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Towards a pragmatic explanation for the prevalence of upward-monotonicity in natural language: some results on communicative stability, the strongest answer condition, and exhaustification

Enguehard, É. (2025). Linguistics and Philosophy, 48, 693–740

Source Bibliography Paper

Enguehard traces the prevalence of upward-monotone logical words to exhaustification. A set of alternatives is strongly stable when every world has a strongest true member, which for finitely many alternatives holds exactly when exhaustifying the non-dominated ones yields the partition they induce. Weak stability of symmetric alternatives and their negations leaves only every and some, strong stability of a lexicon's determiner sentences and their negations makes every determiner monotone in its scope, and among proper clones it singles out the upward-monotone functions. The messages a pragmatic speaker can choose as the prior varies are the minimal true ones, so that weak stability is Bar-Lev and Katzir's communicative stability.

Implementation notes

  • A world is a valuation Set ι of the atoms, as the appendix identifies the cells of the atoms' partition with worlds, and a proposition is the Boolean function of the atoms it expresses. The atom for i is {I | i ∈ I}, the Montague lift Quantifier.individual i of the name in Ann came to the party. Upward monotonicity is IsUpperSet; the conjunction and the disjunction of the atoms in I and outside I are Set.Ici I and (Set.Iic I)ᶜ.
  • Strong stability is Question.IsExhaustivelyResolvable at every world. Weak stability is read as printed: a strongest true member, or only atoms among the minimal true members. At a world where nothing is true this holds vacuously.
  • The theorem relating strong stability to exhaustification assumes finitely many alternatives, which the appendix assumes for innocent exclusion and minimal worlds; its basic form needs the assumption too (see Errata).
  • The paper compares strong stability with the condition of a 2020 manuscript of Fox's, that exhaustification yield exactly the induced partition rather than contain it. Fox's 2018 paper states the same condition, Question Partition Matching, so the comparison runs through Fox2018.QPM: the monotone propositions over two atoms are strongly stable but fail it.
  • A symmetric proposition is one that, read as a quantifier over the atoms, is Quantifier.QuantityInvariant, Mostowski's invariance and the counterpart of Keenan and Stavi's logicality for one argument. A projectively symmetric one lives on a set of atoms and is invariant under the permutations fixing it, which makes it a conservative, quantity-invariant determiner frozen to a restrictor in Peters and Westerståhl's sense (Quantifier.exists_det_apply_eq_iff_livesOn). Bar-Lev and Katzir's result on binary connectives is the result on symmetric functions at two atoms.
  • A clone assigns propositions to every number of atoms, zero included; the paper's clones start at one atom, and the results quantify over positive numbers. Both halves of the clone result are proved without the classification of the maximal clones the paper invokes: conjunction and negation express every proposition, and the two constants with any non-monotone function compose to negation.
  • Optimality in the probabilistic model of the appendix compares the literal listener's mass at the world, RSA.listener μ (RSA.truthful sem), strictly across the true messages with other meanings, which is the paper's assumption that no two messages tie; the utility is its logarithm, which the comparison drops. A prior is a probability measure giving every world positive mass, and the worlds are finitely many, where the paper also allows countably many.

Errata

  • The result on two-place connectives fails when its condition that the atoms are not dominated is read among the alternatives alone: six propositions over two atoms, among them if q then p, are bilaterally strongly stable with neither atom dominated, yet if q then p is not monotone. The result holds when the negated atoms are not dominated among the negations either (repaired).
  • The theorem relating strong stability to exhaustification is stated for any set of alternatives, and the appendix assumes them finite only for innocent exclusion and minimal worlds, but its omitted lemma, that exhaustifying against the non-dominated alternatives changes nothing, fails for infinitely many: over ℕ ∪ {∞} the non-dominated alternatives exhaustify to exactly the induced partition, yet none is strongest at ∞.
  • The results on symmetric and on projectively symmetric functions, and their proofs, assume at least two atoms: p and not p over one atom are bilaterally strongly stable and symmetric, and not p, the negated existential, is not monotone.
  • Weak stability as defined holds vacuously at a world where no alternative is true, whereas the proof of the clone result for weak stability assumes that every world has a true and a false alternative, the coverage the paper asks of the strongest answer condition and assumes in its probabilistic model. Under the definition as printed that result fails: the falsity-preserving functions form a proper clone that contains exclusive disjunction and is bilaterally weakly stable, vacuously so where no atom holds. With the coverage the result holds.

TODO

  • The paper's result on convex functions.

References

Stability

L117–119
L121–124
L126–129
L131–134

Strong stability and exhaustification

(26), with its extension in the appendix: for finitely many alternatives, they are strongly stable iff exhaustifying them yields every cell of the partition they induce, iff exhaustifying the ones not dominated yields exactly that partition, whether the exhaustifier is the basic one, innocent exclusion or minimal worlds.

theorem tfae_stronglyStableL149–222
{W : Type u_1} (A : Set (Set W)) (hA : A.Finite) :
[StronglyStable A, (Question.partition A).classes ⊆ Exhaustification.exh A '' A, Exhaustification.exh {φ : Set W | φ ∈ A ∧ ¬Exhaustification.Dominated A φ} '' {φ : Set W | φ ∈ A ∧ ¬Exhaustification.Dominated A φ} = (Question.partition A).classes, (Question.partition A).classes ⊆ Exhaustification.exhIE A '' A, Exhaustification.exhIE {φ : Set W | φ ∈ A ∧ ¬Exhaustification.Dominated A φ} '' {φ : Set W | φ ∈ A ∧ ¬Exhaustification.Dominated A φ} = (Question.partition A).classes, (Question.partition A).classes ⊆ Exhaustification.exhMW A '' A, Exhaustification.exhMW {φ : Set W | φ ∈ A ∧ ¬Exhaustification.Dominated A φ} '' {φ : Set W | φ ∈ A ∧ ¬Exhaustification.Dominated A φ} = (Question.partition A).classes].TFAE

Built from

L308–316

Two atoms

L345–350
L353–360
L457–493

Symmetric functions

A proposition over the atoms is symmetric (33) when, read as a quantifier over them, it is Quantifier.QuantityInvariant.

L538–555
L559–595

Over the atoms and symmetric functions of them, if the alternatives and their negations are weakly stable, then besides the atoms only the contradiction, the tautology, every and some over the atoms remain: the corners of the square of opposition other than no and the missing O-corner. At two atoms this is Bar-Lev & Katzir 2025's result on binary connectives.

theorem mem_every_some_of_bilaterallyWeaklyStable(34c)L597–637
{ι : Type u_2} {A : Set (Set (Set ι))} [Finite ι] [Nontrivial ι] (hat : ∀ (i : ι), {I : Set ι | i ∈ I} ∈ A) (hsym : ∀ φ ∈ A, (φ ∉ Set.range fun (i : ι) => {I : Set ι | i ∈ I}) → Quantifier.QuantityInvariant fun (x : Set ι) => x ∈ φ) (h : BilaterallyWeaklyStable (Set.range fun (i : ι) => {I : Set ι | i ∈ I}) A) {φ : Set (Set ι)} (hφ : φ ∈ A) (hφa : φ ∉ Set.range fun (i : ι) => {I : Set ι | i ∈ I}) :
φ ∈ {∅, Set.univ, {B : Set ι | Quantifier.Det.every Set.univ B}, {B : Set ι | Quantifier.Det.some Set.univ B}}

Built from

L710–740
L745–778

Projectively symmetric functions

L793–797
L843–897
L900–919

(37b) read off a lexicon: if the atoms and the sentences that conservative, quantity-invariant determiners form with any restrictor are bilaterally strongly stable, then, with at least two atoms, every determiner in the lexicon is monotone in its scope.

theorem scopeMonotone_of_bilaterallyStronglyStableL921–936
{ι : Type u_2} [Nontrivial ι] {D : Set (Quantifier.Det ι)} (hC : ∀ d ∈ D, d.Conservative) (hQ : ∀ d ∈ D, d.QuantityInvariant) (h : BilaterallyStronglyStable ((Set.range fun (i : ι) => {B : Set ι | Quantifier.individual i B}) ∪ ⋃ d ∈ D, Set.range fun (A : Set ι) => {B : Set ι | d A B})) (d : Quantifier.Det ι) :
d ∈ D → d.ScopeMonotone

Built from

L974–1012

Closed classes of functions

L1016–1021
L1067–1070
L1073–1098
L1103–1108
L1117–1173

A proper clone is bilaterally strongly stable at every number of atoms iff it is the clone of upward-monotone functions.

theorem bilaterallyStronglyStable_iff_eq_isUpperSet(40b)L1175–1200
{F : (n : ℕ) → Set (Set (Set (Fin n)))} (hF : IsClone F) (hL : ∃ (n : ℕ), 0 < n ∧ F n ≠ Set.univ) :
(∀ (n : ℕ), 0 < n → BilaterallyStronglyStable (F n)) ↔ ∀ (n : ℕ), 0 < n → F n = {φ : Set (Set (Fin n)) | IsUpperSet φ}

Built from ,

L1204–1225

Bar-Lev and Katzir's communicative stability

L1237–1241

A.3: a message is optimal at w for some prior giving every world positive mass iff its meaning is minimal among the meanings true at w, so that which message the pragmatic speaker chooses does not depend on the prior exactly where a strongest true meaning exists.

theorem exists_prior_iff_minimalL1243–1292
{W : Type u_1} [MeasurableSpace W] [DiscreteMeasurableSpace W] {U : Type u_3} [MeasurableSpace U] [Countable U] [MeasurableSingletonClass U] (sem : U → Set W) [Fintype W] {w : W} {u : U} :
(∃ (μ : MeasureTheory.Measure W), MeasureTheory.IsProbabilityMeasure μ ∧ (∀ (v : W), μ {v} ≠ 0) ∧ IsOptimal sem μ w u) ↔ Minimal (fun (x : Set W) => x ∈ Question.trueAnswers (Set.range sem) w) (sem u)

Built from

L1297–1319
L1323–1348
L1352–1366