Superlative Semantics

@cite{heim-1999} @cite{sharvit-stateva-2002} @cite{szabolcsi-1986}

Compositional semantics for the superlative morpheme -est/most.

@cite{heim-1999}: Absolute vs. Relative

@cite{heim-1999} identifies two readings of superlatives:

• Absolute: "Kim climbed the highest mountain" = the mountain that is highest of all mountains. ⟦-est⟧ = λC.λG.λx. x ∈ C ∧ ∀y ∈ C, y ≠ x → G(x) > G(y)

• Relative: "KIM climbed the highest mountain" = the mountain that Kim climbed is higher than what anyone else climbed. Focus on "Kim" determines the comparison set.

The two readings arise from the scope of -est relative to other operators: wide scope yields relative, narrow scope yields absolute.

def Semantics.Degree.Superlative.absoluteSuperlative {Entity : Type u_1} {D : Type u_2} [LinearOrder D] (μ : Entity → D) (C : Set Entity) (x : Entity) : Prop

Absolute superlative: x is the G-est entity in comparison class C. "The tallest mountain" = the mountain x in C such that for all y ≠ x in C, height(x) > height(y).

Equations

• Semantics.Degree.Superlative.absoluteSuperlative μ C x = (x ∈ C ∧ ∀ y ∈ C, y ≠ x → μ x > μ y)

def Semantics.Degree.Superlative.relativeSuperlative {Alt : Type u_1} {Entity : Type u_2} {D : Type u_3} [LinearOrder D] (μ : Entity → D) (f : Alt → Entity) (focusedAlt : Alt) (alternatives : Set Alt) : Prop

Relative superlative: x has a higher degree than all focus alternatives. "KIM climbed the highest mountain" = Kim's mountain is higher than anyone else's.

f maps each alternative (person) to the relevant entity (the mountain they climbed). Focus alternatives determine the comparison.

Equations

• Semantics.Degree.Superlative.relativeSuperlative μ f focusedAlt alternatives = ∀ a ∈ alternatives, a ≠ focusedAlt → μ (f focusedAlt) > μ (f a)

theorem Semantics.Degree.Superlative.absolute_unique {Entity : Type u_1} {D : Type u_2} [LinearOrder D] (μ : Entity → D) (C : Set Entity) (x y : Entity) (hx : absoluteSuperlative μ C x) (hy : absoluteSuperlative μ C y) : x = y

The absolute superlative is unique (at most one entity satisfies it) when the ordering is strict.

theorem Semantics.Degree.Superlative.absoluteSuperlative_isGreatest {Entity : Type u_1} {D : Type u_2} [LinearOrder D] (μ : Entity → D) (C : Set Entity) (x : Entity) (h : absoluteSuperlative μ C x) : IsGreatest (μ '' C) (μ x)

The absolute superlative entails that μ(x) is the greatest element of the degree image μ '' C. Strict > for distinct entities implies ≥ for all.

The converse fails: IsGreatest allows ties (μ x = μ y), while absoluteSuperlative requires strict dominance.

theorem Semantics.Degree.Superlative.superlative_iff_universal_comparative {Entity : Type u_1} {D : Type u_2} [LinearOrder D] (μ : Entity → D) (C : Set Entity) (x : Entity) : absoluteSuperlative μ C x ↔ x ∈ C ∧ ∀ y ∈ C, y ≠ x → Comparative.comparativeSem μ x y Core.Scale.ScalePolarity.positive

Superlative = universal comparative (@cite{heim-1999}): "x is the tallest in C" iff "x is taller than every other y in C".

The superlative universally quantifies over the comparative: ⟦-est⟧ applies ⟦-er⟧ to every alternative in the comparison class. This semantic decomposition is the reflex of @cite{bobaljik-2012}'s morphosyntactic containment hypothesis ([[[ADJ] CMPR] SPRL]).