exp3Row) => decide (r.verificationTrueRate ≥ 96 ∨ r.verificationTrueRate ≤ 4)) = true

Verification rates perfectly track the classical (non-SI) truth value: when the classical reading is true, verification ≥ 96%; when false, ≤ 4%. Participants do not deploy the local-SI reading at all in the verification task.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.verification_extremal_in_predicted_rows : (exp3Results.all fun (r : Exp3Row) => decide (r.verificationPred = false → r.verificationTrueRate ≥ 96 ∨ r.verificationTrueRate ≤ 4)) = true

Conventionalism's predicted trend for verification is falsified across all non-DE conditions: where Table 3 predicts "verification should be 0" (i.e. "false"), the rate is overwhelmingly 100.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.inference_clusters_around_chance : (exp3Results.all fun (r : Exp3Row) => decide (r.inferenceRate ≥ 38 ∧ r.inferenceRate ≤ 62)) = true

Inference rates clustered around chance (paper §5.2 page 4:23: "all rates, for DE and non-DE items alike, clustered around chance level, give or take 12%" — i.e. roughly 38–62%).

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.deInferenceRates : List ℕ

Inference rates from DE rows of exp3Results.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.nonDeInferenceRates : List ℕ

Inference rates from non-DE rows of exp3Results.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.deInferenceMean : ℚ

Mean inference rate in DE conditions, derived from exp3Results. Paper §5.2 reports 45% from raw counts; integer arithmetic over the filtered rows gives (58 + 46) / 2 = 52. The discrepancy reflects rounding in Table 3's reported proportions, not a substantive difference.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.nonDeInferenceMean : ℚ

Mean inference rate in non-DE conditions, derived from exp3Results. Paper §5.2 reports 51%; integer arithmetic yields 52 (see deInferenceMean docstring).

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theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.inference_means_within_chance_band : 50 - 12 ≤ nonDeInferenceMean ∧ nonDeInferenceMean ≤ 50 + 12 ∧ 50 - 12 ≤ deInferenceMean ∧ deInferenceMean ≤ 50 + 12

The paper's "fails to coalesce into the predicted pattern" disconfirmation (paper §5.2 page 4:23): both DE and non-DE inference means lie within the "around 50%, give or take 12%" band the paper explicitly delimits. The conventionalist-predicted pattern (non-DE preference, DE no-preference) requires non-DE >> DE; instead both means sit near chance. Note: paper reports 51%/45% from raw counts; this file computes 52%/52% from rounded Table-3 percentages — the qualitative "both within ±12 of 50%" claim survives either way.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.think_above_50_others_below : lookupRate exp1aResults EmbeddingType.all ≤ 50 ∧ lookupRate exp1bResults EmbeddingType.want ≤ 50 ∧ lookupRate exp1aResults EmbeddingType.must ≤ 50 ∧ thinkAvgRate > 50

After Exp 3 establishes the ~50% paradigm baseline in inference tasks, the rates from Exp 1a–b for all (27%) and want (32%) are below baseline — they may be entirely paradigm artifacts. Only think (57.5% averaged) survives correction.

§6 Minimal conventionalism setup

Paper §6 (pages 4:23–4:24) sets up Exp 4 by analyzing what minimal conventionalism (the weakest form: "local-SI readings exist") would predict. The argument:

1. In its bare form, minimal conventionalism makes no predictions (claim that (28a) "All the customers shot at some of the salesmen" may or may not be read as (28b) "All the customers shot at some but not all of the salesmen" cannot be falsified).
2. With the auxiliary assumption that native speakers can detect the readings their grammar makes available, minimal conventionalism does predict: in a situation that falsifies (28b), participants should claim either that (28a) is false or that it's ambiguous.
3. Exp 4 tests this by adding a "could be either" response option to Exp 3's verification task.

§7 Experiment 4

Method (paper §7.1, pages 4:25–4:26): 22 first-year linguistics students at University College London. Materials: Exp 3's verification items translated to English, plus a third response option ("could be either"), plus 5 ambiguous control sentences. 30 trials total in 10 pseudo-random orders. Conducted in a classroom with oral instructions to ensure participants understood the notion of ambiguity (using calibration items like "Visiting relatives can be boring" and "The girl hit the boy with the telescope").

Stats (paper §7.2, page 4:26): Wilcoxon's Exact test W = 208, n = 20, p < .0001 (n=20 reflects 2 exclusions from the 22 UCL participants) for the difference between ambiguous-control rates (~70%) and critical non-DE "could be either" rates (~6%).

structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.Exp4Row : Type

Response rates for critical and DE control items in Exp 4 (paper Table 4, page 4:27). The "either" rate is the load-bearing diagnostic: minimal conventionalism predicts it should be high in non-DE cases.

• quantifier : QuantifierContext
• trueRate : ℕ

  % saying "true"

• falseRate : ℕ

  % saying "false"

• eitherRate : ℕ

  % saying "could be either"

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp4Row.repr : Exp4Row → ℕ → Std.Format

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@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp4Row : Repr Exp4Row

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• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp4Row = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprExp4Row.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp4Results : List Exp4Row

Experiment 4 results (paper Table 4, page 4:27, n = 22).

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structure Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.AmbiguityControl : Type

One ambiguous control item from Exp 4 (paper Table 5, page 4:27). Sentences are quoted exactly as in the paper.

• sentence : String
• eitherRate : ℕ

  % saying "could be either"

@[implicit_reducible]

instance Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprAmbiguityControl : Repr AmbiguityControl

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• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprAmbiguityControl = { reprPrec := Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprAmbiguityControl.repr }

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.instReprAmbiguityControl.repr : AmbiguityControl → ℕ → Std.Format

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.genuineAmbiguityResults : List AmbiguityControl

Ambiguous control items (paper Table 5, page 4:27). Sentences are verbatim — note paper uses the circles and the squares with definite articles in (29a), (29c)–(29e).

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.genuineAmbiguityRates : List ℕ

Either-rates from genuineAmbiguityResults, extracted for clean arithmetic in genuineAmbiguityMean.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.genuineAmbiguityMean : ℚ

Mean genuine-ambiguity detection across Table 5 items: (82+73+59+77+59)/5 = 70. Paper §7.2 page 4:26 reports 70%.

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def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp4NonDeTotalResponses : ℕ

Total non-DE trials in Exp 4: 4 critical items × 22 participants (paper §7.2 page 4:26 footnote on the 9-of-88 calculation).

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• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp4NonDeTotalResponses = 4 * 22

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp4NonDeConventionalistConsistent : ℕ

Total non-DE trials in Exp 4 consistent with minimal conventionalism (paper §7.2 page 4:26): "only 9 out of 88 responses".

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• Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.exp4NonDeConventionalistConsistent = 9

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.either_rate_below_genuine_ambiguity : (exp4Results.all fun (r : Exp4Row) => decide (r.eitherRate ≤ 14)) = true

The §7.2 disconfirmation of minimal conventionalism: critical-item "either" rates (Table 4 non-DE rows) all sit at or below 14%, while genuine-ambiguity controls (Table 5) average 70%. Participants reliably detect real ambiguities but not the ones conventionalism predicts.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.nonDe_consistent_with_minimal_at_10_percent : ↑exp4NonDeConventionalistConsistent / ↑exp4NonDeTotalResponses < 11 / 100

Total fraction of non-DE responses consistent with minimal conventionalism (paper §7.2 page 4:26): "only 9 out of 88 responses (i.e. 10%)". Stated as a ℚ inequality — the actual value is 9/88 ≈ 10.227%, which rounds to 10% per the paper's footnote 10.

§8 Gricean camp

Paper §8 (pages 4:28–4:29) sketches the Gricean explanation for the one embedded condition that did yield elevated SI rates: think (57.5%, the only one to survive Exp 3's paradigm correction).

The Gricean derivation proceeds in two steps. First, the global SI for "Bob believes Anna ate some" yields the primary implicature (32):

(32) ¬K(Bob believes Anna ate all)

Second, if Bob is competent on whether Anna ate all (33):

(33) Bob believes all ∨ Bob believes ¬all

then (32) + (33) entail (34): "Bob believes Anna didn't eat all." This looks like a local SI but is derived from global pragmatics + competence. The same derivation does not generalize to universal quantifiers (paper example (35)–(38)) because the analogous auxiliary (38) is a collective uniformity assumption "considerably less plausible" than (33).

Footnote 11 (page 4:28) lists the references for the Gricean derivation of seemingly-embedded SIs that the §8 argument is built on: @cite{recanati-2003}, @cite{sauerland-2004}, @cite{vanrooij-schulz-2004}, @cite{geurts-2006}, @cite{geurts-2009}, @cite{russell-2006}, @cite{spector-2006}. None of @cite{russell-2006}, @cite{spector-2006}, or @cite{geurts-2006} are formalized in linglib yet.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.competence_explains_think_via_processAlternative : have p := Implicature.Competence.processAlternative true Implicature.BeliefState.disbelief; p.weakHolds = true ∧ p.competenceAssumed = true ∧ p.strongDerived = true

The §8 derivation routed through the canonical Sauerland-style competence machinery in Implicature.Competence. The paper's claim is that given the speaker has a determinate negative stance about the alternative (here .disbelief — speaker believes ¬(Bob-believes-all)) and the competence assumption (33), the strong implicature is derived. The spine's processAlternative true .disbelief produces the strong-derived flag exactly when these conditions hold; this theorem exhibits the spine's outcome_ii_strong machinery applied to the §8 think-reading derivation.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.gricean_derivation_for_universals_requires_uniformity {Customer Salesman : Type} (Shot : Customer → Salesman → Prop) (globalSI : ¬∀ (c : Customer) (s : Salesman), Shot c s) (uniformity : (∀ (c : Customer) (s : Salesman), Shot c s) ∨ ∀ (c : Customer), ¬∀ (s : Salesman), Shot c s) (c : Customer) : ¬∀ (s : Salesman), Shot c s

The same derivation chain applied to universal quantifiers (paper §8 (35)–(38)). For "All the customers shot at some of the salesmen" the Gricean primary implicature is (36) "Not all the customers shot at all the salesmen", i.e. ¬ ∀ c s, Shot c s. The conventionalist construal (37) is "All the customers shot at some but not all of the salesmen (hence, none of the customers shot at all the salesmen)". The auxiliary needed to bridge (36) → (37)'s parenthetical is (38):

Either all the customers shot at all the salesmen, or none of the customers shot at all the salesmen.

This is a collective uniformity assumption — (∀c∀s, Shot c s) ∨ (∀c, ¬∀s, Shot c s) — not a per-customer competence. The paper's substantive claim (page 4:29) is that (38) is "considerably less plausible than (33)": (33) only stipulates that one opinion-holder (Bob) has a determinate stance, whereas (38) requires that every customer's behavior partition the same way (all-shot-all or none-shot-all, with no in-between distribution). Implausibility is not formalized; this theorem captures the formal half of the §8 argument, licensing the gap between "the inference goes through under (38)" and "(38) is independently warranted."

Bridge: ∅-condition SI as Implicature SomeAllWorld

Wraps the canonical some-derived SI as an Implicature value over the canonical SomeAllWorld model from Phenomena.ScalarImplicatures.Basic, exercising the spine in Defs.lean and the diagnostics in Diagnostics.lean.

def Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.somePassedSI : Implicature SomeAllWorld Prop

The neo-Gricean SI derived from some students passed in a UE context (paper §1's ∅-condition). Mechanism is the Sauerland Standard Recipe / neo-Gricean derivation; the SI content is notUniversal (= the canonical "not all" inference from SomeAllWorld).

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theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.somePassedSI_isReinforceable : Implicature.IsReinforceable SomeAllWorld.atLeastOne somePassedSI

The SI is reinforceable: there's an assertion-world (.all) where the assertion holds but the inferred content fails — so adding "…but not all" is non-redundant.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.somePassedSI_isCancellable : Implicature.IsCancellable SomeAllWorld.atLeastOne somePassedSI

The SI is cancellable (Sadock 1978's diagnostic), via the spine's IsReinforceable.toCancellable.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.somePassedSI_corresponds_to_unembedded_data : lookupRate exp1aResults EmbeddingType.simple = 93 ∧ lookupRate exp1bResults EmbeddingType.simple = 94

The neo-Gricean SI for somePassedSI corresponds to the ∅-condition tested in Experiments 1a-b, where the empirical SI endorsement rate is 93% (Exp 1a) and 94% (Exp 1b). The spine bridge is anchored in the data.

Bridge: think condition ↔ AttitudeInterpretation.local_

The §1 think condition tests whether participants endorse the attitude-embedded SI ("Betty thinks Fred didn't hear all the Verdi operas" inferred from "Betty thinks Fred heard some of the Verdi operas"). This is exactly the AttitudeInterpretation.local_ reading formalized in Embedded/Attitudes.lean: the strengthened some sits inside Betty's belief-content. The cross-experiment 57.5% endorsement rate measures how often participants compute local_ rather than global.

theorem Phenomena.ScalarImplicatures.Studies.GeurtsPouscoulous2009.think_condition_corresponds_to_local_attitude : thinkAvgRate = 575 / 10 ∧ Embedded.Attitudes.believesSomeMeaning Embedded.Attitudes.AttitudeInterpretation.local_ { actual := SomeAllWorld.all, johnBelieves := SomeAllWorld.all } = false

The paper's think condition tests endorsement of the local attitude reading, which Embedded/Attitudes.lean formalizes as AttitudeInterpretation.local_. The empirical rate (57.5%) measures the local-vs-global preference; the paper's §8 Gricean derivation explains it via Bob-style competence on a single opinion-holder.

§Conclusion: Chierchia et al. (39)

Paper §Conclusion (page 4:30) example (39) is the marked contrastive construction @cite{chierchia-fox-spector-2008} cite as evidence for embedded SI: "If you take a salad OR desert, you pay $20; but if you take BOTH there is a surcharge" (paper's typo "desert" preserved). Geurts & Pouscoulous's punchline: such examples are strongly marked, in which "the contrast between or and both is essential" — not generic embedded-SI evidence. The paper's verdict (page 4:31) is that mainstream conventionalism's defense "primarily relies on data that are strongly marked, like (39), for example."

Appendix

Sample stimulus pairs from Experiments 1a-b live in the paper's appendix (page 4:31) and the separately available background-materials file (doi:10.3765/sp.2.4a); they are not reproduced here.