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Variable-Force Modality in English Infinitival Relatives: A Matter of Degree

Grano, T. (2026). Linguistic Inquiry, 57(4), 711–743

An infinitival relative such as a book for you to read has a deontic modal whose force varies between possibility (could) and weak necessity (should), and the possibility reading needs a weakly interpreted weak quantifier: there is a book for you to read has both readings, every book for you to read only the second (the determiner-modal generalization, §2). Grano puts possibility and weak necessity on one deontic scale and gives the relative a silent gradable modal, a counterpart of should relating a degree to the measure of a proposition (31). The positive morpheme binds its degree argument at should's threshold, giving weak necessity (33), and existential closure strengthened above the scale's least degree gives possibility (48). By Diesing's Mapping Hypothesis existential closure applies only inside the verb phrase, where only weakly interpreted weak quantifiers are interpreted (§5), and Gen or an adverb of quantification binding the degree in its restriction has the same effect (§6). Overt should has dedicated possibility modals beside it, which block existential closure (§4.5).

Here a binder sets a standard on the deontic scale, the positive morpheme that of weak necessity and the other two that of possibility. On a scale on which two propositions meeting should's threshold are jointly permissible, two cats meet a force's standard consistently with you may adopt at most one cat exactly when the force is possibility, and the judgments of (8)–(10) follow from the readings the binders at each position give.

Implementation notes

The paper derives the oddness of (9) from agglomeration, □p ∧ □q implying □(p ∧ q) (p. 716). The degree semantics (33) does not deliver agglomeration, and a scale whose measure of a conjunction is the lesser measure, which delivers it at every threshold, delivers it for possibility too, so that (8) would be odd as well (not_atMostOne_of_inf). The argument needs only that two propositions meeting the weak necessity threshold be jointly permissible (JointlyPermissible), which a counting measure with a threshold above half has (countingModel). Gen and the adverbs of quantification are universal binders, as in (63).

Errata

(72) binds the degree outside Gen with = d, so every relevant firefighter's measure is the same degree (eq72_uniform). Its gloss, "meets or exceeds d", is the reading the text describes and is strictly weaker (eq72_gloss_weaker). The glosses of (62) and (73) also read = d as "meets or exceeds d", a difference a universal or existential binder does not notice but a proportional one would; with = d the pairs counted are as many as the individuals (ncard_pairs).

TODO

The overgeneration of (73), matrix existential closure binding the subject of the relative, which the paper leaves open (§6.2).
The adjacency account of (35), the comparatives (39)–(44), and predicate infinitives (fn. 22).

References

Grano 2026, Variable-Force Modality in English Infinitival Relatives: A Matter of Degree
Diesing 1992, Indefinites
Deal 2011, Modals Without Scales formalized
Kennedy 2007, Vagueness and Grammar: The Semantics of Relative and Absolute Gradable Adjectives formalized
Lassiter 2017, Graded Modality: Qualitative and Quantitative Perspectives

Binders of the degree argument (§4, §6)

L80–82

L85–87

Existential closure above the least degree only just escapes triviality: the standard of possibility covers the one every degree meets (p. 729).

theorem closure_covBy L89–93

{D : Type u_2} [LinearOrder D] [OrderBot D] {θ θ' : D} :

⊥ ⋖ Modality.ModalForce.standard θ θ' Modality.ModalForce.possibility

An unselective universal binding the degree in its restriction, (62), is local existential closure, (63)–(65).

theorem unselective_iff L95–100

{α : Type u_1} {D : Type u_2} [LinearOrder D] [OrderBot D] {θ θ' : D} (μ : α → D) (A B : α → Prop) :

(∀ (x : α) (d : D), A x ∧ μ x = d ∧ ⊥ < d → B x) ↔ ∀ (x : α), A x ∧ μ x ∈ Modality.ModalForce.standard θ θ' Modality.ModalForce.possibility → B x

L103–114

L118–124

L126–130

The preamble of (8)–(10)

Two cats meet a force's standard consistently with the preamble, on a counting scale where propositions meeting the weak necessity threshold are jointly permissible, exactly when the force is possibility (p. 716) ((8), (10)).

theorem consistent_iff L239–247

(f : Modality.ModalForce) :

(∃ (θ : ℕ), JointlyPermissible count θ ∧ AtMostOne count adoptCat ∧ TwoMeet count adoptCat (Modality.ModalForce.standard θ θ f)) ↔ f = Modality.ModalForce.possibility

Built from , , , ,

The judgments

The continuations of (8)–(10) are judged as the readings of their modals predict.

theorem cat_rows L276–280

(e : Datum) :

e ∈ List.filter (fun (x : Datum) => x.readings.isEmpty) Examples.all → ∀ m ∈ e.parse? "modal" modalTable, ∀ ps ∈ e.parse? "interpretation" interpretationTable, e.judgment.band = band m ps

Built from , ,

A reading of (1), (2), (61), (66) or (67) is available exactly when a binder available at the nominal's position gives it.

theorem reading_rows L282–288

(e : Datum) :

e ∈ Examples.all → ∀ m ∈ e.parse? "modal" modalTable, ∀ ps ∈ e.parse? "interpretation" interpretationTable, ∀ r ∈ e.readings, ∀ f ∈ List.lookup r.1 forceTable, r.2 = Judgment.acceptable ↔ ∃ p ∈ ps, f ∈ readings m p

Built from , , ,

Interpretive Economy (§7)

The silent modal's scale has a least degree, impermissibility (p. 728), so Interpretive Economy admits no contextual standard on it: the relative standard of the positive morpheme is a counterexample, as likely is (Kennedy 2007, Lassiter 2017) (§7).

theorem not_admits_contextual L292–298

{D : Type u_1} [LinearOrder D] [OrderBot D] :

¬(Degree.Boundedness.ofOrder D).Admits Degree.PositiveStandard.contextual

Errata: (72)

L306–309

L312–315