Theory Space for the Morphology/Syntax Interface

@cite{kalin-bjorkman-etal-2026}

@cite{kalin-bjorkman-etal-2026} identify four binary dimensions along which theories of the morphology/syntax interface vary. These dimensions are partially independent: some combinations are structurally impossible (a process-based theory cannot be non-lexicalist, since generative syntax is piece-based). The constrained space defines the positions of the major contemporary theories: DM, PFM, Nanosyntax, and MaS.

The four dimensions

1. Lexicalism (§2.1.1): whether the Morphology is a dedicated component separate from the Syntax (lexicalist) or whether morphological and syntactic computation use the same principles (non-lexicalist).

2. Architecture (§2.1.2): the relative ordering of the Morphology and the Syntax. Lexicalist theories use pre-syntactic or parallel architectures; non-lexicalist theories use syntactic (morphology = syntax) or post-syntactic architectures.

3. Pieces vs processes (§2.1.3): whether complex morphological forms result from combining discrete stored pieces (Item-and-Arrangement) or from applying (morpho)phonological rules/transformations to stems (Item-and-Process). Non-lexicalist theories are necessarily piece-based, since syntax is piece-based.

4. Realizational vs incremental (§2.1.4): whether phonological exponents are separated from meanings/functions and matched later (realizational) or built up in lockstep with meaning (incremental).

Structural constraints

Not all 2⁴ = 16 combinations are possible:

• Process-based → lexicalist (syntax is piece-based, so non-lexicalist morphology must also be piece-based)
• Syntactic architecture → non-lexicalist (by definition: morphology is syntax)
• Post-syntactic architecture → non-lexicalist (morphology operates on syntactic output)
• Pre-syntactic architecture → lexicalist (morphology feeds syntax, implying separation)
• Parallel architecture → lexicalist (morphology and syntax are co-present but independent)

inductive Morphology.TheorySpace.Lexicalism : Type

Whether the Morphology is a dedicated component separate from the Syntax (lexicalist) or uses the same computational system (non-lexicalist). @cite{kalin-bjorkman-etal-2026} §2.1.1.

• lexicalist : Lexicalism

  Morphology is a separate component; the Lexical Integrity Hypothesis holds (syntax cannot manipulate sub-word pieces).

• nonLexicalist : Lexicalism

  Morphological and syntactic computation operate with the same kinds of principles and processes.

@[implicit_reducible]

instance Morphology.TheorySpace.instDecidableEqLexicalism : DecidableEq Lexicalism

Equations

• Morphology.TheorySpace.instDecidableEqLexicalism x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Morphology.TheorySpace.instReprLexicalism.repr : Lexicalism → Nat → Std.Format

Equations

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

@[implicit_reducible]

instance Morphology.TheorySpace.instReprLexicalism : Repr Lexicalism

Equations

• Morphology.TheorySpace.instReprLexicalism = { reprPrec := Morphology.TheorySpace.instReprLexicalism.repr }

inductive Morphology.TheorySpace.Architecture : Type

The relative ordering of morphological and syntactic computation. @cite{kalin-bjorkman-etal-2026} §2.1.2.

Lexicalist theories use preSyntactic or parallel architectures. Non-lexicalist theories use syntactic or postSyntactic.

• preSyntactic : Architecture

  Morphology feeds the Syntax (input to syntactic computation). Lexicalist.

• parallel : Architecture

  Morphology and Syntax run independently, mapping to each other. Lexicalist.

• syntactic : Architecture

  Morphology is the Syntax (no separate morphological component). Non-lexicalist.

• postSyntactic : Architecture

  Syntax feeds the Morphology (morphology operates on syntactic output). Non-lexicalist.

@[implicit_reducible]

instance Morphology.TheorySpace.instDecidableEqArchitecture : DecidableEq Architecture

Equations

• Morphology.TheorySpace.instDecidableEqArchitecture x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Morphology.TheorySpace.instReprArchitecture.repr : Architecture → Nat → Std.Format

Equations

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

@[implicit_reducible]

instance Morphology.TheorySpace.instReprArchitecture : Repr Architecture

Equations

• Morphology.TheorySpace.instReprArchitecture = { reprPrec := Morphology.TheorySpace.instReprArchitecture.repr }

inductive Morphology.TheorySpace.Exponence : Type

Whether complex morphological forms result from combining discrete stored pieces (Item-and-Arrangement) or from applying rules to stems (Item-and-Process). @cite{kalin-bjorkman-etal-2026} §2.1.3.

• pieceBased : Exponence

  Complex words = combination of discrete, independently-stored morphemes. Traditional morphemes are primitive.

• processBased : Exponence

  Complex words = result of applying (morpho)phonological modifications (processes) to a stem. Morphemes are not primitive; affixation is one possible output of a rule.

@[implicit_reducible]

instance Morphology.TheorySpace.instDecidableEqExponence : DecidableEq Exponence

Equations

• Morphology.TheorySpace.instDecidableEqExponence x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Morphology.TheorySpace.instReprExponence : Repr Exponence

Equations

• Morphology.TheorySpace.instReprExponence = { reprPrec := Morphology.TheorySpace.instReprExponence.repr }

def Morphology.TheorySpace.instReprExponence.repr : Exponence → Nat → Std.Format

Equations

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

inductive Morphology.TheorySpace.Mapping : Type

Whether phonological exponents are independent of or unified with the meanings/functions they realize. @cite{kalin-bjorkman-etal-2026} §2.1.4.

• realizational : Mapping

  Features/meanings precede or are independent of phonological exponents. Exponents realize already-present features. Late Insertion is the prototypical realizational mechanism.

• incremental : Mapping

  Form and meaning are built up in lockstep. A morpheme is a pairing of form and meaning; adding it adds both simultaneously.

@[implicit_reducible]

instance Morphology.TheorySpace.instDecidableEqMapping : DecidableEq Mapping

Equations

• Morphology.TheorySpace.instDecidableEqMapping x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Morphology.TheorySpace.instReprMapping : Repr Mapping

Equations

• Morphology.TheorySpace.instReprMapping = { reprPrec := Morphology.TheorySpace.instReprMapping.repr }

def Morphology.TheorySpace.instReprMapping.repr : Mapping → Nat → Std.Format

Equations

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

structure Morphology.TheorySpace.TheoryPosition : Type

A position in the four-dimensional theory space. @cite{kalin-bjorkman-etal-2026} Table 2.

• lexicalism : Lexicalism
• architecture : Architecture
• exponence : Exponence
• mapping : Mapping

def Morphology.TheorySpace.instDecidableEqTheoryPosition.decEq (x✝ x✝¹ : TheoryPosition) : Decidable (x✝ = x✝¹)

Equations

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

@[implicit_reducible]

instance Morphology.TheorySpace.instDecidableEqTheoryPosition : DecidableEq TheoryPosition

Equations

• Morphology.TheorySpace.instDecidableEqTheoryPosition = Morphology.TheorySpace.instDecidableEqTheoryPosition.decEq

@[implicit_reducible]

instance Morphology.TheorySpace.instReprTheoryPosition : Repr TheoryPosition

Equations

• Morphology.TheorySpace.instReprTheoryPosition = { reprPrec := Morphology.TheorySpace.instReprTheoryPosition.repr }

def Morphology.TheorySpace.instReprTheoryPosition.repr : TheoryPosition → Nat → Std.Format

Equations

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

def Morphology.TheorySpace.dm : TheoryPosition

Distributed Morphology (@cite{halle-marantz-1993}). Non-lexicalist, post-syntactic, piece-based, realizational.

Equations

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

def Morphology.TheorySpace.pfm : TheoryPosition

Paradigm Function Morphology (@cite{stump-2001}). Lexicalist, parallel, process-based, realizational.

Equations

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

def Morphology.TheorySpace.nanosyntax : TheoryPosition

Nanosyntax (@cite{starke-2009}). Non-lexicalist, post-syntactic, piece-based, realizational. Shares DM's position on all four dimensions; differs in the size of spellout (phrasal, not terminal).

Equations

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

def Morphology.TheorySpace.mas : TheoryPosition

Morphology as Syntax (@cite{collins-kayne-2023}). Non-lexicalist, syntactic (integrated), piece-based, incremental.

Equations

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

def Morphology.TheorySpace.TheoryPosition.wellFormed (t : TheoryPosition) : Bool

A theory position is well-formed if its dimension values respect the structural dependencies identified by @cite{kalin-bjorkman-etal-2026}:

• Process-based → lexicalist (syntax is piece-based)
• Syntactic/post-syntactic architecture → non-lexicalist
• Pre-syntactic/parallel architecture → lexicalist

Equations

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

theorem Morphology.TheorySpace.dm_wellFormed : dm.wellFormed = true

theorem Morphology.TheorySpace.pfm_wellFormed : pfm.wellFormed = true

theorem Morphology.TheorySpace.nanosyntax_wellFormed : nanosyntax.wellFormed = true

theorem Morphology.TheorySpace.mas_wellFormed : mas.wellFormed = true

theorem Morphology.TheorySpace.dm_eq_nanosyntax : dm = nanosyntax

DM and Nanosyntax agree on all four dimensions.

theorem Morphology.TheorySpace.dm_pfm_share_mapping : dm.mapping = pfm.mapping

DM and PFM agree only on mapping (both realizational).

theorem Morphology.TheorySpace.dm_pfm_differ_lexicalism : dm.lexicalism ≠ pfm.lexicalism

theorem Morphology.TheorySpace.dm_pfm_differ_exponence : dm.exponence ≠ pfm.exponence

theorem Morphology.TheorySpace.dm_mas_share_lexicalism : dm.lexicalism = mas.lexicalism

DM and MaS agree only on lexicalism and exponence.

theorem Morphology.TheorySpace.dm_mas_share_exponence : dm.exponence = mas.exponence

theorem Morphology.TheorySpace.dm_mas_differ_architecture : dm.architecture ≠ mas.architecture

theorem Morphology.TheorySpace.dm_mas_differ_mapping : dm.mapping ≠ mas.mapping

theorem Morphology.TheorySpace.pfm_unique_processBased : pfm.exponence = Exponence.processBased ∧ dm.exponence = Exponence.pieceBased ∧ nanosyntax.exponence = Exponence.pieceBased ∧ mas.exponence = Exponence.pieceBased

PFM is the only major theory that is process-based.

theorem Morphology.TheorySpace.mas_unique_incremental : mas.mapping = Mapping.incremental ∧ dm.mapping = Mapping.realizational ∧ nanosyntax.mapping = Mapping.realizational ∧ pfm.mapping = Mapping.realizational

MaS is the only major theory that is incremental.

theorem Morphology.TheorySpace.nonLexicalist_processBased_illFormed (a : Architecture) (m : Mapping) : { lexicalism := Lexicalism.nonLexicalist, architecture := a, exponence := Exponence.processBased, mapping := m }.wellFormed = false

A non-lexicalist, process-based theory is ill-formed: syntax is piece-based, so non-lexicalist morphology (which shares the syntactic computation) must also be piece-based.

theorem Morphology.TheorySpace.lexicalist_syntactic_illFormed (e : Exponence) (m : Mapping) : { lexicalism := Lexicalism.lexicalist, architecture := Architecture.syntactic, exponence := e, mapping := m }.wellFormed = false

A lexicalist theory cannot have syntactic architecture (morphology is syntax contradicts morphology being separate).