[SP21]: quantifier scope ambiguity in truth-value judgments #

"When pragmatics matters more for truth-value judgments: An investigation of quantifier scope ambiguity", Glossa 6(1): 110. A modeling paper: RSA with lifted scope and QUD variables explains the endorsement patterns of [ML03] and successors, separating pragmatic factors (world and QUD priors) from grammatical processing (scope access).

Two models, each in its own namespace and each with two faces — an exact-ℚ≥0 kernel face at the paper's parameters and a general-α structural face on RSA/PMF.posterior operators:

EveryNot: "Every horse didn't jump", n = 2 (3 worlds × 2 utterances × 2 scopes × 3 QUDs; paper §3).
TwoNot: "Two horses didn't jump", n = 4, parameterized by numeral reading (exact vs at-least; paper §4).

Main results #

EveryNot.s2_ordering_*: endorsement orders w0 > w1 > w2 across all Figure 2 prior configurations (endorsement at w=1: .29 at b_suc = 0.1, .80 at b_suc = 0.9).
EveryNot.s2_qud_ordering: favoring none? < how-many? < all? monotonically increases endorsement (.38 < .48 < .63; the adult pattern of the paper's Figure 4).
EveryNot.baseline_L1_zero_gt_one / _one_gt_two: the baseline L₁ ordering for every α > 0.
EveryNot.S1g_all_qud_scope_invariant: under the all? QUD the speaker is scope-invariant — the structural mechanism of pragmatic dominance.
TwoNot.exact_baseline_endorsement_high / atleast_baseline_endorsement_low / exact_vs_atleast_endorsement: the 1-of-2 vs 2-of-4 asymmetry flips with the numeral reading (Figure 7) — the paper's argument that exact numeral semantics is basic.
TwoNot.L0_exact_gt_atLeast_at_w2: the L0 concentration contrast driving that asymmetry, for every α.
every_not_scope_entailment / exact_two_not_scope_independent: universals have nested scope readings, exact numerals independent ones — why numerals are diagnostic for isomorphism.

Implementation notes #

Truth conditions are grounded compositionally (every_sem, ScopeDerivation) and in the named numeral meanings of Semantics.Numerals. The kernel faces use PMF.ofScores over ℚ≥0 score chains at α = 1 (§3.2; costs cancel, fn. 8) with L0 taking no world prior (fn. 6); the structural faces are parametric in α (the parametric-prior layer is TODO). Developmental claim (§3.3): children and adults differ only in these parameter values.

TODO #

General-α, parametric-prior versions of the QUD ordering and of the exact-vs-at-least endorsement divergence (their α = 1 instances at the paper's parameters are kernel-proved), together with the parametric prior layer (PMF.binomial world prior, weighted scope/QUD priors) and the general S2 they require; the emergent b_suc-monotonicity of endorsement as a limit statement.

[SP21]: quantifier scope ambiguity in truth-value judgments #

Main results #

Implementation notes #

TODO #

Shared domain: every-not (n = 2) #

Shared domain: two-not (n = 4) #

Scope-entailment asymmetry ([ML03]) #

Numeral-semantics grounding #

The every-not model (§3) #

Truth conditions and compositional grounding #

The kernel face: exact-ℚ scores at the paper parameters #

L₁ predictions #

S₂ endorsement predictions #

The structural face: general α #

The uniform world prior #

L0 as RSA.L0OfBoolMeaning #

QUD projection facts on ℝ≥0∞ #

S1g — speaker conditional on (latent, world) #

Marginal speaker — PMF.bind over the latent prior #

L1 — Bayesian inversion via PMF.posterior #

The general-α baseline orderings #

Scope collapse under QUD = all? (paper §3.2, restated §5.1) #

Scope-prior irrelevance under QUD = all? #

The general-α QUD ordering (open) #

The two-not model (§4) #

Truth conditions and compositional grounding #

The kernel face: exact-ℚ scores at the paper parameters #

Endorsement predictions (§4.2, Figure 7) #

Informativity contrasts #

The structural face: exact-vs-at-least necessity (§4.2.2) #

Extension of twoNot under each numeral reading #

L0 (literal listener) for the TwoNot model #

The L0 concentration contrast #

Vacuous-zero contrast at the divergence worlds #

The cross-model asymmetry (§4.2.2) #

L0 as `RSA.L0OfBoolMeaning` #

Marginal speaker — `PMF.bind` over the latent prior #

L1 — Bayesian inversion via `PMF.posterior` #

Extension of `twoNot` under each numeral reading #