@cite{rudolph-kocurek-2024}: Metalinguistic Gradability #
@cite{rudolph-kocurek-2024}
Rachel Etta Rudolph & Alexander W. Kocurek. 2024. Metalinguistic gradability. Semantics & Pragmatics 17, Article 7: 1-58.
Overview #
Finite model verification of the semantic expressivist framework for
metalinguistic comparatives ("Ann is more a linguist than a philosopher"),
equatives, degree modifiers, and conditionals. Verifies the paper's
entailment predictions (§4.4), assertoric content (§3.3), nonclassicality
of acceptance-preservation (§4.4), degree modifiers very/sorta/
mostly (§6.1), metalinguistic conditionals (§6.3), the No Reversal
bridge to delineation comparatives (§7), and the revised semantics
for ME transitivity (Supplement §B).
Models #
Model 1 (3 interpretations, linear order i₀ < i₁ < i₂):
- i₀: Ann is a philosopher, not a linguist
- i₁: Ann is a linguist, not a philosopher
- i₂: Ann is both a linguist and a philosopher
Captures "Ann is more a linguist than a philosopher": at i₂ (top-ranked), both hold, and the (La∧¬Pa)-witness i₁ outranks the (Pa∧¬La)-witness i₀.
Model 2 (2 interpretations, tied j₀ ≡ j₁):
- j₀: La true, Pa false
- j₁: La false, Pa true
Witnesses borderline cases, equative satisfaction, and nonclassicality of acceptance-preservation (parallels informational entailment for epistemic modals, @cite{yalcin-2007}).
Model 3 (3 interpretations, linear i₀ < i₁ < i₂, two entities):
- i₀: neither Ann nor Ben is tall
- i₁: Ann is tall, Ben is not
- i₂: both are tall
Satisfies No Reversal for tall: extensions are monotonically nested.
Demonstrates that under NR, the MC simplifies to the delineation
comparative (§7). A counterexample model violating NR shows that
MC and delineation diverge without this constraint.
Model 4 (4 interpretations, 3 propositions, ordering l < j ≡ k < i):
- i: Ann is a linguist, philosopher, and psychologist (1,1,1)
- j: Ann is a linguist and psychologist, not a philosopher (1,0,1)
- k: Ann is a philosopher only (0,1,0)
- l: Ann is a linguist and philosopher, not a psychologist (1,1,0)
Counterexample to ME transitivity in the basic semantics (Supplement §B):
La ≈ Pa and Pa ≈ Ca hold, but La ≈ Ca fails (La ≻ Ca holds vacuously).
The revised semantics (evalRevised) blocks La ≻ Ca and restores
transitivity.
Connections #
- Theory layer:
Theories/Semantics/Comparison/Metalinguistic.lean(SemanticOrdering, MFormula, eval, evalRevised, evalGen, evalMCond, assertoricContent, DistanceFunction, evalVery/evalSorta/evalMostly, noReversal, MetalinguisticCG) - Delineation:
Theories/Semantics/Comparison/Delineation.lean(@cite{klein-1980}'s comparison class approach — connected via No Reversal) - Hierarchy:
Theories/Semantics/Comparison/Hierarchy.lean(Klein ← Kennedy ← Measurement strict hierarchy)
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqW x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Two predicates: linguist and philosopher.
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqPred x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqEntity x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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"Ann is a linguist"
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"Ann is a philosopher"
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"Ann is more a linguist than a philosopher"
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"Ann is (exactly) as much a linguist as a philosopher"
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqI3 x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Linear ordering: i0 ≤ i1 ≤ i2.
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Interpretation function:
- i₀: Ann is a philosopher, not a linguist
- i₁: Ann is a linguist, not a philosopher
- i₂: Ann is both
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Observation 1a: MCs are consistent with truth of both constituents. At i₂, Ann is both a linguist and a philosopher, yet "Ann is more a linguist than a philosopher" is true — the (La∧¬Pa)-interpretation i₁ outranks the (Pa∧¬La)-interpretation i₀.
The MC is also true at i₁ (where Ann is a linguist but not a philosopher).
The MC is false at i₀ (no (La∧¬Pa)-witness at or below i₀).
Observation 2: A ≻ B, B ⊨ A. If the MC holds and Ann is a philosopher, she is a linguist.
(e) Asymmetry: A ≻ B ⊨ ¬(B ≻ A) on this model.
(f) Irreflexivity: ⊨ ¬(A ≻ A). No sentence is more the case than itself.
(k) Equative reflexivity: ⊨ A ≈ A.
(l) Equative symmetry: A ≈ B ⊨ B ≈ A on this model.
Observation 4: A ≈ B ⊨ ¬(A ≻ B) ∧ ¬(B ≻ A). (By definition.)
(g) Transitivity: A ≻ B, B ≻ C ⊨ A ≻ C.
(n) Trichotomy (requires totality of ≤): ⊨ (A ≻ B) ∨ (B ≻ A) ∨ (A ≈ B).
Two interpretations for borderline / nonclassicality witnesses.
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqI2 x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Tied ordering: j0 ≡ j1 (both maximal).
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j₀: La true, Pa false; j₁: La false, Pa true.
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Observation 5: A ≈ ¬A is satisfiable (not contradictory). With tied interpretations where one makes La true and the other makes La false, "Ann is (exactly) as much a linguist as not" is coherent — it expresses a borderline case.
La ≈ Pa holds on the tied model (both MCs are blocked by ties).
(m) Negation equative: A ≈ B ⊨ ¬A ≈ ¬B (on the tied model).
The assertoric content of "Ann is a linguist" holds on ord₃: the top-ranked interpretation (i₂) makes La true.
The assertoric content of "Ann is a philosopher" also holds on ord₃ (i₂ makes Pa true).
The assertoric content of La_mc_Pa holds on ord₃: at the unique maximal interpretation i₂, the MC evaluates to true.
For substantive Obs 3: i₂ is linguist only.
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The tautology La ∨ ¬La has assertoric content 1 on the tied model.
Nonclassicality of acceptance-preservation: (La ≻ ¬La) ∨ (¬La ≻ La) does NOT have assertoric content 1 on the tied model. At both maximal interpretations (j₀ ≡ j₁), neither direction of MC holds because the interpretations are tied.
This is the key result paralleling informational entailment for epistemic modals (@cite{yalcin-2007}).
Distance function for ord₃: close(i) includes interpretations at the same level or one level below.
- d(i₀) = {i₀}
- d(i₁) = {i₀, i₁}
- d(i₂) = {i₁, i₂}
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very La is true at i₂: all interpretations reasonably close to i₂ (namely i₁ and i₂) make La true.
very La is false at i₁: i₀ is reasonably close to i₁ but makes La false.
very A ⊨ A: if very La holds, then La holds (centeredness).
sorta La holds at i₁: some close interpretation (i₁ itself) makes La true, even though another close interpretation (i₀) does not.
sorta La is false at i₀: d(i₀) = {i₀}, and La is false at i₀.
mostly La is true at i₂: there is a reasonably high level (i₁) where La is uniformly true, and the only A-false level (i₀) is below it.
mostly La is false at i₁: i₀ is the only candidate level below i₁ in d(i₁), but La is false at i₀.
mostly A ⊨ sorta A (on this model): if A is mostly the case, then it is sorta the case.
very A ⊨ mostly A (on this model).
No Reversal holds trivially on single-entity models (the antecedent e₁ ≠ e₂ can never be satisfied). For a substantive NR test, a two-entity model is needed.
The key consequence of NR: when it holds, Fa ≻ Gb simplifies to the
delineation comparative ∃ i' ≤ i : Fa∧¬Gb — the universal clause
of eq. (48) becomes redundant. This connects metalinguistic
comparatives to @cite{klein-1980}'s supervaluation comparative
(see Theories/Semantics/Comparison/Delineation.lean) and
@cite{kamp-1975}'s completion-based approach.
Two entities for gradable adjective models.
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqEntity2 x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqPred1 x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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NR model for "Ann is taller than Ben":
- i₀: neither Ann nor Ben is tall (empty extension)
- i₁: Ann is tall, Ben is not (Ann enters the extension first)
- i₂: both are tall
Satisfies No Reversal: extensions are monotonically nested
({} ⊆ {ann} ⊆ {ann, ben}). Uses the same 3-interpretation
linear order ord₃ from Model 1.
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"Ann is tall"
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"Ben is tall"
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No Reversal holds for tall on the two-entity model.
Since Ann enters the extension before Ben (at i₁ vs i₂), there is no
interpretation where Ben is tall but Ann is not. NR is satisfied
because the extensions are monotonically nested.
Compare with nr_trivial_single_entity above, which holds vacuously
on a single-entity model. Here NR constrains the relationship between
two distinct entities' extensions across the ordering.
Ann is taller than Ben: the MC tall(ann) ≻ tall(ben) is true
at i₁ and i₂. Witness: i₁ (Ann is tall, Ben is not).
The MC is false at i₀ (no witness: Ann is not tall at i₀).
The reverse MC — Ben taller than Ann — is false everywhere. There is no interpretation where Ben is tall but Ann is not.
Bridge: under NR, the full MC (eq. 48, with domination clause) gives the same result as the delineation comparative (eq. 128, just ∃ i' ≤ i : Fa ∧ ¬Fb, without domination clause).
NR makes clause (ii) of the MC redundant: if Fa is true and Fb is false at i, then for any i' ≤ i where Fb becomes true, Fa must also be true — so there are no (Fb∧¬Fa)-witnesses to worry about.
This connects metalinguistic comparatives to @cite{klein-1980}'s
delineation comparative (see Delineation.lean).
NR-violating model: Ann and Ben "swap" across interpretations.
- i₀: Ann tall, Ben not
- i₁: Ben tall, Ann not (reversal!)
- i₂: both tall
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No Reversal fails on the violating model (for e₁=ben, e₂=ann): Ben is tall at i₁ and Ann is not, but at i₀ ≤ i₁ where Ann is tall, Ben is not — a reversal.
Without NR, MC and delineation diverge: the MC Ta ≻ Tb is
FALSE at i₂ (the (Tb∧¬Ta)-witness i₁ outranks the (Ta∧¬Tb)-witness
i₀, violating the domination clause), but the simple delineation
condition (∃ Fa∧¬Fb) is TRUE (i₀ is a witness).
Reflexivity: A → A is true everywhere (every A-interpretation trivially makes A true on any ordering).
La → Pa fails at i₂ on Model 1: the La-restricted ordering ≤_{La} includes i₁ (where La is true but Pa is false), so there exists a ranked La-interpretation that does not satisfy Pa.
This shows → is NOT the material conditional — La and Pa are both true at i₂, but the conditional is false because it quantifies over all ranked La-interpretations, not just the current one.
Observation M1 (§6.3): ⊨ A → (A ≻ ¬A).
"If Pluto is a planet, then it's more a planet than not" is classically valid. On ≤_A (restricted to A-interpretations), A ≻ ¬A trivially holds: every A-interpretation makes A true, so the (A∧¬(¬A))-witness exists, and there are no (¬A∧¬A)-witnesses in the restricted ordering.
This parallels the validity of epistemic "if p then probably p" (@cite{yalcin-2007}).
ME transitivity counterexample #
The basic semantics fails to validate ME transitivity:
A ≈ B, B ≈ C ⊭ A ≈ C (Supplement §B). Model 4 provides a minimal
counterexample.
Key insight: the (La∧¬Ca)-witness l has no matching (Ca∧¬La)-witness,
so La ≻ Ca holds vacuously under the basic semantics. The revised
semantics blocks this: l must dominate either all Ca-interpretations
(i and j are above it) or all ¬La-interpretations (k is above it),
and it can do neither.
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- Phenomena.Gradability.RudolphKocurek2024.instDecidableEqPred3 x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Ordering: l < j ≡ k < i (three levels). j and k are tied at the middle level — this is essential for the equatives La ≈ Pa and Pa ≈ Ca to hold (witnesses block each other).
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Interpretation function for transitivity counterexample:
- i: all three true (linguist, philosopher, psychologist)
- j: linguist and psychologist only (philosopher false)
- k: philosopher only (linguist and psychologist false)
- l: linguist and philosopher only (psychologist false)
The (La∧¬Pa)-witness j and (Pa∧¬La)-witness k are at the same level, ensuring neither MC direction holds for La vs Pa (and similarly for Pa vs Ca). But the only (La∧¬Ca)-witness is l at the bottom, with no (Ca∧¬La)-witness anywhere.
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"Ann is a linguist" (3-predicate model)
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"Ann is a philosopher" (3-predicate model)
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"Ann is a psychologist"
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A ≈ B holds: Ann is as much a linguist as a philosopher. The (La∧¬Pa)-witness j and (Pa∧¬La)-witness k are tied at the middle level, blocking both MC directions.
B ≈ C holds: Ann is as much a philosopher as a psychologist. The (Pa∧¬Ca)-witnesses k, l and (Ca∧¬Pa)-witness j are balanced: k is tied with j, blocking both MC directions.
A ≈ C FAILS: basic semantics predicts Ann is MORE a linguist than a psychologist. ME transitivity is violated.
The failure mechanism: La ≻ Ca holds in the basic semantics. The (La∧¬Ca)-witness l dominates all (Ca∧¬La)-interpretations vacuously — there are none (Ca is true only at i and j, where La is also true).
Under the revised semantics, La ≻ Ca is blocked: the (La∧¬Ca)- witness l cannot dominate all Ca-interpretations (i and j are above it) or all ¬La-interpretations (k is above it).
ME transitivity is restored: A ≈ C holds under the revised semantics, as required by transitivity from A ≈ B and B ≈ C.
The revised semantics preserves A ≈ B (no regression).
The revised semantics preserves B ≈ C (no regression).
On Model 1 (linear ordering), the revised MC agrees with the basic MC. The two diverge only when the revised semantics' stronger domination clause matters — on a linear ordering with no ties at critical levels, the basic witnesses satisfy the revised conditions too.
Degree theory on finite models #
Finite model verifications that the degree-theoretic formulation
(Supplement §C) correctly tracks the revised semantics on Models 1–4.
The key bridge theorems (Facts 9–10) are stated in
MetalinguisticDegree.lean; here we verify their instances on
concrete models via native_decide.
The denotation of La on Model 1: {i₁, i₂} (La true at i₁ and i₂).
The denotation of Pa on Model 1: {i₀, i₂} (Pa true at i₀ and i₂).
∼ reflexivity: ⟦La⟧ ∼ ⟦La⟧ (verified by decidable computation).
La ≻ Pa on Model 1 under revised semantics: the denotation sets are NOT degree-equivalent (La outranks Pa).
⊐ confirms: ⟦La⟧ ⊐ ⟦Pa⟧ (La's denotation strictly better).
The reverse direction fails: ⟦Pa⟧ ⋣ ⟦La⟧.
La ≈ Pa on Model 4: denotation sets are degree-equivalent.
Pa ≈ Ca on Model 4: denotation sets are degree-equivalent.
La ≈ Ca on Model 4 (revised): denotation sets are degree-equivalent. This is the key test: ∼ transitivity gives us deg(La) = deg(Ca) even though the basic MC incorrectly predicts La ≻ Ca.
⊐ correctly blocks La ⊐ Ca on Model 4.