Documentation

theorem Minimalism.theta_iff_lexical (c : Cat) :

canAssignTheta c = true ↔ isLHead c = true

Theta assignment is exactly the lexical heads.

def Minimalism.isEPInternal (daughter parent : Cat) :

A daughter is EP-internal (complement) if it shares [±V, ±N] features with its parent AND has a strictly lower F-value.

EP-internal elements are inside the extended projection:

VP is EP-internal to vP (both verbal, F0 < F1)
NP is EP-internal to DP (both nominal, F0 < F1)

EP-external elements (specifiers) are outside:

DP in Spec-vP is EP-external (nominal ≠ verbal)

Equations

Minimalism.isEPInternal daughter parent = (Minimalism.categoryConsistent daughter parent && decide (Minimalism.fValue daughter < Minimalism.fValue parent))

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def Minimalism.isEPExternal (daughter parent : Cat) :

EP-external: either different family or not lower F-value. Specifiers are typically EP-external to the projection they sit.

Equations

Minimalism.isEPExternal daughter parent = !Minimalism.isEPInternal daughter parent

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def Minimalism.fullVerbalEP :

Full verbal EP: V → v → T → C. This is the standard clausal spine for finite clauses.

Equations

Minimalism.fullVerbalEP = [Minimalism.Cat.V , Minimalism.Cat.v , Minimalism.Cat.T , Minimalism.Cat.C ]

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def Minimalism.fullNominalEP :

Full nominal EP: N → n → Q → Num → D. Q (classifier / individuation) is below Num (number / counting) per @cite{borer-2005}: individuation must precede counting.

Equations

Minimalism.fullNominalEP = [Minimalism.Cat.N , Minimalism.Cat.n , Minimalism.Cat.Q , Minimalism.Cat.Num , Minimalism.Cat.D ]

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def Minimalism.smallClauseVerbalEP :

Small clause EP: just the lexical head, no functional layers. E.g., "consider [SC him intelligent]" — the SC has no T or C.

Equations

Minimalism.smallClauseVerbalEP = [Minimalism.Cat.V ]

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def Minimalism.adjectivalEP :

Adjectival EP: A → a. The minimal adjectival extended projection, parallel to the verbal (V → v) and nominal (N → n) categorizer layers. Further adjectival functional structure (DegP, etc.) is language-dependent and not included here.

Equations

Minimalism.adjectivalEP = [Minimalism.Cat.A , Minimalism.Cat.a ]

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def Minimalism.smallClauseAdjectivalEP :

Adjectival small clause EP: just A. E.g., "consider [SC him happy]"

Equations

Minimalism.smallClauseAdjectivalEP = [Minimalism.Cat.A ]

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def Minimalism.locationalPP :

Locational adpositional EP: P → Place. @cite{dendikken-2010}: locational PPs project PlaceP but not PathP. E.g., Dutch preP op de heuvel 'on the hill' (locational).

Equations

Minimalism.locationalPP = [Minimalism.Cat.P , Minimalism.Cat.Place ]

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def Minimalism.directionalPP :

Directional adpositional EP: P → Place → Path. @cite{dendikken-2010}: directional PPs project PathP above PlaceP. E.g., Dutch postP de heuvel op 'onto the hill' (directional).

Equations

Minimalism.directionalPP = [Minimalism.Cat.P , Minimalism.Cat.Place , Minimalism.Cat.Path ]

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def Minimalism.infinitivalEP :

Infinitival EP: V → v → T (no C). E.g., "want [to leave]" — truncated at T, no complementizer.

Equations

Minimalism.infinitivalEP = [Minimalism.Cat.V , Minimalism.Cat.v , Minimalism.Cat.T ]

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def Minimalism.isTruncated (spine : List Cat) :

Is this EP truncated (missing functional layers compared to the full EP)?

Equations

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

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def Minimalism.argumentDomainCat (topCat : Cat) :

Cat

The highest F-value that still denotes a property ⟨e,t⟩ or is in the intermediate zone. This defines the argument domain boundary.

For verbal EPs: the argument domain extends to vP (F1)

vP still denotes ⟨e,t⟩ (property of events)
TP (F2) starts binding tense → no longer ⟨e,t⟩

For nominal EPs: the argument domain extends to nP (F1)

nP still denotes ⟨e,t⟩ (property of entities)
NumP (F2) starts binding number → no longer ⟨e,t⟩ This parallels the verbal domain: v ↔ n at the same F-level.

Equations

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def Minimalism.isInArgumentDomain (c topCat : Cat) :

Is a category within the argument domain of a given top category? The argument domain includes all F-levels ≤ the boundary.

Equations

Minimalism.isInArgumentDomain c topCat = decide (Minimalism.fValue c ≤ Minimalism.fValue (Minimalism.argumentDomainCat topCat))

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epSemanticType Cat.v = EPSemanticType.intermediate

theorem Minimalism.v_is_intermediate :

v denotes an intermediate type (event quantification domain).

epSemanticType Cat.C = EPSemanticType.proposition

theorem Minimalism.c_is_proposition :

C denotes a proposition (clausal force).

epSemanticType Cat.D = EPSemanticType.entity

theorem Minimalism.d_is_entity :

D denotes an entity (determination/referentiality).

epSemanticType Cat.n = EPSemanticType.intermediate

theorem Minimalism.n_is_intermediate :

Nominal categorizer n is intermediate (still ⟨e,t⟩-ish).

epSemanticType Cat.Q = EPSemanticType.intermediate

theorem Minimalism.q_is_intermediate :

Q is intermediate (classifier/individuation: CUM → QUA).

epSemanticType Cat.Num = EPSemanticType.intermediate

theorem Minimalism.num_is_intermediate :

Num is intermediate (number/counting: QUA → measured).

All lexical heads denote properties.

theorem Minimalism.vp_internal_to_vp :

isEPInternal Cat.V Cat.v = true

VP is EP-internal to vP (complement position).

theorem Minimalism.np_internal_to_np :

isEPInternal Cat.N Cat.n = true

NP is EP-internal to nP (complement of categorizer).

theorem Minimalism.np_internal_to_qp :

isEPInternal Cat.n Cat.Q = true

nP is EP-internal to QP (complement of classifier/individuation).

theorem Minimalism.qp_internal_to_nump :

isEPInternal Cat.Q Cat.Num = true

QP is EP-internal to NumP (complement of number/counting).

theorem Minimalism.nump_internal_to_dp :

isEPInternal Cat.Num Cat.D = true

NumP is EP-internal to DP (complement of determiner).

theorem Minimalism.np_internal_to_dp :

isEPInternal Cat.N Cat.D = true

NP is EP-internal to DP (transitively).

theorem Minimalism.dp_external_to_vp :

isEPExternal Cat.D Cat.v = true

DP is EP-external to vP (specifier position): different [±V, ±N] features (nominal ≠ verbal).

argumentDomainCat Cat.C = Cat.v

theorem Minimalism.full_clause_argdomain :

The argument domain of a full clause (C) is vP.

argumentDomainCat Cat.T = Cat.v

theorem Minimalism.tp_argdomain :

The argument domain of a TP is also vP.

argumentDomainCat Cat.D = Cat.n

theorem Minimalism.full_dp_argdomain :

The argument domain of a full DP is nP (parallel to vP for clauses).

argumentDomainCat Cat.Q = Cat.n

theorem Minimalism.qp_argdomain :

The argument domain of QP is nP.

argumentDomainCat Cat.Num = Cat.n

theorem Minimalism.nump_argdomain :

The argument domain of NumP is nP.

theorem Minimalism.v_in_argdomain :

isInArgumentDomain Cat.V Cat.C = true

V is within the argument domain of a full clause.

theorem Minimalism.v_head_in_argdomain :

isInArgumentDomain Cat.v Cat.C = true

v is within the argument domain of a full clause.

theorem Minimalism.t_not_in_argdomain :

isInArgumentDomain Cat.T Cat.C = false

T is NOT within the argument domain of a full clause.

theorem Minimalism.c_not_in_argdomain :

isInArgumentDomain Cat.C Cat.C = false

C is NOT within the argument domain (it's the top).

theorem Minimalism.full_verbal_ep_wellformed :

allCategoryConsistent fullVerbalEP = true ∧ allFMonotone fullVerbalEP = true

Full verbal EP is well-formed: consistent and monotone.

allCategoryConsistent fullNominalEP = true ∧ allFMonotone fullNominalEP = true

theorem Minimalism.full_nominal_ep_wellformed :

Full nominal EP is well-formed: consistent and monotone.

theorem Minimalism.adjectival_ep_wellformed :

allCategoryConsistent adjectivalEP = true ∧ allFMonotone adjectivalEP = true

Adjectival EP is well-formed: consistent and monotone.

allCategoryConsistent infinitivalEP = true ∧ allFMonotone infinitivalEP = true

theorem Minimalism.infinitival_ep_wellformed :

Infinitival EP is well-formed.

isTruncated smallClauseVerbalEP = true

theorem Minimalism.small_clause_is_truncated :

A small clause is truncated relative to a full verbal EP.

isTruncated fullVerbalEP = false

theorem Minimalism.full_verbal_not_truncated :

A full verbal EP is not truncated.

F1+ heads cannot assign theta roles (@cite{grimshaw-2005} Definition 10). Note: Panagiotidis (2015 §4.5) argues categorizers (v, n, a) are lexical, not functional — but in Grimshaw's F-value system they are F1 (non-lexical). The theta restriction here follows Grimshaw, not Panagiotidis.

theorem Minimalism.lexical_heads_assign_theta :

canAssignTheta Cat.V = true ∧ canAssignTheta Cat.N = true ∧ canAssignTheta Cat.A = true ∧ canAssignTheta Cat.P = true

Lexical heads can assign theta roles.

theorem Minimalism.place_path_no_theta :

canAssignTheta Cat.Place = false ∧ canAssignTheta Cat.Path = false

Place and Path are functional heads: they do NOT assign theta roles.

theorem Minimalism.p_internal_to_place :

isEPInternal Cat.P Cat.Place = true

P is EP-internal to PlaceP: same [-V,-N], F0 < F1. @cite{dendikken-2010}: P is the lexical complement of Place.

theorem Minimalism.place_internal_to_path :

isEPInternal Cat.Place Cat.Path = true

PlaceP is EP-internal to PathP: same [-V,-N], F1 < F2. @cite{dendikken-2010}: Place is the complement of Path in directional PPs.

theorem Minimalism.locational_pp_ep_wellformed :

allCategoryConsistent locationalPP = true ∧ allFMonotone locationalPP = true

Locational PP EP is well-formed: consistent and monotone.

allCategoryConsistent directionalPP = true ∧ allFMonotone directionalPP = true

theorem Minimalism.directional_pp_ep_wellformed :

Directional PP EP is well-formed: consistent and monotone.

isTruncated locationalPP = true

theorem Minimalism.locational_pp_truncated :

A locational PP [P, Place] is truncated relative to the full adpositional EP [P, Place, Path]. @cite{dendikken-2010}: locational PPs lack the directional PathP layer.

allCategoryConsistent splitCPVerbalEP = true ∧ allFMonotone splitCPVerbalEP = true

theorem Minimalism.splitCP_ep_wellformed :

The split-CP spine is well-formed (consistent and monotone).

theorem Minimalism.fin_internal_to_foc :

isEPInternal Cat.Fin Cat.Foc = true

Fin is EP-internal to Foc: same [+V,-N], F3 < F4. The IP/CP boundary (Fin) is properly dominated by focus.

theorem Minimalism.foc_internal_to_top :

isEPInternal Cat.Foc Cat.Top = true

Foc is EP-internal to Top: same [+V,-N], F4 < F5. Focus is below topic in the C-domain hierarchy.

theorem Minimalism.top_internal_to_c :

isEPInternal Cat.Top Cat.C = true

Top is EP-internal to C: same [+V,-N], F5 < F6. Topic is below the complementizer (= Force in unsplit contexts).

theorem Minimalism.t_internal_to_fin :

isEPInternal Cat.T Cat.Fin = true

T is EP-internal to Fin: same [+V,-N], F2 < F3. Tense is properly dominated by finiteness.