On Necessity and Comparison #
Supports [Slo70]'s proposal that a comparative semantics is the defining property of weak necessity. Weak necessity modals (ought, should) and evaluative comparative predicates (good, better, preferable, worthwhile) form a semantic natural class: both are relativized to negotiable ideals — priorities not endorsed by all discourse participants.
Rubinstein splits [Kra81]/[Kra91] ordering-source material into two
kinds. Non-negotiable priorities are promoted to modal-base status, restricting
the favored worlds via [Fra96]'s compatibility-restricted union; the
remaining negotiable priorities stay as an ordering source. Strong necessity
quantifies over all favored worlds (no ordering, [Fra96]'s non-comparative
analysis); weak necessity over the best favored worlds, ranked by the negotiable
ideals — the comparative component. The reduction weakNecessityR ≅ necessity ties
this back to [vFI08]-style Kratzer semantics.
Main definitions #
PriorityTypology— modal base split into circumstances, non-negotiable priorities, and a negotiable ordering source.favoredWorlds,strongNecessityR,weakNecessityR— the favored-worlds set and the strong/weak necessity operators over it.WeakNecessityStrategy— the three crosslinguistic routes to weak necessity.
Main results #
strong_entails_weak_R,weak_not_entails_strong_R— the strong/weak asymmetry.strongR_eq_simpleNecessity,weakR_eq_necessity— reductions to Kratzer semantics when no priorities are promoted.should_to_haveto_shift— the tax-report promotion (§3.3): the same prejacent shifts from weak-only to strong necessity when a negotiable ideal is endorsed.negRaising_iff_fragment_weak,evaluatives_neg_raise— neg-raising tracks the English fragment's weak-necessity marking ([Hor78] diagnostics).hebrew_strategy_evaluative,weak_rarity— Hebrew expresses weak necessity only through evaluative comparison; only 62/200 languages grammaticalize it ([Nar12]).
Empirical stimuli (Hebrew, Spanish, English diagnostics) live as typed
LinguisticExample rows in Data.Examples.Rubinstein2014; theorems quantify over
their judgment/readings/paperFeatures.
Priority reconceptualization (§3.2) #
Rubinstein's reconceptualized modal backgrounds (§3.2): [Kra81]'s single ordering source is split by negotiability, with the non-negotiable part promoted to modal-base status.
- circumstances : Semantics.Modality.Kratzer.ModalBase World
Factual circumstances: the Kratzer modal base f(e).
- nonNegotiable : Semantics.Modality.Kratzer.ModalBase World
Non-negotiable priorities h(e): endorsed by all participants, promoted to modal-base status.
- negotiable : Semantics.Modality.Kratzer.OrderingSource World
Negotiable priorities g(e): not endorsed by all participants (§3.3, def 49), promoted by an opinionated assessor; remain as ordering source.
Instances For
Favored worlds (§3.2, definitions 39–40) #
Favored worlds (def 40), consistent case: worlds satisfying both the circumstances f(w) and the non-negotiable priorities h(w). This is the intersection to which [Fra96]'s compatibility-restricted union (def 39) reduces when h(w) is consistent with f(w) — the case the paper's examples use.
Equations
- Rubinstein2014.favoredWorlds pt w = Intensional.Premise.propIntersection (pt.circumstances w ++ pt.nonNegotiable w)
Instances For
With no non-negotiable priorities, favored worlds are the standard Kratzer accessible worlds.
Strong and weak necessity (§3.2, definitions 41–42) #
Strong necessity (def 41): universal quantification over the favored worlds. No ordering source is consulted, so strong necessity is non-comparative.
Equations
- Rubinstein2014.strongNecessityR pt p w = ∀ w' ∈ Rubinstein2014.favoredWorlds pt w, p w'
Instances For
Weak necessity (def 42): universal quantification over the best favored worlds, ranked by the negotiable ordering source g(e) — the comparative component.
Equations
- Rubinstein2014.weakNecessityR pt p w = ∀ w' ∈ Semantics.Modality.Kratzer.bestAmong (Rubinstein2014.favoredWorlds pt w) (pt.negotiable w), p w'
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Strong necessity entails weak necessity (§1) #
Strong necessity entails weak necessity, since BEST(Fav, g) ⊆ Fav
(bestAmong_sub). Parallel to Directive.strong_entails_weak.
Equations
The converse fails: weak necessity does NOT entail strong necessity. If p holds at all BEST favored worlds but not at all favored worlds, weak necessity holds but strong necessity does not.
Concretely: with circumstances = nonNegotiable = ∅ and
negotiable = [λw => w = w₁], we have
favoredWorlds ce_pt w₀ = Set.univ and
bestAmong univ [λw => w = w₁] = {w₁}. Thus ce_p (which says
w = w₁) holds at all best worlds but not at all favored worlds.
Reduction to standard Kratzer semantics (bridge to Directive.lean) #
When no priorities are promoted to modal-base status (h = ∅), Rubinstein's strong necessity reduces to simple Kratzer necessity (no ordering), and her weak necessity reduces to standard Kratzer necessity with the negotiable ordering source.
With no promoted priorities, Rubinstein's strong necessity equals simple Kratzer necessity (no ordering).
With no promoted priorities, Rubinstein's weak necessity equals standard Kratzer necessity under the negotiable ordering.
This is not the same as Directive.weakNecessity — Rubinstein's
"weak" with h=∅ equals Directive's "strong" (with g). The analyses
differ structurally: Directive treats all priorities as ordering-source
material; Rubinstein promotes some to modal-base status.
When no priorities are promoted AND no negotiable ordering exists, strong and weak necessity coincide (both = simple necessity).
The tax report scenario (§3.3, examples 45–47, 51) #
The paper's central worked example (§3.3): An accountant says "We should report all our revenue" — promoting a negotiable ideal about international revenue. The law about domestic revenue is non-negotiable. Later, if the manager endorses the ideal, the international-revenue clause is promoted to non-negotiable status, making "We have to report all our revenue" appropriate.
We model this with two propositions:
- reportDomestic: a non-negotiable legal obligation (in h)
- reportInternational: a negotiable ideal promoted by the speaker (in g)
- reportAll: the conjunction (the prejacent of should/have-to)
Equations
- Rubinstein2014.instDecidablePredWorldReportDomestic = id inferInstance
Equations
- Rubinstein2014.instDecidablePredWorldReportInternational = id inferInstance
Equations
- Rubinstein2014.instDecidablePredWorldReportAll = id (id (id inferInstance))
In scenario A, weak necessity holds: all BEST favored worlds satisfy reportAll (the ordering picks out worlds where international revenue is also reported).
The single negotiable ideal reportInternational holds at w₀ (which is
in favored worlds and satisfies all of reportInternational), so any
"best" favored world must also satisfy it. The only favored world
satisfying both is w₀, so reportAll holds at all best favored worlds.
In scenario A, strong necessity FAILS: not all favored worlds satisfy reportAll (worlds reporting only domestic revenue survive).
w₁ is favored (satisfies reportDomestic) but does not satisfy reportInternational, so reportAll fails at w₁.
In scenario B (after promotion), strong necessity holds: all favored worlds now satisfy reportAll.
With both reportDomestic and reportInternational non-negotiable,
favored worlds must satisfy both, so reportAll holds trivially.
The should→have-to shift: the SAME proposition goes from weak-only to strong necessity when the negotiable ideal is promoted.
The evaluative comparative natural class (§2.1, §2.1.3) #
The central empirical thesis: ought, should, good, better,
preferable, and worthwhile share a semantic core — quantification over
BEST worlds ranked by negotiable ordering sources. Class membership is
diagnosed by two felicity tests (Test 1: "x E q, but doesn't have to q";
Test 2 with an exclusive: "y has to q, x only E q"). The stimuli are typed
LinguisticExample rows in Data.Examples.Rubinstein2014; we read their
felicity off judgment.
Weak-necessity ought passes Test 1 (§2.1, ex 8a): "I ought to do the dishes but I don't have to" is felicitous because weak necessity is strictly weaker than have-to.
The Hebrew evaluative comparatives all pass Test 1 in the bribe scenario (§2.1.3, ex 21): yoter tov, 'adif, and kday are felicitous translations of priority-type ought.
carix 'need' fails both strength tests (§2.1.2, ex 16a, 19): substituting it for ought is infelicitous, aligning it with strong necessity (xayav 'must') rather than the comparative class.
The morphological comparative better supports an explicit than-clause (§2.1.3, ex 24), making the comparative backbone overt. Modal ought (positive/superlative) only selects the overall best — see [Rub14].
Neg-raising and negotiability (§2.2, §3.4) #
Rubinstein connects the evaluative comparative class to neg-raising (§3.4):
predicates relativized to negotiable ordering sources have an "opinionated"
alternative □.¬p available, enabling the excluded-middle inference that
underlies neg-raising. Strong necessity modals, which quantify over favored
worlds WITHOUT ordering, lack this alternative. Horn's ([Hor78]) cyclicity
diagnostic ("I don't think you should leave" ≅ "I think you should stay")
splits the modals; the stimuli carry the lower-negation reading on readings.
The weak/strong split for English modal verbs is derived from the
English fragment (Auxiliaries.lean) rather than re-stipulated: a verb
neg-raises iff the fragment marks it weak necessity.
Stimuli testing neg-raising under higher negation.
Equations
- Rubinstein2014.negRaisingRows = List.filter (fun (x : Data.Examples.LinguisticExample) => List.lookup "diagnostic" x.paperFeatures == some "negRaising") Rubinstein2014.Examples.all
Instances For
The neg-raising split, derived from the fragment. For every English modal-verb stimulus, the lower-negation (neg-raising) reading is available iff the English fragment marks the verb as weak necessity. Flipping either the fragment's force assignment or the recorded reading breaks this — the classification is derived, not stipulated.
Every evaluative-comparative stimulus shows the neg-raising reading (good, better, 'adif), the empirical core of the natural-class claim (§2.2, ex 30, 31a, 33).
Strong necessity modal verbs (must, have to) lack the neg-raising reading (§2.2, ex 31b).
Headline: neg-raising is subsingleton-ness of the modal's domain (§3.4) #
Rubinstein ties neg-raising to negotiability via an "opinionated alternative".
The structural content is sharper and fully general, and lives as substrate:
Homogeneity.negRaising_iff_subsingleton shows a universal modal neg-raises iff
its domain is a subsingleton. The weak/strong split below is then exactly whether
the ordering source collapses that domain — a negotiable ideal can, the bare
favored set cannot. (That subsingleton/decidedness property is also what
[AJ22] call homogeneity, so the shared lemma is substrate both
analyses consume.)
Strong necessity neg-raises at (pt, w) iff the favored-worlds domain is a
subsingleton. Generically it is not (favored worlds are mixed), so the
non-comparative modal is not a neg-raiser.
Weak necessity neg-raises at (pt, w) iff its BEST domain is a subsingleton.
A decisive negotiable ordering source collapses the favored set to one best
class — so the comparative modal neg-raises exactly where the strong one,
ranging over the un-collapsed favored set, cannot.
Rubinstein-unique: the comparative split. There is a priority background
whose favored set is not a subsingleton but whose BEST set is — a negotiable
ordering source collapsing a mixed favored set to a single best world. By the
two corollaries, weak necessity then neg-raises (its BEST domain is decided)
exactly where strong necessity cannot (its favored domain is not). A&J have no
analogue: their homogeneity is intrinsic to should, not produced by a
promotable ordering source. Witness: ce_pt, where all of Fin 4 is favored
but the negotiable ideal (· = w₁) makes w₁ the unique best.
Membership in Rubinstein's comparative natural class: an evaluative comparative predicate, or a modal verb the English fragment marks weak necessity. The strong modal verbs (must, have to, need) are excluded.
Equations
- One or more equations did not get rendered due to their size.
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The English/Hebrew judgments confirm the structural prediction: across every
neg-raising stimulus, the lower-negation reading is available iff the
expression is in the comparative class (the evaluative comparatives plus the
fragment's weak-necessity verbs), excluding must/have to. A finite data
check — its explanation is negRaising_iff_subsingleton.
Cross-linguistic typology of weak necessity (§2.1; Table 1, [Nar12]) #
There are three crosslinguistic routes to weak necessity: a dedicated lexical
item (English should/ought), compositional weakening of a strong modal
(Spanish debería = deber+COND), or evaluative-comparative language
(Hebrew yoter tov). Hebrew lacks the first two; this supports the claim that
weak necessity is comparative — where the comparative route is the only route,
it surfaces overtly. Data imported from Studies/Narrog2010.lean.
Rubinstein's three routes to expressing weak necessity (§2.1).
- lexical : WeakNecessityStrategy
- compositional : WeakNecessityStrategy
- evaluativeComparative : WeakNecessityStrategy
Instances For
Equations
- Rubinstein2014.instDecidableEqWeakNecessityStrategy 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
English has a lexical weak-necessity strategy (§2.1, ex 8a).
Spanish has a compositional strategy: deber + conditional (§2.1, ex 8b).
Hebrew has neither lexical nor compositional weak necessity (§2.1.1–2.1.2): no Hebrew stimulus uses those strategies.
The Hebrew route to weak necessity is exclusively evaluative-comparative (§2.1.3, ex 21): every Hebrew strategy-bearing stimulus uses it.
Only 62 of the 200 surveyed languages grammaticalize weak deontic necessity (Table 1), supporting Rubinstein's claim that weak necessity is not a universal grammatical category. The Table-1 row totals exceed the 131 languages with deontic necessity because some have modals of multiple types.
Bridge to the English fragment (Auxiliaries.lean) #
The English fragment classifies modals by force; we verify these match Rubinstein's force assignments: should/ought are weak necessity (comparative class), must/need are strong necessity (non-comparative).
The English fragment classifies should as weak necessity.
The English fragment classifies ought as weak necessity.
The English fragment classifies must as strong necessity.
must is NOT classified as weak necessity — confirming it is outside the evaluative comparative natural class.
should is NOT classified as strong necessity — confirming the asymmetry: comparative class members have strictly weaker force.
need is classified as strong necessity — matching its exclusion from the evaluative comparative class (§2.1.2, note 14).
need is NOT classified as weak necessity — confirming it fails the scalar tests (examples 16, 18–19).