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On the variable strength of only

Crnič, L. (2026). Semantics of Natural Languages, 1(1), 1–46

Source Bibliography Paper

Crnič analyses only as two operators over focus alternatives. Only asserts that no relevant innocently excludable alternative holds, and a presuppositional exh above it presupposes that the excludable alternatives of the only-sentence are false, so the positive inference of only varies with the domain of exh. Unpruned, the sentence is Horn's entry; pruned to one alternative outside the domain of only, it is the weak entry. For an atomic prejacent only the full domain satisfies the Weakening Constraint, so pruning surfaces where the full domain leaves the assertion uninformative. Unpruned, the analysis of exceptives is von Fintel's least exception set.

Implementation notes

  • A world is the set of atoms that hold, the guests at the party or the ranks Gal holds, so the alternative that every atom of the plurality X holds is the interval Set.Ici X. Focus alternatives and resource domains are sets of pluralities, and the domain of exh is the set of pluralities its alternatives are built on.
  • only and exhOnly exclude innocently, through Exhaustification.excludesIE. In these models the prejacent's own world falsifies every alternative the prejacent does not entail, so innocent exclusion is exclusion (Exhaustification.excludesIE_eq_excludes).
  • pex is the partial entry (9), undefined where its presupposition fails. Under the Strong Kleene Exhaustification.pexIE of Bassi et al. 2021, which is false wherever its prejacent is, Not only Gal was at the party would not presuppose that Gal was there.
  • The Weakening Constraint ranges over domains of excludable alternatives, as the resolutions (41b) and (50b) are written. Read literally, (56) compares a domain with every larger set of alternatives, and a domain lacking a non-excludable alternative fails it, since adding one changes nothing (exhOnly_inter_excludable).
  • Relevance over the context is Alternatives.IsAllowableRestriction over the subtype of the context, with each only-alternative restricted to it by preimage (onlyCtx).
  • The ranking (58) of Informativity, Presupposition.PostAccommodationInformative, over Weakening is read off the constraints each domain satisfies; the accommodation operator of (56) is posSupport.

Errata

  • (6b) and (32c) give the excludable alternatives of Gal was at the party as those about individuals disjoint from Gal. With the plural alternatives of (12), every alternative other than the prejacent is excludable, Gal and Tal were at the party among them (isInnocentlyExcludable_Ici_iff, isInnocentlyExcludable_galTal); the exhaustive inference is unaffected.
  • (110b) fails with the plural alternatives of (12), on which the strong presupposition (13) rests; Crnič 2026a shows that singular ones give a weak presupposition. Someone but Gal has the excludable alternative someone but everyone except Gal, that Gal was there (isInnocentlyExcludable_but_some), so exh in (109) is not vacuous and the Economy Constraint (108) does not exclude (107) (judgment_107).
  • (40), (56) and (58) do not single out the resolution (41) of Gal is only a grad student: pruning to only [postdoc] satisfies and violates the same constraints, and under negation conveys that Gal is a professor (pruned_postdoc). The closure of relevance under the Boolean operations, set aside in fn. 3, does single it out, but only computed over the context: there (41) is the one informative pruning that is an allowable restriction in the sense of Fox & Katzir 2011 (isAllowableRestriction_onlyCtx_iff), while over all worlds pruning to only [non-academic] and only [postdoc] is allowable too (isAllowableRestriction_only_ranks).

TODO

  • The sufficiency modal constructions of §3.4, the embedded exhaustification of §3.5, the integrated rivals of §4, the conjunctive closure of fn. 4, and the exceptives of §5.2–5.3.

References

Pluralities and the distributed only

L98–101
L105–109
L127–129
L131–132
L138
L140–142
L154–160
L162–164
L166–168
L170–172
L179–185
L188–196

Without pruning, exh over the only-sentence presupposes the prejacent, and the sentence is the entry of Horn 1969 for only on the entailment scale, (13).

theorem exhOnly_eq_onlyL198–207
{α : Type u_1} {F C : Set (Finset α)} {X : Finset α} [Fintype α] [DecidableEq α] (hX : X ∈ C) (hC : C ⊆ F) (hsingle : ∀ x ∈ X, {x} ∈ C) (hco : ∀ x ∈ X, Finset.univ.erase x ∈ F) :
exhOnly F C F X = Focus.only (fun (x1 x2 : Set (Finset α)) => x1 ⊆ x2) (Set.Ici '' C) id (Set.Ici X)

Built from

Pruning the domain of exh to one alternative outside the domain of only leaves the presupposition that some alternative in the domain of only holds, the weak entry of Horn 1996, (44), (54), (82).

theorem exhOnly_singleton_eqL210–223
{α : Type u_1} {F C : Set (Finset α)} {X Y : Finset α} (hX : X ∈ C) (hC : C ⊆ F) (hY : Y ∈ F) (h : ∀ Z ∈ C, ¬Z ⊆ Y) :
exhOnly F C {Y} X = (⋃₀ (Set.Ici '' C)).indicator fun (w : Finset α) => SignType.ofProp (w ∈ Exhaustification.excludes (Set.Ici '' C) (Set.Ici X))

Built from

The Weakening Constraint

L234–243

For an atomic prejacent every nonempty domain of excludable alternatives conveys the same after accommodation, so the Weakening Constraint admits the full domain alone, (55), (57).

theorem weakening_iffL245–252
{α : Type u_1} {F C : Set (Finset α)} (R : Set (Finset α)) {x : α} (hR : R.Nonempty) (hRE : R ⊆ excludable F C {x}) :
Weakening F C R {x} ↔ R = excludable F C {x}

Built from ,

Exceptives

L268–274

Without pruning, exh over no A but X is true exactly where X is the exception set of Von Fintel 1993, given that X lies within the restrictor, (102)–(106), (114).

theorem pex_but_no_eq_one_iffL276–294
{α : Type u_1} {F : Set (Finset α)} {A X w : Finset α} [Fintype α] [DecidableEq α] (hF : ∀ (Z : Finset α), Z.Nonempty → Z ∈ F) (hXA : X ⊆ A) (hco : ∀ x ∈ X, (Finset.univ.erase x).Nonempty) :
pex F F (but Quantifier.Det.no A) X w = 1 ↔ Quantifier.Exceptive.IsExceptionSet Quantifier.Det.no ↑A ↑X ↑w

Built from ,

L300–305

The party

The academic ladder

L407–410
L428–429

For Gal is only a grad student, the presupposition of the unpruned domain entails the assertion, (39), (40), while pruning to only [non-academic] leaves the presupposition that Gal is an academic, which does not, at the cost of the Weakening Constraint, (41)–(44).

theorem table_2Table 2L448–465
:
¬Presupposition.PostAccommodationInformative oneRank (exhOnly ranks ladder ranks {Rank.grad}) ∧ Presupposition.PostAccommodationInformative oneRank (exhOnly ranks ladder {{Rank.nonAcademic}} {Rank.grad}) ∧ ¬Weakening ranks ladder {{Rank.nonAcademic}} {Rank.grad}

Built from , , , ,

Relevance closed under the Boolean operations