Sauerland (2003): A New Semantics for Number #
@cite{sauerland-2003}
Sauerland, U. (2003). A new semantics for number. Proceedings of SALT 13, 258–275.
See also @cite{sauerland-anderssen-yatsushiro-2005} for the companion chapter "The Plural is Semantically Unmarked."
Core Insight #
Number features are presuppositional partial identity functions on the
entity domain, ordered by PrivativePair.specLevel. The general
presuppositional framework (phiPresup, sgSem, plSem, etc.) is in
Theories.Semantics.Presupposition.PhiFeatures; this file applies it to
Sauerland's specific arguments about number semantics.
Key Results (study-specific) #
- Feature-Subset Principle =
specLevellinear order (§2) - Maximize Presupposition derives singular preference (§3)
- Coordination forces plural on non-atomic sums (§4)
every = JE ∘ DER: decomposition into definite + atomic universal, with JE presupposing scope predicate definedness (§5)- Existential ⟦a⟧ projects presuppositions existentially, contrasting with JE's universal projection (§5b)
- Multiplicity as presuppositional competition, not at-issue Horn scale (§6);
bridge to
Multiplicity.PluralTheory.implicature - Negated Pl entails negated Sg: "didn't harvest tomatoes" → "didn't harvest a tomato" (§6b)
- Czech gender coordination data: masculine=vacuous, feminine=non-masc, neuter=inanimate; parallels number resolution (§7)
- Politeness prediction: unmarked phi-values used for honorification (§8)
The presupposition of [Sg] is mereological atomicity.
[Pl] is always defined — its presupposition is vacuous.
[Sg] is defined exactly at atoms.
§2: Feature-Subset Principle #
Sauerland's principle (46): if F₁ and F₂ are presuppositional features that can be inserted in the same syntactic position, their domains must stand in a subset relationship.
This is a consequence of the PrivativePair structure: the three well-formed
cells are linearly ordered by specLevel, so their presuppositional domains
are necessarily nested (atoms ⊂ all entities).
Presuppositional domain nesting: the domain of [Sg] (atoms) is contained in the domain of [Pl] (all entities). This is the Feature-Subset Principle for number: dom(Sg) ⊆ dom(Pl).
The containment is strict: there exist non-atomic entities in dom(Pl) \ dom(Sg).
The specLevel ordering on PrivativePair is the Feature-Subset
Principle: more specified cells have strictly smaller presuppositional
domains.
§3: Maximize Presupposition Derives Singular Preference #
When the referent is atomic, [Sg]'s presupposition is satisfied. Since [Sg] and [Pl] have the same assertive content (both are identity), and [Sg] has a strictly stronger presupposition, Maximize Presupposition (@cite{heim-1991}) blocks [Pl]: the speaker must use [Sg].
When the referent is non-atomic, [Sg]'s presupposition fails. [Pl] (with its vacuous presupposition) is the only available option.
The formal derivation uses phi_mp_selects_maximal from
MaximizePresupposition.lean: the OT constraint phiMP assigns 0
violations to the maximal cell (= [Sg]) and penalizes weaker cells
(= [Pl]). When [Sg] is among the candidates (i.e., its presupposition
is satisfied), the optimal candidate must have maximal presupStrength.
Same assertive content: [Sg] and [Pl] both assert True.
Instance of the general phiPresup_same_assertion.
[Sg]'s presupposition is strictly stronger than [Pl]'s: dom(Sg) ⊂ dom(Pl). Given any non-atom, it witnesses the strict inclusion.
MP selects [Sg] over [Pl] via OT. When [Sg] (= .maximal) is
among the candidates and phiMP is the top-ranked constraint, the
optimal form has maximal presuppositional strength.
This is a direct application of phi_mp_selects_maximal. The
entity-level precondition is that [Sg]'s presupposition (atomicity)
is satisfied — see mp_blocks_plural_at_atom.
Maximize Presupposition for number: at an atomic referent,
[Sg] is defined (its presupposition is satisfied), making .maximal
a viable candidate. By mp_selects_sg, MP selects [Sg] over [Pl].
This is why "*John are here" is ungrammatical.
At a non-atomic referent, [Sg] is undefined — only [Pl] is available. This is why "The books were lying on the table" requires plural.
§4: Coordination Forces Plural #
For "Kai and Lina are playing": each conjunct is atomic (gets [Sg]), but their mereological sum is non-atomic. The φ-head above the coordination must bear [Pl] because [Sg]'s presupposition fails on the non-atomic sum.
A coordination of two distinct atoms produces a non-atom.
Each conjunct individually satisfies [Sg].
The coordination as a whole requires [Pl]: [Sg] fails on the sum, [Pl] is the only option.
§5: Decomposition of every #
Sauerland decomposes every into two morphemes:
- DER: the definite determiner — selects the maximal element of the
star-closed restrictor
*R(=AlgClosure R). - JE: a universal quantifier restricted to atoms (= mereological parts) of a group individual, with an existence presupposition.
⟦JE⟧(X)(P) is defined iff every atom of X is in the domain of P;
where defined, it asserts that P holds of every atom of X.
The [Sg] feature on φ within JE's scope restricts quantification to atoms — this is why every quantifies over atomic individuals.
Connection to QForall #
@cite{haslinger-etal-2025-nllt}'s unified universal Q_∀ and Sauerland's
JE ∘ DER compute the same truth conditions for singular (atomic) NP
restrictors: both reduce to ∀x[R(x) → Q(x)]. The derivation paths
differ — Sauerland goes through DER (maximal *R element) + JE (atoms
of the result), while Haslinger et al. go through maxNonOverlap on
the restrictor directly — but the results coincide. See
je_der_agrees_with_qforall.
JE (30b): universal quantifier over atoms of a group individual.
⟦JE⟧(X)(P):
- presupposes: every atom of X is in the domain of P (the scope predicate's presupposition projects universally — this derives presupposition projection under "every", cf. (33a–b))
- asserts: P holds of every atom of X
The atomicity restriction comes from the [Sg] feature in φ below JE.
The atom relation is mereological: a is an atom of X iff
Atom a ∧ a ≤ X.
Equations
- Sauerland2003.JE X P domP = { presup := fun (x : E) => ∀ (a : E), Mereology.Atom a → a ≤ X → domP a, assertion := fun (x : E) => ∀ (a : E), Mereology.Atom a → a ≤ X → P a }
Instances For
JE presupposes that the scope predicate is defined at every atom of X. This derives presupposition projection under "every": "Every boy invited his sister" presupposes each boy has a sister.
With a total scope predicate (no presupposition), JE's presupposition is trivially satisfied.
every = JE ∘ DER (30): Sauerland decomposes every into two
morphemes. DER (the definite article) selects the maximal element
of *R via Mereology.isMaximal. JE distributes over atoms of
the result.
⟦every⟧(R)(P) = ⟦JE⟧(max(*R))(P)
Equations
- Sauerland2003.everySem maxStarR P domP = Sauerland2003.JE maxStarR P domP
Instances For
DER is well-defined: *R is CUM, so it has at most one maximal
element (cum_maximal_unique). DER's uniqueness presupposition is
automatically satisfied for star-closed predicates.
JE's assertion = standard universal when the atoms below maxR
are exactly the R-elements (the normal case for atomic restrictors:
*R = R ∪ sums, and atoms of max(*R) = R-atoms).
This makes explicit what je_der_agrees_with_qforall leaves
implicit: JE's truth conditions at max(*R) are ∀x, R(x) → Q(x).
JE ∘ DER agrees with Q_∀ on atomic restrictors.
Both Sauerland's JE∘DER and @cite{haslinger-etal-2025-nllt}'s QForall
reduce to ∀x[R(x) → Q(x)] when the restrictor is atomic with no
overlap. Combined with je_assertion_eq_forall, this gives the full
equivalence chain:
JE(DER(R))(Q) ↔ ∀x, R(x) → Q(x) ↔ QForall R Q
Pl has no presupposition to project (33b).
"Every boy should invite his sisters" — the [Pl] on "sisters" has a vacuous presupposition, so JE's universal projection is trivially satisfied. The sentence is compatible with boys having different numbers of sisters, including just one (though MP implicates ≥ 2 for at least one).
§5b: Existential Quantifier ⟦a⟧ (§5, (35)) #
Indefinites project their scope's presupposition existentially:
⟦a⟧(R)(S) is defined iff ∃x: R(x) ∧ x ∈ domain(S) where defined: ⟦a⟧(R)(S) = 1 iff ∃x: R(x) ∧ S(x) = 1
Unlike JE (which projects universally), the indefinite only requires that some restrictor individual satisfy the scope presupposition. This is why "A boy invited his sister" presupposes there exists a boy with a sister, not that every boy has one.
⟦a⟧ (35): existential quantifier with existential projection.
The presupposition of ⟦a⟧(R)(S) is that some R-individual is in the domain of S. The assertion is that some R-individual satisfies S.
Equations
- Sauerland2003.aSem R S domS = { presup := fun (x : E) => ∃ (x : E), R x ∧ domS x, assertion := fun (x : E) => ∃ (x : E), R x ∧ S x }
Instances For
The existential projects presuppositions existentially: ⟦a⟧(R)(S) only requires some restrictor individual to be in dom(S), unlike JE which requires all atoms.
Universal vs existential projection contrast.
JE projects universally (every boy must have a sister). ⟦a⟧ projects existentially (some boy must have a sister).
This asymmetry follows from the quantificational force of the determiner, not from any property of the presupposition itself.
§6: Multiplicity Inference #
The "more than one" reading of bare plurals ("Emily fed giraffes" → more than one giraffe) arises because [Sg] and [Pl] are presuppositional alternatives with the same assertion. Using [Pl] when [Sg]'s presupposition would be satisfied violates Maximize Presupposition. So using [Pl] implicates that [Sg]'s presupposition fails — i.e., the referent is non-atomic (more than one).
On the modern (2020s) account (@cite{delpinal-bassi-sauerland-2024}),
this is derived via presuppositional exhaustification (pex): pex over
{⟦Sg⟧, ⟦Pl⟧} presupposes the negation of [Sg]'s presupposition (= ¬Atom).
See Theories/Semantics/Exhaustification/Presuppositional.lean.
Bridge to Phenomena.Plurals.Multiplicity #
This is the implicature theory of multiplicity
(Multiplicity.PluralTheory.implicature): plural literally means "one or
more," and "more than one" arises as a pragmatic inference — specifically,
an inference from presuppositional competition (Maximize Presupposition),
not from at-issue scalar competition (Horn scales).
Presuppositional vs at-issue competition #
Alternatives.Number.numberScale models sg/pl as a Horn scale where
singular is the "stronger" alternative. But this is misleading on
Sauerland's analysis: [Sg] and [Pl] have identical at-issue content
(both are identity functions — sg_pl_same_assertion). There is no
at-issue strength difference. The competition is entirely in the
presuppositional dimension: [Sg] has the stronger presupposition
(atomicity), while [Pl]'s is vacuous.
This means number competition falls under violatesMP (presuppositional
competition: same assertion, different presupposition strength), NOT
under violatesConversationalPrinciple (at-issue competition: different
truth conditions). Both operators are defined in
Theories/Semantics/Alternatives/Structural.lean, and the distinction
between them is exactly Sauerland's theoretical contribution.
The competition between [Sg] and [Pl] is presuppositional, not at-issue: they share their assertive content but differ in presupposition. This is the structural signature of Maximize Presupposition (same assertion + strictly ordered presuppositions), as opposed to scalar implicature (different assertions).
The same-assertion condition follows from phiPresup_same_assertion;
the strength ordering is presupStrength .maximal > presupStrength .minimal
(= 2 > 0). Together with mp_selects_sg, this shows number
competition is governed by phiMP, not by at-issue scalar
implicature.
§6b: Negated Indefinites #
(41a) "Lina didn't harvest a tomato" / "Lina harvested no tomato" (41b) "Lina didn't harvest tomatoes" / "Lina harvested no tomatoes"
(41b) entails (41a): if Lina didn't harvest any tomatoes (Pl), she certainly didn't harvest a single tomato (Sg). This follows from the weak semantics of [Pl]: its assertion includes atomic referents.
The entailment goes: Pl's assertion is ¬∃x [*tomato](x) ∧ harvest(x).
Sg's assertion is ¬∃x atom(x) ∧ [*tomato](x) ∧ harvest(x).
Since atoms are in *tomato, the Pl assertion is strictly stronger.
Crucially, adding ¬atom(x) to the assertion of (41b) — which would block the entailment — is ruled out because the negation of the Sg presupposition is itself negated (the two premises of the derivation are the Sg presupposition and the Pl assertion, but the assertion is negated, blocking the reasoning from (40)).
Pl negation entails Sg negation (41).
If no plurality of tomatoes was harvested, then no atomic tomato was harvested either. This follows from the inclusive semantics of [Pl] (which covers atoms) and holds because [Pl]'s assertion subsumes [Sg]'s assertion.
§7: Gender Agreement in Czech Coordinations (§6, (45)) #
Czech gender agreement on the verb is determined by the entire coordination, not just one conjunct. This parallels number: the φ-head above the coordination must have a gender feature whose presupposition is satisfied by the sum of the conjuncts.
- (45a) Jan a Petr šli (went-masc-pl): both male → masc ✓
- (45b) Věra a Barbara šly (went-fem-pl): both female → fem ✓
- (45c) Jan a Věra šli/*šly (went-masc-pl, NOT fem): mixed gender → masc
- (45d) Matka a její dítě šly (went-fem-pl): mother + neut child → fem
- (45e) Otec a jeho dítě šli (went-masc-pl): father + neut child → masc
Sauerland's analysis: masculine is vacuous (minimal cell), feminine
presupposes non-masculinity (intermediate), neuter presupposes
inanimate/genderless (maximal). These are precisely the presuppositional
denotations mascSem, femSem, neutSem from PhiFeatures.lean §3b.
The Czech facts follow from Maximize Presupposition over the sum:
- (45c): The sum includes a male referent (Jan), so fem's presupposition (non-masculine) fails → only masc (vacuous) is available.
- (45d): The sum includes a female referent and an inanimate referent. Neut presupposition (all inanimate) fails, but fem presupposition (all non-masculine) succeeds → MP selects fem.
A referent's sex/animacy category (for Czech gender agreement).
- male : ReferentGender
- female : ReferentGender
- inanimate : ReferentGender
Instances For
Equations
- Sauerland2003.instDecidableEqReferentGender x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
isNonMasculine predicate: inanimate → non-masculine (containment).
(45c): Jan (male) + Věra (female) → fem FAILS. The sum includes a male referent, so femSem's presupposition (all non-masculine) is not satisfied. Only mascSem (vacuous) works.
(45d): Matka (female) + dítě (inanimate) → neut FAILS, fem SUCCEEDS. Neut presupposition (all inanimate) fails because mother is not inanimate. Fem presupposition (non-masculine) succeeds: mother is female and child is inanimate (hence non-masculine).
(45e): Otec (male) + dítě (inanimate) → only masc. Both neut and fem fail on the male referent.
Gender resolution parallels number resolution.
Just as [Sg] fails on a coordination of two distinct atoms (→ [Pl]), [Fem] fails on a coordination containing a male referent (→ [Masc]). Both follow from the same mechanism: the presupposition of the more specified feature is not satisfied by the sum, so Maximize Presupposition selects the less specified (vacuous) alternative.
§8: Politeness Prediction #
Sauerland §2.2 predicts that polite address forms should recruit the less specified phi-feature values — precisely those with weaker (or vacuous) presuppositions. The prediction: as polite forms replacing 2nd person singular, only 2nd/3rd person plural and 3rd person singular should be possible. German uses 3rd person plural "Sie" (14a) and historically 2nd person plural "Ihr".
This prediction follows from the general principle that vacuous
presuppositions (specLevel 0) are compatible with any context —
they impose no domain restriction and hence no falsehood risk.
The formal content of this prediction is captured by
isSemanticUnmarked in PhiFeatures.lean §5 and verified by
@cite{wang-r-2023} (see Phenomena/Politeness/Studies/Wang2023.lean),
which proves that all three cross-linguistic honorific strategies
(plural, 3rd person, indefinite) recruit the minimal (unmarked) cell.
The three semantically unmarked phi-values correspond to the three cross-linguistic politeness strategies.
The three semantically marked phi-values are NOT used for politeness — their presuppositions would impose unwanted restrictions on the addressee.