[Boc15a] Degree Semantics Parameter and Washo

[Boc15a] The Degree Semantics Parameter and cross-linguistic variation (Semantics and Pragmatics 8(6): 1–48, doi:10.3765/sp.8.6) argues that Washo (Hokan isolate, California/Nevada) systematically lacks degree morphology: no comparatives, no measure phrases, no degree adverbs, no equatives, no superlatives. The proposal is that Washo gradable predicates are degree-free vague predicates in the [Kle80] style — type ⟨e, t⟩ relative to a contextually- supplied comparison class, with no degree variable. This positions Washo as an empirical attestation of the negative setting of [BKF+09]'s Degree Semantics Parameter (DSP), and as a counterexample to the universalist (English-projective) view that all natural-language gradable predicates introduce degree arguments.

Why this paper grounds linglib's substrate

Linglib's [Kle80]/[Ken07] comparison hierarchy (Semantics/Degree/Gradability/Hierarchy.lean) already proves degree_characterization: degree semantics is exactly the monotone fragment of Klein's delineation framework. [Boc15a]'s Washo data sits inside this monotone fragment (Washo tall, long, bent are single-criterion monotone predicates), so the empirical interest is NOT that Washo motivates the strict-generality results (delineation_strictly_more_general, nlDel_not_degree_representable) — those are about non-monotone clever-style predicates, a different phenomenon. The empirical interest is that Bochnak shows the truth-conditional equivalence of degree-based and Klein-based comparatives does NOT entail a particular LEXICAL TYPE: a language can have monotone delineations as its lexical entries WITHOUT exposing degrees in those entries' types or admitting degree morphology.

Sections

1. The Degree Semantics Parameter as a typed parameter on languages (Bochnak eq. 7, after [BKF+09]).
2. English vs Washo lexical-entry TYPES (eqs. 1, 5/11) — the central contrast is at the level of types, not denotations.
3. The conjoined-comparison construction (eq. 14, eq. 27) and its per-context truth conditions; existential closure recovers [Kle80]'s comparativeSem.
4. Absolute-standard incompatibility (eq. 23–24a/b/c): all three sub-cases mechanically derived. This is one of [Boc15a]'s two diagnostics that conjoined comparison is implicit ([Ken07]'s sense).
5. Crisp-judgment effect under the Similarity Constraint (eqs. 20–22): stipulated as a felicity predicate distinct from truth conditions (the constraint is pragmatic per [Kle80], [Gra00]). The granularity threshold is a linglib modeling choice, not from the paper.

§2.2 Consistency Constraints — substrate in Delineation.lean §13

[Boc15a]'s eq. (28a/b) Consistency Constraints — his "strongest formal result" per the linguistics audit — are substrate- level. Delineation.lean §13 houses IsSoundDelineation (CC-b shape, generalising eq. 28b to abstract scalar relations R per the paper's "the scalar concept encoded by G" wording) and IsCompleteDelineation (the converse direction, NOT in Bochnak — closer to [Bur17]'s Plenitude / Granularity axioms). CC-a (eq. 28a) is exactly IsMonotoneDelineation _ Set.univ — no separate substrate needed.

The §3 comparison-entailment theorem factors through comparativeSem_iff_of_sound_and_complete: the equivalence comparativeSem del a b ↔ height b < height a is a one-line corollary firing via typeclass synthesis from instSoundMeasureDelineation / instCompleteMeasureDelineation. The smuggled-measure workaround flagged by the 0.230.434 audit is closed.

The load-bearing footnote-11 caveat (single shared comparison class) is formalised as cc_b_requires_shared_class in §6 below — paper-anchored here since the footnote is Bochnak-specific.

Future work still flagged

• §3–§4 van Rooij-style degree-free alternative analysis. Bochnak ultimately rejects it on parsimony grounds (§4: requires unifying differential MPs and crisp-judgment witnesses). A faithful formalization should engage [vR03a] (and successors not yet in the bib) and prove Bochnak's parsimony argument.
• §4.3 Wellwood (2014) much-based middle-ground analysis and the Washo t'e:k'e' counterevidence (eqs. 64–68). Bochnak's most original cross-linguistic argument lives here.

def Bochnak2015.DegreeSemanticsParameter : Type

[BKF+09] (DSP), as adopted by [Boc15a] eq. 7. The DSP records whether a language's gradable lexicon introduces degree arguments. true for English-type, false for Washo-type. A language-level setting per [Boc15a], not per-predicate.

Equations

• Bochnak2015.DegreeSemanticsParameter = Bool

def Bochnak2015.english : DegreeSemanticsParameter

English: positive DSP setting (degree arguments + degree morphology).

Equations

• Bochnak2015.english = true

def Bochnak2015.washo : DegreeSemanticsParameter

Washo: negative DSP setting per [Boc15a].

Equations

• Bochnak2015.washo = false

def Bochnak2015.motu : DegreeSemanticsParameter

Motu (Austronesian, Papua New Guinea): also negative DSP per [BKF+09]'s appendix and [Sta85] typology. Bochnak's eq. 6 records the conjoined-comparison construction.

Equations

• Bochnak2015.motu = false

theorem Bochnak2015.english_pos : english = true

theorem Bochnak2015.washo_neg : washo = false

theorem Bochnak2015.motu_neg : motu = false

[Boc15a]'s central proposal is at the level of LEXICAL TYPES, not denotations:

- English `[[tall]]` (eq. 1): type ⟨d, ⟨e, t⟩⟩ — takes a degree
  argument that comparative/measure morphology binds.
- Washo `[[tall_Washo]]^c` (eq. 5/11): type ⟨e, t⟩ relative to a
  comparison class. **No degree argument; no measure function in
  the denotation.**

Bochnak (p. 6:4): *"The semantics in (5) contains no measure
function, and no degree variable at all."*

In linglib types: English entries have shape `ℕ → E → Prop`
(degree-saturated ⟨e,t⟩); Washo entries have shape
`ComparisonClass E → E → Prop` — the type signature of any
Klein-style delineation (`Delineation.lean` line 72ff).

We define `tallEnglish` to demonstrate the English shape. We do
NOT define a `tallWasho` constant: the Washo lexicon's entry IS
just an arbitrary delineation; theorems below quantify over
`del : ComparisonClass E → E → Prop` to keep the no-measure
discipline visible at the type level. 

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

[Boc15a] eq. 1: standard English-style degree-based [[tall]] = λdλx. height(x) ≽ d. Type ⟨d, ⟨e, t⟩⟩.

Equations

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

def Bochnak2015.washoConjoined {E : Type u_1} (del : Semantics.Gradability.Delineation.ComparisonClass E → E → Prop) (C : Semantics.Gradability.Delineation.ComparisonClass E) (x y : E) : Prop

[Boc15a] eq. 14: the Washo conjoined-comparison construction juxtaposes a positive form and a negated antonymic form (typically with the negation suffix -eːs). Bochnak argues NO comparative morpheme is involved — overt or covert.

[Boc15a] eq. 27 gives the truth conditions for a specific context C: x is more G than y iff x counts as G in C and y does not. Per-context Boolean conjunction of the lexical delineation and its negation.

Equations

• Bochnak2015.washoConjoined del C x y = (del C x ∧ ¬del C y)

theorem Bochnak2015.washoConjoined_witnesses_comparativeSem {E : Type u_1} (del : Semantics.Gradability.Delineation.ComparisonClass E → E → Prop) {C : Semantics.Gradability.Delineation.ComparisonClass E} {x y : E} (h : washoConjoined del C x y) : Semantics.Gradability.Delineation.comparativeSem del x y

[Kle80]'s existential comparativeSem is the existential closure of the Washo conjoined construction over comparison classes. [Boc15a]'s eq. 27 is the per-context form; Klein's is the "exists a discriminating context" form.

theorem Bochnak2015.comparativeSem_measureDelineation_iff_degree {E : Type u_1} (height : E → ℕ) (a b : E) : Semantics.Gradability.Delineation.comparativeSem (Semantics.Gradability.Delineation.measureDelineation height) a b ↔ height b < height a

Truth-conditional equivalence to height comparison for a measure-induced delineation. The Washo LEXICON exposes no degree variable (§2 above), but its truth conditions in a measure-induced model coincide with English's -er. The cross-linguistic divergence is at the level of TYPE and construction, not truth-conditional content.

Substrate-grounded derivation. Now factors through Delineation.lean §13's IsSoundDelineation / IsCompleteDelineation typeclasses (paper-anchored in ConsistencyConstraints.lean to [Boc15a] §2.2 eq. 28). The instSoundMeasureDelineation and instCompleteMeasureDelineation instances fire by typeclass synthesis; the equivalence is a one-line corollary of comparativeSem_iff_of_sound_and_complete. The lexical entry no longer needs to expose height for the comparison entailment to go through — height participates only via instance synthesis.

[Boc15a] eq. 23 (English) and eq. 24 (Washo, three sub-cases) record the diagnostic: conjoined comparison fails with absolute-standard predicates. The minimal scenario uses two slightly bent rods where one is more bent than the other; ALL three Washo conjoined attempts fail:

- eq. 24a: *bent ∧ straight* — fails because *straight* requires
  zero curvature, but both rods are bent.
- eq. 24b: *bent ∧ ¬bent* — fails because *both* rods are bent.
- eq. 24c: *straight ∧ ¬straight* — fails because *both* rods are
  bent (so neither is straight, but the form requires one to be).

These failures are construction-level (the conjoined form requires
an antonym OR negation to hold absolutely), not pragmatic. We
derive all three sub-cases mechanically. 

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

[Ken07] min-standard absolute-degree predicate: holds iff the measure exceeds the scale's bottom endpoint (here 0). Models bent, wet, dirty. The standard is fixed at the scale endpoint, not contextually supplied — so this predicate has no ComparisonClass parameter, unlike Klein-style delineations.

Equations

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

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

[Ken07] max-standard absolute-degree predicate: holds iff the measure is at the scale endpoint (0 for curvature). Models straight, dry, clean. The lexical antonym of bentPred.

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)

[Boc15a] eq. 24a: bent ∧ straight fails when 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)

[Boc15a] eq. 24b: bent ∧ ¬bent fails when both rods are bent — the second conjunct is false.

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)

[Boc15a] eq. 24c: straight ∧ ¬straight fails when both rods are bent — the first conjunct is false.

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 English degree-based comparative SUCCEEDS in the same scenario where all three Washo conjoined attempts fail. Witnesses Bochnak's diagnostic that conjoined comparison is implicit: same model, conjoined fails on every antonym pairing, comparative succeeds.

[Boc15a] eq. 20 records the Similarity Constraint (parenthetically attributed in the paper to [Kle80] and [Gra00]): when x and y differ only minimally in the property G, speakers are unable or unwilling to judge x is G ∧ y is not G as true.

This is a FELICITY constraint, not a truth-conditional one. The
Washo conjoined construction inherits it (eq. 21: the ladder
example is judged infelicitous in a minimal-difference context),
which is [bochnak-2015]'s second diagnostic that Washo
comparison is *implicit* in [kennedy-2007]'s sense.
[bochnak-2015] eq. 22 (the *wewš* "almost" hedge) shows the
construction can be salvaged in crisp contexts via hedges.

Linglib stipulates the Similarity Constraint as a felicity
predicate distinct from the truth conditions established in §3.
The empirical content is Bochnak's claim that Washo speakers obey
it where English `-er` users do not.

**Modeling-choice flag.** The granularity threshold `ε` and the
specific `ε ≥ 2` instantiation in `crisp_judgment_blocks_conjoined`
are linglib's parameterization; [bochnak-2015] gives no
numerical `ε`. The Lean theorem captures the SHAPE of the
constraint, not the paper's quantitative content. 

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

The granularity-ε distinguishability predicate underlying the Similarity Constraint: x and y are crisply distinguishable on measure μ at granularity ε iff their measure difference is at least ε.

Equations

• Bochnak2015.crisplyDistinguishable μ ε x y = (μ x ≥ μ y + ε ∨ μ y ≥ μ x + ε)

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

The conjoined-comparison felicity predicate. Felicitous on measure μ at granularity ε only if x exceeds y AND the pair is crisply distinguishable. SEPARATES truth conditions (the height inequality) from felicity (the granularity threshold).

Equations

• Bochnak2015.conjoinedFelicitous μ ε x y = (μ y < μ x ∧ Bochnak2015.crisplyDistinguishable μ ε x y)

theorem Bochnak2015.crisp_judgment_blocks_conjoined {E : Type u_1} (μ : E → ℕ) (ε : ℕ) (x y : E) (hgap : ε ≥ 2) (hclose : μ x = μ y + 1) : ¬conjoinedFelicitous μ ε x y

[Boc15a] eq. 21 (the ladder scenario) made formal under linglib's ε ≥ 2 modeling: a 1-unit measure difference cannot satisfy the Similarity Constraint, so the conjoined form is infelicitous.

theorem Bochnak2015.large_gap_restores_felicity {E : Type u_1} (μ : E → ℕ) (ε : ℕ) (x y : E) (hbigger : μ x ≥ μ y + ε) (hpos : μ y < μ x) : conjoinedFelicitous μ ε x y

[Boc15a] eq. 22 (the wewš "almost" hedge) shows the conjoined form CAN be salvaged in crisp contexts via hedges. Formally: a sufficiently large measure gap restores felicity even at large ε. The hedge effectively raises the granularity threshold the speaker considers crisp.

theorem Bochnak2015.cc_b_requires_shared_class : ∃ (Entity : Type) (del : Semantics.Gradability.Delineation.ComparisonClass Entity → Entity → Prop) (R : Entity → Entity → Prop), Semantics.Gradability.Delineation.IsSoundDelineation del R ∧ ∃ (C₁ : Semantics.Gradability.Delineation.ComparisonClass Entity) (C₂ : Semantics.Gradability.Delineation.ComparisonClass Entity) (x : Entity) (y : Entity), del C₁ x ∧ ¬del C₂ y ∧ ¬R x y

[Boc15a] fn 11 (p. 6:16): the comparison entailment requires a SINGLE comparison class shared by both conjuncts of the conjoined construction. Weakening to "exists C₁ that makes x positive AND exists C₂ (possibly distinct) that makes y negative" does NOT suffice.

Counterexample model: Entity := Bool. The delineation del C x := x = true ∧ true ∈ C makes true positive in any CC containing true, and false negative everywhere. Soundness w.r.t. R a b := a = true ∧ b = false holds (the only positive discrimination is x = true ∧ y = false). But the SPLIT-CC weakened form admits (x, y) = (true, true) via C₁ := {true} and C₂ := {false} — del C₁ true ∧ ¬ del C₂ true holds, while R true true = false.

Paper-anchored here (rather than substrate-level) because the footnote-11 caveat is Bochnak-specific: it explains why eq. (27) threads a single shared C through both conjuncts.