Bochnak (2015): The Degree Semantics Parameter and cross-linguistic variation

Bochnak argues that Washo lacks degree morphology altogether, with no comparatives, measure phrases, equatives, superlatives or degree adverbs, and analyzes its gradable predicates as Klein-style vague predicates relative to a comparison class with no degree variable, the negative setting of Beck's Degree Semantics Parameter. Comparison is a conjoined construction, one clause asserting the positive and the other denying it, whose truth conditions entail a comparison through Klein's consistency constraints, the sound and complete delineations of Semantics/Degree/Delineation.lean. Kennedy's two diagnostics of implicit comparison follow: incompatibility with absolute-standard predicates, and the crisp-judgment effect under the similarity constraint, stated with van Rooij's margin-of-error order.

Main statements

• comparative_ofMeasure_height_iff: the conjoined comparison closes to Klein's comparative, which for a measure-induced delineation is height comparison, with no degree variable.
• eq24a_bent_straight_fails, english_more_bent_succeeds: on two bent rods every conjoined pairing fails where the explicit comparative succeeds.
• crisp_pair_not_marginOrder: a minimally different pair defeats implicit comparison at any positive margin.
• cc_b_requires_shared_class: the comparison entailment needs one comparison class shared by both conjuncts.

References

• [Boc15a]
• [BKF+09]
• [Kle80]
• [Ken07a]
• [Ken07b]
• [Gra00]
• [vR11a]

The two lexical shapes ((1), (5))

The paper's proposal is a contrast in lexical type. English tall (1) takes a degree argument for degree morphology to bind; Washo entries (5) are delineations, Degree.Delineation E, with — p. 6:4 — "no measure function, and no degree variable at all". No Washo constant is defined: the theorems below quantify over delineations, keeping the no-measure discipline visible in the types.

def Bochnak2015.tallEnglish {E : Type u_1} (height : E → ℕ) (d : ℕ) (x : E) : Prop

The degree-based English entry (1) is [[tall]] = λd λx. height(x) ≥ d, of type ⟨d,⟨e,t⟩⟩.

Equations

• Bochnak2015.tallEnglish height d x = (height x ≥ d)

Conjoined comparison ((14), (27)–(29))

def Bochnak2015.washoConjoined {E : Type u_1} (del : Degree.Delineation E) (C : Set E) (x y : E) : Prop

The conjoined comparison (14), with truth conditions (27), holds in context C when x counts as G and y does not. No comparative morpheme, overt or covert, is involved.

Equations

• Bochnak2015.washoConjoined del C x y = (x ∈ del.toFun C ∧ y ∉ del.toFun C)

theorem Bochnak2015.washoConjoined_comparative {E : Type u_1} (del : Degree.Delineation E) {C : Set E} {x y : E} (h : washoConjoined del C x y) : del.Comparative x y

Klein's existential comparative is the closure of (27) over contexts, since a conjoined comparison witnesses the discriminating class.

theorem Bochnak2015.washoConjoined_norm_related {E : Type u_1} (del : Degree.Delineation E) {C : Set E} {x y : E} (h : x ∉ del.toFun C) : ¬washoConjoined del C x y

Where the target does not count as G, both individuals being short in the paper's context (29), the conjoined comparison is simply false, so conjoined comparison is obligatorily norm-related.

theorem Bochnak2015.comparative_ofMeasure_height_iff {E : Type u_1} (height : E → ℕ) (a b : E) : (Degree.Delineation.ofMeasure height).Comparative a b ↔ height b < height a

For a measure-induced delineation the closure of (27) coincides with height comparison — the (28) entailment, through Delineation.IsSoundFor and Delineation.IsCompleteFor (the paper's Consistency Constraints in Semantics/Degree/Delineation.lean), with no degree variable in the entry.

Test 1: absolute standards ((23)–(24))

Two rods, both slightly bent, one more than the other. The English explicit comparative (23a) is true; each Washo conjoined attempt (24a–c) requires straight or not bent to hold of one rod, which is false.

def Bochnak2015.bentPred {E : Type u_1} (curvature : E → ℕ) (x : E) : Prop

A minimum-standard predicate such as bent or wet holds when the measure exceeds the scale's bottom. The standard is the endpoint, not a comparison class.

Equations

• Bochnak2015.bentPred curvature x = (curvature x > 0)

def Bochnak2015.straightPred {E : Type u_1} (curvature : E → ℕ) (x : E) : Prop

A maximum-standard predicate such as straight or dry, the lexical antonym of bentPred, holds when the measure sits at the endpoint.

Equations

• Bochnak2015.straightPred curvature x = (curvature x = 0)

theorem Bochnak2015.eq24a_bent_straight_fails {E : Type u_1} (curvature : E → ℕ) (x y : E) (_hx : bentPred curvature x) (hy : bentPred curvature y) : ¬(bentPred curvature x ∧ straightPred curvature y)

Bent ∧ straight (24a) fails, since both rods have nonzero curvature.

theorem Bochnak2015.eq24b_bent_notbent_fails {E : Type u_1} (curvature : E → ℕ) (x y : E) (_hx : bentPred curvature x) (hy : bentPred curvature y) : ¬(bentPred curvature x ∧ ¬bentPred curvature y)

Bent ∧ ¬bent (24b) fails, since the second rod is bent too.

theorem Bochnak2015.eq24c_straight_notstraight_fails {E : Type u_1} (curvature : E → ℕ) (x y : E) (hx : bentPred curvature x) (_hy : bentPred curvature y) : ¬(straightPred curvature x ∧ ¬straightPred curvature y)

Straight ∧ ¬straight (24c) fails, since the first rod is not straight.

theorem Bochnak2015.english_more_bent_succeeds {E : Type u_1} (curvature : E → ℕ) (x y : E) (hmore : curvature x > curvature y) : ∃ (d : ℕ), tallEnglish curvature d x ∧ ¬tallEnglish curvature d y

The explicit comparative (23a) succeeds in the same scenario, since any nonzero difference suffices.

Test 2: crisp judgments and the margin of error ((20)–(22), (58)–(60))

The Similarity Constraint (20) bars separating a minimally-different pair, so the conjoined form is infelicitous on two nearly-equal ladders (21) — salvageable with hedges like wewš 'almost' (22). The paper's van Rooij section makes the margin formal: implicit comparison rests on the margin-of-error order (60), a semi-order (58); explicit comparison is its ε = 0 case, the strict weak order (59).

def Bochnak2015.marginOrder {E : Type u_1} (μ : E → ℕ) (ε : ℕ) (x y : E) : Prop

In the margin-of-error order (60), x exceeds y by more than the margin of error ε.

Equations

• Bochnak2015.marginOrder μ ε x y = (μ y + ε < μ x)

theorem Bochnak2015.marginOrder_irrefl {E : Type u_1} (μ : E → ℕ) (ε : ℕ) {x : E} : ¬marginOrder μ ε x x

The margin-of-error order is irreflexive (58a).

theorem Bochnak2015.marginOrder_intervalOrder {E : Type u_1} (μ : E → ℕ) (ε : ℕ) {x y v w : E} (h₁ : marginOrder μ ε x y) (h₂ : marginOrder μ ε v w) : marginOrder μ ε x w ∨ marginOrder μ ε v y

The margin-of-error order is an interval order (58b).

theorem Bochnak2015.marginOrder_semitransitive {E : Type u_1} (μ : E → ℕ) (ε : ℕ) {x y z : E} (h₁ : marginOrder μ ε x y) (h₂ : marginOrder μ ε y z) (v : E) : marginOrder μ ε x v ∨ marginOrder μ ε v z

The margin-of-error order is semi-transitive (58c).

theorem Bochnak2015.marginOrder_zero_iff {E : Type u_1} (μ : E → ℕ) {x y : E} : marginOrder μ 0 x y ↔ μ y < μ x

At ε = 0 the margin order is plain measure comparison.

theorem Bochnak2015.marginOrder_zero_almostConnected {E : Type u_1} (μ : E → ℕ) {x y : E} (h : marginOrder μ 0 x y) (z : E) : marginOrder μ 0 x z ∨ marginOrder μ 0 z y

At ε = 0 the order is almost connected (59c), and so, with irreflexivity and transitivity, the strict weak order of explicit comparison.

theorem Bochnak2015.crisp_pair_not_marginOrder {E : Type u_1} (μ : E → ℕ) (ε : ℕ) {x y : E} (hε : 1 ≤ ε) (hclose : μ x = μ y + 1) : ¬marginOrder μ ε x y

A minimally different pair never clears a positive margin, the crisp-judgment infelicity of the conjoined form (21).

theorem Bochnak2015.crisp_pair_marginOrder_zero {E : Type u_1} (μ : E → ℕ) {x y : E} (hclose : μ x = μ y + 1) : marginOrder μ 0 x y

The same pair satisfies explicit comparison, since -er needs only a nonzero difference.

Footnote 11: the shared comparison class

theorem Bochnak2015.cc_b_requires_shared_class : ∃ (Entity : Type) (del : Degree.Delineation Entity) (R : Entity → Entity → Prop), del.IsSoundFor R ∧ ∃ (C₁ : Set Entity) (C₂ : Set Entity) (x : Entity) (y : Entity), x ∈ del.toFun C₁ ∧ y ∉ del.toFun C₂ ∧ ¬R x y

The (28) entailment requires one comparison class shared by both conjuncts of (27), as footnote 11 notes. Soundness constrains only same-class separations, so a delineation sound for R can separate x from y across two distinct classes while R x y fails.