Docs / Studies / Studies.Spector2025

An Alternative to Presuppositional Exhaustification: Supervaluationist RSA

Spector, B. (2025). In Proceedings of Sinn und Bedeutung 29 (pp. 1501–1517). University of Messina, C.U.M.O.

Source Bibliography Paper

Spector argues against reading exhaustification as presuppositional, Bassi, Del Pinal and Sauerland's trivalent pex under Strong Kleene projection with Doron and Wehbe's Post-Accommodation Informativity. In the marbles context of (11), where the marbles are expected to behave alike, some of the marbles sank passes that constraint yet is odd, while some but not all of the marbles sank is fine. He replaces pex by a bivalent exhaustivity operator and lets every sentence be underspecified between a parse with the operator and one without. A speaker who is maximally averse to the risk of being misread values a message at a world by its least utility over its parses, the log of the literal listener's probability of the world less the message's cost (the MaxiMin rule (28)), and chooses by a softmax of that value.

The parse of a scalar item with the operator is the K3 reading of its pex meaning and the parse without it the LP reading, and Strong Kleene logic is supervaluation over these readings, each occurrence read at its own standard. So the model is the MaxiMin speaker over the two readings of a trivalent meaning. She uses a message exactly where it is true, its K3 reading, and she ranks the messages true at a world by the surprisal of their LP reading less the cost, the LP reading being the weaker, whatever the world and the rationality. Silence beats a message whose LP reading is too probable, and where some but not all is improbable some but not all beats some and not all.

Implementation notes

  • The speaker is RSA.maximinSpeaker against the literal listeners of the parses, taken as one listener on pairs of a parse and a world, entertaining every parse of her world; her weight is the least literal-listener mass over the parses, to the power of the rationality, times the cost weight (speaker_eq_ofWeights). The speaker takes any finite family of parses; the model's parses are the readings of the pex meaning at the two designation standards.
  • The worlds are those of Spector2017, no marble, some but not all, or all of them sank. Some is exhaustified against all and not all against no; the other messages have no other alternatives.
  • Priors, rationality and costs are arguments, of which the paper's tables are instances.

Errata

  • Footnote 13 says that the normalizing sum is positive exactly when at each world some message is true "under a certain parse". The MaxiMin speaker gives no weight to a message false under one of its parses, nor at a world of prior probability zero (speaker_apply_singleton_ne_zero_iff), so the condition is that at each world of positive prior some message be true under all its parses.

TODO

  • Under a downward-entailing operator the least informative parse reads the item with the operator ((22), (25)): the weakest bivalent variant of a sentence reads each occurrence by its polarity. The Magri effects (18) and (26) fit a three-world model of no, one and several pets or hats; (17) needs a quantifier, and the gradience of (29) a model with degrees.
  • Footnote 12 also says that the expected-utility speaker makes the other predictions less robust: she ranks by a geometric mean of the parses' probabilities, which depends on the prior over the parses (speaker_eq_maximinSpeaker: the MaxiMin speaker does not).
  • The contrast of (11a) with only some, (11c), and the QUD-sensitive refinement of §8.

References

Readings of a trivalent meaning

L88–91

A sentence whose scalar items are read by pex is true by Strong Kleene exactly where each of its bivalent variants is true, the variant reading each item at its own standard, by its parse with the operator at K3 (readings_pexIE_k3) and without it at LP (readings_pexIE_lp) (§5) ((21), (24)).

theorem eval_strongKleene_pexIE_eq_one_iffL118–127
{W : Type u_1} {U : Type u_2} {w : W} {ALT : U → Set (Set W)} {lit : U → Set W} [DecidableEq U] (φ : Trivalent.Formula U) :
Trivalent.Formula.eval Trivalent.Connectives.strongKleene (fun (σ : U) => Exhaustification.pexIE (ALT σ) (lit σ) (ALT σ) w) φ = 1 ↔ ∀ ψ ∈ φ.bivalentVariants, Trivalent.Formula.evalBool (fun (x : U × Trivalent.Designation) => decide (w ∈ readings (fun (σ : U) => Exhaustification.pexIE (ALT σ) (lit σ) (ALT σ)) x.2 x.1)) ψ = true

Built from

Quantifiers on the composition engine (§5)

Spector says that Truth on all parses extends to quantifiers. A parse resolves each occurrence of a word before interpretation (PhraseStructure.Tree.readings), so a raised quantifier's parses read its scalar predicate at one standard for every entity it ranges over. Strong Kleene reads each entity at a standard of its own, as the set effect does. The two agree under a monotone or antitone quantifier and come apart under exactly two.

L160–164

The MaxiMin speaker (§6)

L254–259
L261–263
L268–270
L272–283
L287–291
L295–301
L303–306
L340–359
L362–370
L374–380

The readings of a trivalent meaning as the parses

L386–388
L390–392

Among the messages true at a world, the speaker ranks by the surprisal of their LP reading less the cost.

theorem speaker_readings_real_lt_iffL400–408
{W : Type u_1} {U : Type u_2} [Fintype W] [MeasurableSpace W] [DiscreteMeasurableSpace W] [Fintype U] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure W} {α : ℝ} {C : U → ℝ} {w : W} {σ τ : U} [MeasureTheory.IsFiniteMeasure μ] {m : U → W → SignType} (hα : 0 < α) (hw : μ {w} ≠ 0) (hσ : m σ w = 1) (hτ : m τ w = 1) :
((speaker μ α C (readings m)) w).real {σ} < ((speaker μ α C (readings m)) w).real {τ} ↔ -Real.log (μ.real (readings m Trivalent.Designation.lp σ)) - C σ < -Real.log (μ.real (readings m Trivalent.Designation.lp τ)) - C τ

Built from ,

The model of §6

L430
L432–439
L447–449

Exactly two students solved some of the problems, some read by pex, is false on both parses (one student solved some but not all, three solved some), so false by Truth on all parses, and undefined entity-wise: reading some with the operator for one of the two who solved every problem leaves exactly two.

theorem exactlyTwo_parses_false_entitywise_undefinedL487–514
(g : Assignment (Fin 3)) {X : SetM Prop} (hX : PhraseStructure.Tree.interp (lexicon (fun (P : Fin 3 → Prop) => {x : Fin 3 | P x}.ncard = 2) fun (x : Fin 3) => meaning Message.some (solved x)) g raised = some ⟨Montague.Ty.t, X⟩) :
SignType.supervaluationOn (parses (lexicon (fun (P : Fin 3 → Prop) => {x : Fin 3 | P x}.ncard = 2) fun (x : Fin 3) => meaning Message.some (solved x)) g raised) id = -1 ∧ SignType.supervaluationOn X.run id = 0

Built from , , , ,

A speaker who knows that some but not all of the marbles sank prefers some but not all to some, and to not all, exactly when its prior mass, weighted by its extra cost, is below that of the literal meaning of the other: in the marbles context some but not all is improbable next to both none and all, so she says some but not all (§6), while some, read by pex, is informative after accommodation, false where no marble sank (§2).

theorem judgments_11L528–551
{μ : MeasureTheory.Measure Spector2017.World} [MeasureTheory.IsFiniteMeasure μ] {α : ℝ} {C : Message → ℝ} (hα : 0 < α) (hsome : μ {Spector2017.World.justSome} ≠ 0) (hnone : μ {Spector2017.World.none} ≠ 0) :
Examples.ex_11a.judgment = Judgment.unacceptable ∧ Examples.ex_11b.judgment = Judgment.acceptable ∧ (((speaker μ α C (readings meaning)) Spector2017.World.justSome).real {Message.some} < ((speaker μ α C (readings meaning)) Spector2017.World.justSome).real {Message.someButNotAll} ↔ μ.real {Spector2017.World.justSome} * Real.exp (C Message.someButNotAll - C Message.some) < μ.real (literal Message.some)) ∧ (((speaker μ α C (readings meaning)) Spector2017.World.justSome).real {Message.notAll} < ((speaker μ α C (readings meaning)) Spector2017.World.justSome).real {Message.someButNotAll} ↔ μ.real {Spector2017.World.justSome} * Real.exp (C Message.someButNotAll - C Message.notAll) < μ.real (literal Message.notAll)) ∧ Presupposition.PostAccommodationInformative {w : Spector2017.World | μ {w} ≠ 0} (meaning Message.some)

Built from , , , ,

L554–569
L572–581

Two marbles, (19)

The model of Cremers, Wilcox and Spector: the blue marble sank (A), the blue and the red marbles sank (A and B) and only the blue marble sank (A and not B), with A exhaustified against A and B.

L603–605
L633–651