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Atomic Plurals: Partee’s Puzzle and the Decomposition of Grammatical Number

Minor, S. (2026). Journal of Semantics

Source Bibliography Paper

Bare plurals take narrowest scope (Miles wants to meet policemen), but in the scope of a plural licensor they can scope out of an intensional context (All the boys want to meet policemen), as Partee observed. Minor splits the singular into ATOM, atomicity at every assignment, and UNIQUE, one value across the state. The plural form spells out either neither feature, a concept that takes no scope, or ATOM alone, an atomic plural, which is an indefinite and can scope. Exh against the singular leaves an atomic plural non-unique, which no state reached from one assignment satisfies unless a weakly distributive licensor has split it. Minor's Plural Compositional Logic is the static counterpart of the dynamic system of Studies/Minor2022: a formula holds of a state where the corresponding update succeeds.

Main definitions

Implementation notes

  • Intensionalized uniqueness unique_w(y) (103b) is Väänänen's dependence atom Team.dep (val w) (val y), which holds when assignments agreeing on the world w agree on y.
  • Events are folded into the lexical relations, as in Studies/Minor2022, and the maximal sum of worlds compatible with the experiencer's wishes (100) is a function bul of the world and the experiencer.
  • The exhaustified atomic plural is atomicity and non-uniqueness, the consequence (121) of its strengthened meaning that fn. 29 states.
  • All the boys is weak distributivity over the sum of the boys, as in Studies/Minor2022.

TODO

  • The narrow-scope readings of (93), the singular and plural under want ((106), (114)), and the narrow-scope dependent plural (124), (132).
  • The referent-set reformulation of §9, which drops the Minimal Context Conjecture (143).

References

  • Minor 2026, Atomic Plurals: Partee’s Puzzle and the Decomposition of Grammatical Number
  • Minor 2022, Dependent Plurals and Three Levels of Multiplicity formalized
  • Magri 2009, A theory of individual-level predicates based on blind mandatory scalar implicatures formalized
  • Väänänen 2007, Dependence Logic: A New Approach to Independence Friendly Logic

Decomposing number (§6.1)

By the Subset Principle (67), the singular spells out only the full head (68), and /s/ spells out both the atomic and the plain plural ((71), (73a)).

theorem spellOutL109–115
:
DistributedMorphology.subsetPrinciple exponents ↑{NumFeature.num, NumFeature.atom, NumFeature.unique} = some Exponent.null ∧ DistributedMorphology.subsetPrinciple exponents ↑{NumFeature.num, NumFeature.atom} = some Exponent.s ∧ DistributedMorphology.subsetPrinciple exponents ↑{NumFeature.num} = some Exponent.s

Built from

L129–135

The head [#, ATOM, UNIQUE] denotes instantiation with the singular's atomicity and uniqueness (74), and the atomic plural [#, ATOM] the same without uniqueness.

theorem denote_headsL137–144
{R S E : Type w} [DynamicSemantics.RegisterStructure R S E] [PartialOrder E] (P : E → Prop) (u : R) :
NumFeature.unique.denote (NumFeature.atom.denote (NumFeature.num.denote P)) u = Minor2022.lex P u ∩ (Minor2022.atomic u ∩ Minor2022.unique u) ∧ NumFeature.atom.denote (NumFeature.num.denote P) u = Minor2022.lex P u ∩ Minor2022.atomic u

Built from

Intensionalized uniqueness (§7.2)

A referent required not to depend on w cannot be introduced where w distinguishes the assignments, so an exhaustified atomic plural fails there ((113)–(121)).

theorem notMem_dom_intro_test_of_injOnL162–167
{R : Type u_1} {S : Type u_2} {E : Type u_3} [DynamicSemantics.RegisterStructure R S E] {w y : R} {H : Set S} {C : DynamicSemantics.Condition (Set S)} {D : DynamicSemantics.Update (Set S)} (hwy : w ≠ y) (hH : Set.InjOn (DynamicSemantics.RegisterStructure.val w) H) (hC : C ⊆ (Team.dep (DynamicSemantics.RegisterStructure.val w) (DynamicSemantics.RegisterStructure.val y))ᶜ) :
H ∉ ((SetRel.comp (Minor2022.intro y) (DynamicSemantics.Update.test C)).comp D).dom

Atomic plurals and Exh (§6.2)

Exh above a VP with an atomic-plural object negates the singular alternative (79), leaving non-uniqueness (87).

theorem exh_atomicPluralL203–211
{R : Type u_1} {S : Type u_2} {E : Type u_3} [DynamicSemantics.RegisterStructure R S E] (u u' : R) [PartialOrder E] {a b : E} (ha : Mereology.Atom a) (hb : Mereology.Atom b) (hab : a ≠ b) (i : S) :
Exhaustification.exh {Minor2022.acrossModels fun (M : Minor2022.Model E) => (SetRel.comp (Minor2022.vp u u' M) (DynamicSemantics.Update.test (Minor2022.atomic u'))).comp (DynamicSemantics.Update.test (Minor2022.unique u'))} (Minor2022.acrossModels fun (M : Minor2022.Model E) => SetRel.comp (Minor2022.vp u u' M) (DynamicSemantics.Update.test (Minor2022.atomic u'))) = Minor2022.acrossModels fun (M : Minor2022.Model E) => (SetRel.comp (Minor2022.vp u u' M) (DynamicSemantics.Update.test (Minor2022.atomic u'))).comp (DynamicSemantics.Update.test (Minor2022.unique u')ᶜ)

The non-uniqueness implicature of an atomic plural is mandatory under any relevance property obeying Magri 2009's postulates (§6.2).

theorem disjoint_exhR_atomicPluralL214–235
{R : Type u_1} {S : Type u_2} {E : Type u_3} [DynamicSemantics.RegisterStructure R S E] (u u' : R) [PartialOrder E] {a b : E} (ha : Mereology.Atom a) (hb : Mereology.Atom b) (hab : a ≠ b) (i : S) {Rel : Set (Minor2022.Model E × Set S × Set S) → Prop} (hR : Magri2009.IsRelevance {w : Minor2022.Model E × Set S × Set S | w.2.1.Subsingleton} (Minor2022.acrossModels fun (M : Minor2022.Model E) => SetRel.comp (Minor2022.vp u u' M) (DynamicSemantics.Update.test (Minor2022.atomic u'))) Rel) :
Disjoint (Magri2009.exhR Rel {Minor2022.acrossModels fun (M : Minor2022.Model E) => SetRel.comp (Minor2022.vp u u' M) (DynamicSemantics.Update.test (Minor2022.atomic u')), Minor2022.acrossModels fun (M : Minor2022.Model E) => (SetRel.comp (Minor2022.vp u u' M) (DynamicSemantics.Update.test (Minor2022.atomic u'))).comp (DynamicSemantics.Update.test (Minor2022.unique u'))} (Minor2022.acrossModels fun (M : Minor2022.Model E) => SetRel.comp (Minor2022.vp u u' M) (DynamicSemantics.Update.test (Minor2022.atomic u')))) {w : Minor2022.Model E × Set S × Set S | w.2.1.Subsingleton}

Without a weakly distributive licensor an atomic plural is contradictory (88), since from a single assignment the state stays a single assignment, where uniqueness holds.

theorem notMem_dom_sentence_atomicPluralL237–242
{R : Type u_1} {S : Type u_2} {E : Type u_3} [DynamicSemantics.RegisterStructure R S E] {u u' : R} {i : S} {P : E → Prop} {M : Minor2022.Model E} [PartialOrder E] (huu' : u ≠ u') :
{i} ∉ SetRel.dom (Minor2022.sentence u u' P id M (Minor2022.atomic u' ∩ (Minor2022.unique u')ᶜ))

Under a weakly distributive licensor an atomic plural is co-distributive (90), each atomic part of the subject carrying a single box, not all the same.

theorem atomicPlural_distW_iffL252–265
{R : Type u_1} {S : Type u_2} {E : Type u_3} [DynamicSemantics.RegisterStructure R S E] {u u' : R} {i : S} {P : E → Prop} {M : Minor2022.Model E} [PartialOrder E] (huu' : u ≠ u') :
{i} ∈ SetRel.dom (Minor2022.sentence u u' P (Minor2022.distW u) M (Minor2022.atomic u' ∩ (Minor2022.unique u')ᶜ)) ↔ ∃ (x : E), P x ∧ ∃ (b : E → E), (Mereology.atomsBelow x).Nonempty ∧ (∀ a ∈ Mereology.atomsBelow x, M.noun (b a) ∧ M.verb a (b a)) ∧ (∀ a ∈ Mereology.atomsBelow x, Mereology.Atom (b a)) ∧ ¬(b '' Mereology.atomsBelow x).Subsingleton

Attitudes and Partee's generalization (§7)

L272–276
L285–289

With a flat complement want is Hintikka's, holding of a state exactly when the complement holds at every world compatible with each experiencer's wishes.

theorem mem_dom_want_test_iffL291–317
{R : Type u_1} {S : Type u_2} {E : Type u_3} [DynamicSemantics.RegisterStructure R S E] {w w' x : R} {J : Set S} [PartialOrder E] {bul : E → E → E} (hxw' : x ≠ w') (hww' : w ≠ w') {Q : Set S} :
J ∈ SetRel.dom (want w w' x bul (DynamicSemantics.Update.test (𝒫 Q))) ↔ ∀ j ∈ J, ∀ v ∈ Mereology.atomsBelow (bul (DynamicSemantics.RegisterStructure.val w j) (DynamicSemantics.RegisterStructure.val x j)), DynamicSemantics.RegisterStructure.extend j w' v ∈ Q

Built from

L319–331
L334–339

On the wide dependent plural reading of All the boys want to meet policemen ((133), (134)), each boy has an atomic policeman he wants to meet, not all the same (§8).

theorem wideScope_iffL348–379
{R : Type u_1} {S : Type u_2} {E : Type u_3} [DynamicSemantics.RegisterStructure R S E] {w w' x y : R} {i : S} [PartialOrder E] (hxy : x ≠ y) (hwx : w ≠ x) (hwy : w ≠ y) (hxw' : x ≠ w') (hww' : w ≠ w') (hyw' : y ≠ w') (M : IModel E) (P : E → Prop) :
{i} ∈ SetRel.dom (wide w w' x y M P (Minor2022.distW x) (Minor2022.atomic y ∩ (Team.dep (DynamicSemantics.RegisterStructure.val w) (DynamicSemantics.RegisterStructure.val y))ᶜ)) ↔ ∃ (s : E), P s ∧ ∃ (b : E → E), (∀ a ∈ Mereology.atomsBelow s, M.inst (DynamicSemantics.RegisterStructure.val w i) (b a) ∧ Mereology.Atom (b a) ∧ ∀ v ∈ Mereology.atomsBelow (M.bul (DynamicSemantics.RegisterStructure.val w i) a), M.meet v a (b a)) ∧ ¬(b '' Mereology.atomsBelow s).Subsingleton

Built from ,

Partee's paradigm (93)

L383–385
L387–394
L396–403

A wide-scope reading of (93) is available exactly when satisfiable, as for the singular (109) and dependent plural (133) but not the plain plural (117), matching Partee's generalization (36).

theorem parteeL405–443
:
(List.lookup "∃ > want" Examples.ex_93a.readings = some Judgment.acceptable ↔ ∃ (M : IModel (Finset Thing)) (i : Dref → Finset Thing), {i} ∈ SetRel.dom (wide Dref.world Dref.bul Dref.subj Dref.obj M (fun (x : Finset Thing) => x = M.miles) id (Minor2022.atomic Dref.obj ∩ Team.dep (DynamicSemantics.RegisterStructure.val Dref.world) (DynamicSemantics.RegisterStructure.val Dref.obj)))) ∧ (List.lookup "∃ > want" Examples.ex_93b.readings = some Judgment.acceptable ↔ ∃ (M : IModel (Finset Thing)) (i : Dref → Finset Thing), {i} ∈ SetRel.dom (wide Dref.world Dref.bul Dref.subj Dref.obj M (fun (x : Finset Thing) => x = M.miles) id (Minor2022.atomic Dref.obj ∩ (Team.dep (DynamicSemantics.RegisterStructure.val Dref.world) (DynamicSemantics.RegisterStructure.val Dref.obj))ᶜ))) ∧ (List.lookup "∃ > want" Examples.ex_93c.readings = some Judgment.acceptable ↔ ∃ (M : IModel (Finset Thing)) (i : Dref → Finset Thing), {i} ∈ SetRel.dom (wide Dref.world Dref.bul Dref.subj Dref.obj M (fun (x : Finset Thing) => x = M.boys) (Minor2022.distW Dref.subj) (Minor2022.atomic Dref.obj ∩ (Team.dep (DynamicSemantics.RegisterStructure.val Dref.world) (DynamicSemantics.RegisterStructure.val Dref.obj))ᶜ)))

Built from , , ,