Decision-Theoretic Semantics: "Also" ([Mer99a] §5.2–5.4)

[Mer99a]

Merin's DTS account of additive particles. Presupposition is modeled as i-irrelevance: a presupposed proposition is one whose conditional is the prior itself — conditioning on it changes nothing. "Also" requires topic-anaphoric salience: the antecedent D must have been relevant before becoming presupposed.

Key Definitions

• presupposedIrrelevant (Def. 12): presupposition as informational inertness, μ[|a] = μ
• TopicAnaphoricSalience (Def. 13): conditions for anaphoric antecedent
• AlsoFelicitous (Hypothesis 8): felicity conditions for "and also"
• properlyAccommodable (partial Def. 14): accommodable propositions

Main Results

• Corollary 15 (also_nonidentity): "also" requires non-identity (a ≠ b)
• Fact 17 (presuppositional_independence_additivity): presupposition implies multiplicativity of the Bayes factor without CIP

Presupposition as irrelevance (Def. 12)

def DTS.Also.presupposedIrrelevant {W : Type u_1} [MeasurableSpace W] (μ : MeasureTheory.Measure W) (a : Set W) : Prop

Definition 12: A proposition A is i-presupposed iff conditioning on it changes nothing: the conditional measure is the prior itself.

This is stronger than P(A) = 1 in spirit: A is informationally inert — P(X∣A) = P(X) for every X at once.

Equations

• DTS.Also.presupposedIrrelevant μ a = (μ[|a] = μ)

Topic-anaphoric salience (Def. 13)

structure DTS.Also.TopicAnaphoricSalience {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (d e : Set W) : Type

Definition 13: Topic-anaphoric salience.

D is topic-anaphorically salient for E in context iff: (i) E is relevant to the current issue H, (ii) D is presupposed (informationally inert), (iii) D was recently relevant — before becoming presupposed, D bore on the issue.

• eRelevant : posRelevant ctx e ∨ negRelevant ctx e

  E is relevant to the current issue.

• dPresupposed : presupposedIrrelevant ctx.prior d

  D is currently presupposed (informationally inert).

• dWasRelevant : ScalarImplicature.RelevanceSign

  D was previously relevant (before becoming presupposed).

"Also" felicity (Hypothesis 8)

structure DTS.Also.AlsoFelicitous {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (qa qb : Set W) : Type

Hypothesis 8: Felicity conditions for "and also(b, B)".

For "Q(a) and also Q(b)": Q(a) and Q(b) have the same relevance sign (both support or both oppose H). This distinguishes "and also" from "but also" (opposite signs).

The sameSign field records the relevance sign of Q(a) before it was presupposed; signMatches requires Q(b) to bear it now.

• qaPresupposed : presupposedIrrelevant ctx.prior qa

  Q(a) is presupposed.

• qbRelevant : posRelevant ctx qb ∨ negRelevant ctx qb

  Q(b) is relevant.

• sameSign : ScalarImplicature.RelevanceSign

  Same relevance sign: Q(a) had the same sign as Q(b) before presupposition.

• signMatches : ScalarImplicature.sgnRelevance ctx qb = self.sameSign

  The sign matches Q(b)'s current relevance direction.

structure DTS.Also.ButAlsoFelicitous {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (qa qb : Set W) : Type

"But also" variant: opposite relevance signs.

"Q(a) but also Q(b)": Q(a) had the opposite relevance sign from Q(b). This combines adversativity ("but") with additivity ("also").

• qaPresupposed : presupposedIrrelevant ctx.prior qa

  Q(a) is presupposed.

• qbRelevant : posRelevant ctx qb ∨ negRelevant ctx qb

  Q(b) is relevant.

• previousSign : ScalarImplicature.RelevanceSign

  Previous sign of Q(a).

• currentSign : ScalarImplicature.RelevanceSign

  Current sign of Q(b).

• oppositeSigns : self.previousSign = ScalarImplicature.RelevanceSign.pos ∧ self.currentSign = ScalarImplicature.RelevanceSign.neg ∨ self.previousSign = ScalarImplicature.RelevanceSign.neg ∧ self.currentSign = ScalarImplicature.RelevanceSign.pos

  Signs are opposite.

Accommodation (partial Def. 14)

def DTS.Also.properlyAccommodable {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (φ : Set W) : Prop

Partial Definition 14: Properly accommodable propositions.

A proposition φ is properly accommodable iff: (i) 0 < P(φ) (non-trivially satisfiable), (ii) P(φ) < 1 (not already known), (iii) φ is irrelevant to the current issue.

Equations

• DTS.Also.properlyAccommodable ctx φ = (0 < ctx.prior φ ∧ ctx.prior φ < 1 ∧ DTS.irrelevant ctx φ)

Theorems

theorem DTS.Also.also_nonidentity {W : Type u_1} [MeasurableSpace W] {E : Type u_2} (ctx : Context W) [MeasureTheory.IsProbabilityMeasure ctx.prior] (Q : E → Set W) (a b : E) (hQa : MeasurableSet (Q a)) (hAlso : AlsoFelicitous ctx (Q a) (Q b)) (hH : ctx.prior ctx.topic ≠ 0) (hNH : ctx.prior ctx.topicᶜ ≠ 0) : a ≠ b

Corollary 15: "Also" requires non-identity.

If Q(a) is presupposed and "Q(a) and also Q(b)" is felicitous, then a ≠ b: a presupposed proposition has Bayes factor 1, but felicity requires Q(b) to be relevant.

theorem DTS.Also.presuppositional_independence_additivity {W : Type u_1} [MeasurableSpace W] (ctx : Context W) [MeasureTheory.IsProbabilityMeasure ctx.prior] {a b : Set W} (ha : MeasurableSet a) (hbm : MeasurableSet b) (hp : presupposedIrrelevant ctx.prior a) (hNotH' : ctx.prior[b | ctx.topicᶜ] ≠ 0) : bayesFactor ctx (a ∩ b) = bayesFactor ctx a * bayesFactor ctx b

Fact 17: Presupposition implies multiplicativity without CIP.

If A is presupposed, then BF(A∧B) = BF(A)·BF(B) with no independence assumption: inertness supplies the factorization.

Prediction 4 (not formalized): "Also" removes causal implicature.

In "Kim fell and she also broke her arm", the additive particle "also" enforces presuppositional independence of the antecedent ("Kim fell"), removing the default causal reading that plain "and" would carry ("Kim fell and [as a result] broke her arm").

This connects to Causation — the causal reading arises from non-independence of the conjuncts, and "also" explicitly marks the antecedent as presupposed (hence independent).