Decision-Theoretic Semantics: "But" ([Mer99a] §4)

[Mer99a]

Merin's DTS account of adversative conjunction. The felicity of "A but B" requires that A and B have opposite relevance signs, and that the conjunction A∧B is negatively relevant (the "but"-clause wins). The default interpretation sets H = B, yielding unexpected-B-given-A.

Key Definitions

• butFelicitous (Hypothesis 4): felicity conditions for "A but B"
• NNIR (Def. 10): Non-Negative Instantial Relevance
• defaultButCtx: the default interpretation where H = B

Main Results

• Theorem 8: CIP + contrariness → unexpectedness (P(B∣A) < P(B))
• Theorem 9: When H = B, CIP holds automatically
• Theorem 10: Negative relevance implies unexpectedness in default-but
• Corollary 11 (Harris universal): NNIR prevents "Qa but Qb"

Felicity conditions for "but"

def DTS.But.butFelicitous {W : Type u_1} [MeasurableSpace W] (ctx : Context W) (a b : Set W) : Prop

Hypothesis 4: Felicity conditions for "A but B".

"A but B" is felicitous iff: (i) A is positively relevant to H, (ii) B is negatively relevant to H, (iii) A∧B is negatively relevant to H (B "wins").

Equations

• DTS.But.butFelicitous ctx a b = (DTS.posRelevant ctx a ∧ DTS.negRelevant ctx b ∧ DTS.negRelevant ctx (a ∩ b))

Non-Negative Instantial Relevance (NNIR)

def DTS.But.NNIR {W : Type u_1} [MeasurableSpace W] (E : Type u_2) (μ : MeasureTheory.Measure W) (Q : E → Set W) : Prop

Definition 10: Non-Negative Instantial Relevance (NNIR).

For a predicate Q over entities, observing Q(a) never makes Q(b) less probable: P(Q(b)∣Q(a)) ≥ P(Q(b)) for all a, b.

This captures a cross-linguistic universal: properties are positively correlated across instances (knowing one dog is friendly makes it more likely another is).

Equations

• DTS.But.NNIR E μ Q = ∀ (a b : E), μ (Q b) ≤ μ[Q b | Q a]

Default but (H = B)

@[reducible, inline]

abbrev DTS.But.defaultButCtx {W : Type u_1} [MeasurableSpace W] (μ : MeasureTheory.Measure W) (b : Set W) (hb : MeasurableSet b) : Context W

Default "but" context: the issue is identified with the but-clause itself (H = B).

Merin argues this is the preferred interpretation when no explicit issue is provided.

Equations

• DTS.But.defaultButCtx μ b hb = { topic := b, topicMeasurable := hb, prior := μ }

Theorems

theorem DTS.But.cip_contrariness_implies_unexpectedness {W : Type u_1} [MeasurableSpace W] (ctx : Context W) [MeasureTheory.IsProbabilityMeasure ctx.prior] [ctx.Nondegenerate] {a b : Set W} (ham : MeasurableSet a) (hcip : CondIndepIssue ctx a b) (hcontr : hContrary ctx a b) (ha0 : ctx.prior a ≠ 0) (hb0 : ctx.prior b ≠ 0) : ctx.prior[b | a] < ctx.prior b

Theorem 8: CIP + contrariness implies unexpectedness.

If A and B are conditionally independent given H and ¬H, and have opposite relevance signs, then P(B∣A) < P(B) — B is unexpected given A.

CIP turns the total-probability decompositions of P(A∧B), P(A), and P(B) into a factorized cross-product whose factors contrariness makes jointly positive.

theorem DTS.But.condIndepIssue_defaultButCtx {W : Type u_1} [MeasurableSpace W] (μ : MeasureTheory.Measure W) [MeasureTheory.IsFiniteMeasure μ] (a b : Set W) (ham : MeasurableSet a) (hbm : MeasurableSet b) : CondIndepIssue (defaultButCtx μ b hbm) a b

Theorem 9: When H = B, issue-conditional independence holds automatically for any A.

P(A∧B∣B) = P(A∣B)·P(B∣B) because B∧(A∧B) = B∧A and P(B∣B) = 1, and P(A∧B∣¬B) = P(A∣¬B)·P(B∣¬B) because both sides vanish on ¬B.

theorem DTS.But.default_but_properties {W : Type u_1} [MeasurableSpace W] (μ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure μ] {a b : Set W} (ham : MeasurableSet a) (hbm : MeasurableSet b) (hNegA : negRelevant (defaultButCtx μ b hbm) a) (ha0 : μ a ≠ 0) (hB : μ b ≠ 0) (hNB : μ bᶜ ≠ 0) : μ[b | a] < μ b

Theorem 10: Negative relevance implies unexpectedness in default-but.

When the issue is B itself and A is negatively relevant to H = B, then P(B∣A) < P(B) — B is unexpected given A.

theorem DTS.But.harris_universal {W : Type u_1} [MeasurableSpace W] {E : Type u_2} (μ : MeasureTheory.Measure W) [MeasureTheory.IsFiniteMeasure μ] (Q : E → Set W) (a b : E) (hQb : MeasurableSet (Q b)) (hb : μ (Q b) ≠ 0) (_hnnir : NNIR E μ Q) : ¬butFelicitous (defaultButCtx μ (Q b) hQb) (Q a) (Q b)

Corollary 11 (Harris universal): NNIR prevents "Qa but Qb".

In the default-but interpretation with a live antecedent Q(b), the issue is Q(b) itself: P(Q(b)∣Q(b)) = 1 while P(Q(b)∣¬Q(b)) = 0, so the Bayes factor is genuinely infinite — Q(b) cannot be negatively relevant to itself, violating butFelicitous.

Theorem 13 (not formalized): Savage-Kemeny-Gaifman-Humburg theorem.

Symmetric probability on finite models extends to infinite models only if NNIR holds, providing a foundational justification for NNIR as a rationality constraint. Requires de Finetti-style exchangeability arguments.

Reference: Gaifman, H. & Snir, M. (1982). Probabilities over rich languages.