Decision-Theoretic Semantics: Scalar Implicature ([Mer99a] §3)

[Mer99a]

Merin's DTS account of scalar implicature via protentive speaker meaning and relevance-ordered alternatives. The key insight: scalar implicature arises because conjunction is more relevant than disjunction (Theorem 6a), so a speaker who says "A or B" implicates ¬(A ∧ B).

Key Definitions

• sgnRelevance — Protentive Speaker Meaning (Def. 7): the hypothesis supported by an utterance's relevance sign
• upwardCone / downwardCone — alternatives ordered by Bayes factor
• ScalarInterpretation — claim/counterclaim structure for scalar alternatives (Hypothesis 1)

Main Results

• Prediction 1 (not_if_not_indeed_disjunct): a disjunct does not always dominate its disjunction
• Prediction 2 (if_not_indeed_conjunction): under CIP, conjunction dominates both conjuncts and disjunction

Protentive Speaker Meaning (Def. 7)

inductive DTS.ScalarImplicature.RelevanceSign : Type

Sign of relevance: positive (supports H), negative (supports ¬H), or neutral.

• pos : RelevanceSign
• neg : RelevanceSign
• neutral : RelevanceSign

@[instance_reducible]

instance DTS.ScalarImplicature.instDecidableEqRelevanceSign : DecidableEq RelevanceSign

Equations

• DTS.ScalarImplicature.instDecidableEqRelevanceSign x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance DTS.ScalarImplicature.instReprRelevanceSign : Repr RelevanceSign

Equations

• DTS.ScalarImplicature.instReprRelevanceSign = { reprPrec := DTS.ScalarImplicature.instReprRelevanceSign.repr }

def DTS.ScalarImplicature.instReprRelevanceSign.repr : RelevanceSign → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

noncomputable def DTS.ScalarImplicature.sgnRelevance {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (e : Set W) : RelevanceSign

Protentive Speaker Meaning (Def. 7): the hypothesis supported by an utterance's relevance sign.

Equations

• One or more equations did not get rendered due to their size.

Relevance-ordered alternatives (Def. 8)

def DTS.ScalarImplicature.upwardCone {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (alts : Set (Set W)) (σ : Set W) : Set (Set W)

Upward cone: alternatives at least as relevant as σ.

Equations

• DTS.ScalarImplicature.upwardCone ctx alts σ = {a : Set W | a ∈ alts ∧ DTS.bayesFactor ctx σ ≤ DTS.bayesFactor ctx a}

def DTS.ScalarImplicature.downwardCone {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (alts : Set (Set W)) (σ : Set W) : Set (Set W)

Downward cone: alternatives at most as relevant as σ.

Equations

• DTS.ScalarImplicature.downwardCone ctx alts σ = {a : Set W | a ∈ alts ∧ DTS.bayesFactor ctx a ≤ DTS.bayesFactor ctx σ}

structure DTS.ScalarImplicature.ScalarInterpretation (W : Type u_2) : Type u_2

Hypothesis 1: Claim/counterclaim structure for scalar alternatives.

The claim is the disjunction of upward-cone members (what the speaker means to convey). The counterclaim is the disjunction of downward-cone members (what the speaker implicates is false).

• uttered : Set W

  The scalar alternative uttered.

• claim : Set W

  The claim: disjunction of upward cone members.

• counterclaim : Set W

  The counterclaim: disjunction of downward cone members.

Predictions

theorem DTS.ScalarImplicature.not_if_not_indeed_disjunct : ¬∀ (ctx : Context World4) (a b : Set World4), posRelevant ctx a → posRelevant ctx b → bayesFactor ctx (a ∪ b) < bayesFactor ctx a

Prediction 1: It is NOT the case that a disjunct always strictly dominates its disjunction in Bayes factor: a disjunction with an absorbed disjunct is exactly as relevant as the dominant disjunct.

theorem DTS.ScalarImplicature.if_not_indeed_conjunction {W : Type u_1} [MeasurableSpace W] (ctx : Context W) [MeasureTheory.IsFiniteMeasure ctx.prior] [ctx.Nondegenerate] (a b : Set W) (hbm : MeasurableSet b) (hcip : CondIndepIssue ctx a b) (hPosA : posRelevant ctx a) (hPosB : posRelevant ctx b) (hNotH : ctx.prior[a | ctx.topicᶜ] ≠ 0) (hNotH' : ctx.prior[b | ctx.topicᶜ] ≠ 0) : bayesFactor ctx a < bayesFactor ctx (a ∩ b) ∧ bayesFactor ctx (a ∪ b) < bayesFactor ctx (a ∩ b)

Prediction 2: Under CIP with both A, B positively relevant, conjunction dominates both conjuncts and disjunction.

This is the core of Merin's scalar implicature account: "A and B" is strictly more relevant than "A or B", explaining why "or" implicates ¬∧.