Haug & Dalrymple (2020) @cite{haug-dalrymple-2020} #
Reciprocity: Anaphora, scope, and quantification. Semantics & Pragmatics 13:10, 1–62. doi:10.3765/sp.13.10.
The relational analysis of reciprocals in Partial Plural Compositional DRT (PPCDRT). Each other is a pronoun bearing an anaphoric relation (reciprocity R) to its antecedent; the narrow/wide scope ambiguity reduces to the choice of antecedent relation between the matrix subject and the embedded local antecedent (group identity ∪ vs. binding =).
What is formalized #
Witness-based formalisation of the paper's empirical contributions over
the PPCDRT substrate (Theories/Semantics/Dynamic/PPCDRT/):
| Paper § | Topic | Witness type |
|---|---|---|
| §3 | Scope readings (narrow / wide) | PluralAssign Person |
| §3.3 | Crossed readings (4-cell classification) | RecipReading triples |
| §4.2 | Underspecified RECIP/REFL | underspecifiedCond lattice |
| §4.4 | Multiple reciprocals | Two-reciprocal witness |
| §4.5 | Subgroup readings (forks, gravity) | Weak-vs-strong contrast |
| §4.6 | Collective antecedents | Distinctness neutralization |
| §5 | Quantified antecedents + truth-value gap | Truth3 via removeGap |
| §6 | Maximize Anaphora as a principle | R_u + maximizeAnaphora |
| §6.2 | Multi-reciprocal pairwise prediction | R_u over two reciprocals |
| §6.3 | MA interacting with scope | Tracy/Matty/Chris case |
Sections paper-acknowledged but not formalised (out of scope for a
study-file size budget): the full §2.3 Δ-relativised distribution
machinery (deferred — the substrate-trimming pass removed the prototyped
delta/⨟/∂/max^u operators since no consumer exercised them; they
will return alongside a Brasoveanu 2007 / Dotlačil 2013 study file);
the §5.2 empirical-fit table; the §7 typological excursus.
Connections to existing linglib substrate #
- @cite{champollion-bumford-henderson-2019} for the §5 supervaluationist
truth-value-gap analysis — realised via
Theories/Semantics/Homogeneity/Basic.lean'sremoveGap/Truth3.metaAssert. - @cite{kriz-2015} for the homogeneity background; same substrate.
- @cite{langendoen-1978} for the reciprocity-as-cumulativity link —
realised via
PPCDRT/Cumulativity.lean'sgroupIdentityCond_iff_cumulativeOp_eqbridge theorem. - @cite{murray-2008}, @cite{cable-2014} for the §4.2 underspecification examples.
Source-paper attribution note #
The §4.2 paragraph in @cite{haug-dalrymple-2020} attributes the German sich / Romance reflexive examples to @cite{cable-2014} (paper p. 32), not to @cite{murray-2008} alone — the latter focuses on Cheyenne. The docstrings here follow that attribution.
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- HaugDalrymple2020.instDecidableEqPerson x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- HaugDalrymple2020.instReprPerson = { reprPrec := HaugDalrymple2020.instReprPerson.repr }
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- HaugDalrymple2020.instInhabitedPerson = { default := HaugDalrymple2020.Person.tracy }
Standard dref indices used throughout.
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A partial assignment with u₁ ↦ a, u₂ ↦ b, u₃ ↦ c.
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- HaugDalrymple2020.assign3 a b c = (HaugDalrymple2020.assign2 a b).update HaugDalrymple2020.u₃ c
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Narrow-scope state for paper eq 49: two assignments where each girl has herself as the embedded subject pronoun. The matrix subject (u₁) and the embedded subject pronoun (u₂) have IDENTICAL value-sets: {Tracy, Chris}. This is group identity (∪u₂ = ∪u₁).
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Membership lemma.
The summed value of u₁ across the narrow-scope state is {tracy, chris}.
The summed value of u₂ across the narrow-scope state is {tracy, chris}.
Narrow scope is group identity (paper §3, eq 49). The matrix
subject (u₁) and the embedded subject pronoun (u₂) have the same
value-set, witnessing the groupIdentityCond of the relational
analysis.
Wide-scope state for paper eq 51: u₂ is bound by u₁ — pointwise identity of values. Each girl thought only of herself as the winner.
UNVERIFIED collapse. The paper distinguishes narrow eq 49 (a
4-row table including a doxastic-world column w) from wide eq 51
(a 2-row table without w). The empirical contrast lives in the
presence/absence of the doxastic-alternative column, which the
intensional δ_w machinery (paper §3.1) makes visible. The current
narrowScopeState/wideScopeState encoding flattens both to a
2-row table — the pointwise-vs-coverage distinction is correct at
each row but the row-multiplicity contrast is lost. Setting
wideScopeState := narrowScopeState reflects this collapse honestly:
until the substrate exposes δ_w, the two states are extensionally
identical for the 2-element domain.
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Wide scope is binding (paper §3, eq 51). In every state of the
plural information state, the embedded subject pronoun's value
equals the matrix subject's value as Option E.
Wide scope also satisfies group identity (binding ⊆ group identity).
This is the substrate-level fact binding_implies_groupIdentity
applied to this concrete case.
Narrow-scope reciprocity state for paper eq 53: u₁ (matrix subject) and u₂ (embedded subject) are group-identical (each girl thought of the group); u₃ (reciprocal) takes the other girl's value at each state. Tracy saw Chris, Chris saw Tracy.
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Reciprocity satisfies group identity between subject pronoun and reciprocal (∪u₃ = ∪u₂).
Reciprocity satisfies per-state distinctness (∂(u₂ ≠ u₃)).
The full reciprocity condition holds for this state.
The two-parameter classification: locus × antecedent relation. Three cells are attested; the (low, bound) cell is empirically empty per paper p. 24 — bound antecedents force high locus.
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The three attested cells correspond to the three RecipReadings.
The empty fourth cell: there is no RecipReading with low locus
and binding antecedent. Paper p. 24: "the bound reading of the
reciprocal's antecedent cannot cooccur with a low locus for the
reciprocal, because it does not make available the plurality that
the reciprocal needs."
Crossed readings (paper §3.3, eq 56): high locus + group-identity
antecedent + group-identity reciprocal slot. The reciprocity comes
from the DRS distinctness presupposition ∂(u₃ ≠ u₂), not from an
anaphoric reciprocity relation. Empirically attested via the
Jennifer Lawrence interview headline (paper p. 25, ex. 57) and
related corpus examples.
Underspecified anaphors (German sich, Cheyenne REFL/RECIP affix)
contribute group identity without the distinctness presupposition.
They permit reflexive (binding-style), reciprocal, and mixed
readings. The semantic core is just groupIdentityCond —
reciprocity is one specialization among others.
Underspecified anaphors also admit reciprocity readings — reciprocity strengthens underspecified by adding distinctness.
Multiple-reciprocal state for paper (85a) reading "where the second reciprocal takes the first one as its antecedent" — semantically interpretation (84b): "Tracy gave Chris a picture of Tracy, and Chris gave Tracy a picture of Chris." Per paper eq 85 (p. 35–36):
- u₁ = subject (each girl, the giver)
- u₂ = first reciprocal (each other₁), antecedent u₁
- u₃ = pictures
- u₄ = second reciprocal (each other₂), antecedent u₂; per the reading, u₄ takes the value of "the other member of u₂'s antecedent group" = u₁ (the giver).
Distinctness conditions per eq 85b: ∂(u₁ ≠ u₂) and ∂(u₂ ≠ u₄). Each reciprocal is distinct from its OWN antecedent. There is no constraint between u₃ (pictures) and any reciprocal.
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Both reciprocals satisfy the paper's distinctness conditions per eq 85b: u₂ ≠ u₁ (first reciprocal distinct from antecedent u₁) and u₄ ≠ u₂ (second reciprocal distinct from its antecedent u₂).
"The forks are propped against each other" (paper eq 88b): each
fork is supported by a group containing one or more of the others
— possibly all, but not necessarily. This is weak reciprocity:
R_u need not be the full Cartesian product minus the diagonal.
Implementation: a 3-fork example where fork₁ leans on fork₂, fork₂
on fork₃, fork₃ on fork₁ (a chain). R_u = {(1,2), (2,3), (3,1)} —
NOT the full strong-reciprocity {(1,2), (2,1), (1,3), (3,1), (2,3), (3,2)}.
Note on paper eq 92. The paper's actual analysis (eq 92) wraps
the support relation in δ_{u_1} distribution and uses sum-dref
∪u_2 on the supporter side. Our flat 3-row encoding shows the
R_u-shape weak-reciprocity property at the value-set level but does
not exercise the equivalence-class structure that distribution
builds. The δ-side of the analysis is substrate-deferred (PPCDRT
delta was trimmed in P6).
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The fork chain has 3 supporting pairs in R_u(u₁, u₂).
Strong reciprocity would require each fork to lean on EVERY other —
e.g., (tracy, matty) would also need to be in R_u. The fork-chain
state does NOT satisfy strong reciprocity.
Collective antecedents NEUTRALIZE the distinctness condition: when
the predicate work-together is interpreted collectively on
∪u₁, the per-state distinctness u₁ ≠ u₃ becomes vacuous.
The state below has u₁ = sailor at each row, with u₃ = ship-of pointing to (possibly the same!) sailor as u₁. Reciprocity-as-stated fails (no per-state distinctness), but the sentence is felicitous because the collective interpretation of u₁ does the predicational work. Paper p. 39.
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The collective state satisfies group identity (∪u₁ = ∪u₃) but FAILS
per-state distinctness — formal reciprocity does not hold. The
sentence is felicitous because work-together is collective.
Paper §5 introduces max^u(K) and shows that quantified antecedents
of reciprocals (most members know each other; few have spoken to each
other) give rise to a truth-value gap: True iff true on both the
maximal-set reading and the reference-set reading; False iff false on
both; Neither otherwise (paper eq 109). The paper invokes
@cite{champollion-bumford-henderson-2019}, following @cite{kriz-2015},
for the supervaluationist machinery.
Here we encode the truth-value gap directly via `Truth3`, exploiting
the existing `Theories/Semantics/Homogeneity/Basic.lean` substrate
(`removeGap`, `Truth3.metaAssert`).
The truth value of a quantified-antecedent reciprocal sentence, given its truth on the maximal-set reading and on the reference-set reading. Paper eq 109.
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The two precisifications H&D §5 makes available for a
quantified-antecedent reciprocal sentence: the maximal set
reading (u ranges over the largest restrictor-satisfying set) and
the reference set reading (u ranges over the largest set such
that the scope-plus-reciprocal relation holds). Paper §5.1, eq 99.
- maximalSet : HDPrecisification
- referenceSet : HDPrecisification
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- HaugDalrymple2020.instDecidableEqHDPrecisification x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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The two-element specification space for H&D §5: both precisifications are admissible.
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Lift a (maximal-set-reading, reference-set-reading) pair of Props to a Prop-valued evaluation over the H&D precisification space.
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- HaugDalrymple2020.hdEval maximalSetReading refSetReading HaugDalrymple2020.HDPrecisification.maximalSet = maximalSetReading
- HaugDalrymple2020.hdEval maximalSetReading refSetReading HaugDalrymple2020.HDPrecisification.referenceSet = refSetReading
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Bridge theorem (P8) — H&D §5's truth-value gap (paper eq 109)
instantiates a 2-precisification supervaluation construction. Paper
§5 footnote 23 cites Križ 2015 and Champollion-Bumford-Henderson 2019
as inspirations for the gap shape; this theorem makes the structural
correspondence Lean-checkable: quantifiedReciprocalTV m r agrees
with superTrue (hdEval m r) hdSpec over the two-element
{maximalSet, referenceSet} precisification space.
The paper itself is more guarded than "the §5 gap is CBH 2019" —
it says §5 is inspired by CBH/Križ. The theorem here exhibits the
truth-table reproducibility (paper eq 109 ↔ superTrue on a
2-element space), not a deeper claim about identity of analyses.
The R_u set for the narrow-scope reciprocity state. Reciprocity
means R_u(u₂, u₃) is the full off-diagonal pair set on the value
range — for a 2-element range {Tracy, Chris}, this is exactly two
pairs: (Tracy, Chris) and (Chris, Tracy). Paper eq 127.
A "diagonal" pair like (Tracy, Tracy) is NOT in R_u for reciprocity: the per-state distinctness condition rules it out.
Paper §6.1 (p. 55) argues SMH over-strengthens. The substrate-level
refutation lives in Reciprocals.lean as SMH_diverges_from_relational
— the relational analysis with MA leaves both readings available on
the default property bundle, while SMH commits to narrow only.
Related principles cited by paper §6: the Maximal Interpretation
Hypothesis of @cite{sabato-winter-2012} and @cite{winter-2001}
(p. 54), the typicality-constrained MA of @cite{poortman-struiksma-kerem-friedmann-winter-2018}
(p. 54), the anaphora-as-exhaustive principle of @cite{kadmon-1990}
(p. 54), and the experimental evidence of @cite{majewski-2014}
(paper §6 docstring reference). The trio MIH/MA/SMH form the natural
scaffold for a principled treatment of the §4.5 reciprocal-strength
typology — see the open work in the future-directions note below.
Paper §6.2 (p. 56): for "The classmates gave each other pictures of each other," Maximize Anaphora predicts pairwise maximization (each classmate gave a picture-of-each-other to each other classmate), NOT the all-triples reading.
The state here witnesses the pairwise reading: in each state, u₃ (the picture's subject) and u₄ (the receiver) form a swap pair.
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The pairwise R_u u₂ u₁ for the first reciprocal (each other₁,
antecedent u₁) has exactly two pairs: (Chris, Tracy) and (Tracy, Chris)
— read as (receiver, giver) per the paper's pair convention.
Witnesses: row 1 (Tracy gave Chris) gives (u₂=Chris, u₁=Tracy)
and row 2 (Chris gave Tracy) gives (u₂=Tracy, u₁=Chris).
Paper eq 135: "Tracy, Matty and Chris think they praised each other." Three-element antecedent group; the wide-scope reading on the matrix plural information state is witnessed below. Paper eq 136 shows the sample output state with three girls × two complement-mate pairs.
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The three-way state has six R_u pairs (each girl paired with each other in both directions): the wide-scope MA prediction is that the full off-diagonal pair-set is realized.
The bridge theorem from PPCDRT/Cumulativity.lean is consumable here:
group identity reduces to Beck-Sauerland ** over the Finset of
value pairs. This is the formal realization of @cite{langendoen-1978}'s
reciprocity-as-cumulativity claim, asserted in the original
Reciprocals.lean docstring as prose (audit finding 4).