Borer 2013: categorial flexibility and the adjective asymmetry

[Bor13]

Roots as phonological indices realized in categorial frames, with no zero categorizer heads: a bare root inserted in a nominal frame is N-equivalent, in a verbal frame V-equivalent — categorization by environment, against DM's merger with a (possibly null) categorizing head. On Morphology.Realization the exoskeletal claim fixes Ctx as the frame itself, with no head in it; the DM rival puts a categorizer in Ctx — a contrast in what instantiates Ctx, not in the shared core.

The book's categorial puzzle: noun-verb flexibility is systematic (chair, walk occur in both frames with no added morphology), but adjectives do not participate — property roots do not extend to nominal or verbal frames (to red, to fat are out), and the verbal uses that do exist (to thin, to yellow) are unpredictable and listed. Adjectives are complex and derived, not bare roots in frames.

Main results

• english — a mini root realization over N/V/A frames.
• nv_flexibility — chair and walk are licensed in both the nominal and verbal frames.
• adjective_rigidity — red and fat are licensed in no frame beyond the adjectival one.
• listedness — thin and red are both property roots, yet only thin has a verbal cell: no frame-level generalization predicts the difference, so the verbal uses are listed.

inductive Borer2013.Frame : Type

The three categorial frames.

• nFrame : Frame
• vFrame : Frame
• aFrame : Frame

@[implicit_reducible]

instance Borer2013.instDecidableEqFrame : DecidableEq Frame

Equations

• Borer2013.instDecidableEqFrame x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Borer2013.instFintypeFrame : Fintype Frame

Equations

• Borer2013.instFintypeFrame = { elems := { val := ↑Borer2013.Frame.enumList, nodup := Borer2013.Frame.enumList_nodup }, complete := Borer2013.instFintypeFrame._proof_1 }

def Borer2013.instReprFrame.repr : Frame → ℕ → Std.Format

Equations

• Borer2013.instReprFrame.repr Borer2013.Frame.nFrame prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Frame.nFrame")).group prec✝
• Borer2013.instReprFrame.repr Borer2013.Frame.vFrame prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Frame.vFrame")).group prec✝
• Borer2013.instReprFrame.repr Borer2013.Frame.aFrame prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Frame.aFrame")).group prec✝

@[implicit_reducible]

instance Borer2013.instReprFrame : Repr Frame

Equations

• Borer2013.instReprFrame = { reprPrec := Borer2013.instReprFrame.repr }

inductive Borer2013.Lex : Type

The sample roots.

• chair : Lex
• walk : Lex
• red : Lex
• fat : Lex
• thin : Lex
• yellow : Lex

@[implicit_reducible]

instance Borer2013.instDecidableEqLex : DecidableEq Lex

Equations

• Borer2013.instDecidableEqLex x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Borer2013.instFintypeLex : Fintype Lex

Equations

• Borer2013.instFintypeLex = { elems := { val := ↑Borer2013.Lex.enumList, nodup := Borer2013.Lex.enumList_nodup }, complete := Borer2013.instFintypeLex._proof_1 }

@[implicit_reducible]

instance Borer2013.instReprLex : Repr Lex

Equations

• Borer2013.instReprLex = { reprPrec := Borer2013.instReprLex.repr }

def Borer2013.instReprLex.repr : Lex → ℕ → Std.Format

Equations

• Borer2013.instReprLex.repr Borer2013.Lex.chair prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Lex.chair")).group prec✝
• Borer2013.instReprLex.repr Borer2013.Lex.walk prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Lex.walk")).group prec✝
• Borer2013.instReprLex.repr Borer2013.Lex.red prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Lex.red")).group prec✝
• Borer2013.instReprLex.repr Borer2013.Lex.fat prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Lex.fat")).group prec✝
• Borer2013.instReprLex.repr Borer2013.Lex.thin prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Lex.thin")).group prec✝
• Borer2013.instReprLex.repr Borer2013.Lex.yellow prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Borer2013.Lex.yellow")).group prec✝

def Borer2013.english : Morphology.Realization Lex Frame String

The mini English system: flexible chair/walk, frame-bound red/fat, and the listed verbal uses of thin/yellow.

Equations

• One or more equations did not get rendered due to their size.

theorem Borer2013.nv_flexibility : english.IsLicensed Lex.chair Frame.nFrame ∧ english.IsLicensed Lex.chair Frame.vFrame ∧ english.IsLicensed Lex.walk Frame.nFrame ∧ english.IsLicensed Lex.walk Frame.vFrame

Noun-verb flexibility is systematic: the bare roots chair and walk are licensed in both frames, with identical spellout — no zero head, no added morphology.

theorem Borer2013.adjective_rigidity : ¬english.IsLicensed Lex.red Frame.nFrame ∧ ¬english.IsLicensed Lex.red Frame.vFrame ∧ ¬english.IsLicensed Lex.fat Frame.nFrame ∧ ¬english.IsLicensed Lex.fat Frame.vFrame

The adjective asymmetry: red and fat occur in the adjectival frame only — property roots do not extend to nominal or verbal frames.

theorem Borer2013.listedness : english.IsLicensed Lex.thin Frame.aFrame ∧ english.IsLicensed Lex.red Frame.aFrame ∧ english.IsLicensed Lex.thin Frame.vFrame ∧ ¬english.IsLicensed Lex.red Frame.vFrame

Listedness: thin and red are both adjectival-frame roots, yet only thin has a verbal cell — the difference tracks no frame-level property, so the verbal uses of to thin, to yellow are listed, not generated.