Heim and Kratzer (1998): Semantics in Generative Grammar #
Heim and Kratzer interpret a tree by a few general rules: Functional Application, Predicate
Modification, Predicate Abstraction over a binder index, and, from Chapter 4, partial
denotations, with a Fregean definite article that may leave a node undefined. The library's
engines Tree.interp and Partial.interp implement these rules. Here the book's rules are
transcribed as interpretation relations that the engines extend, and the engines are run on the
toy model for the book's definite descriptions (Chapter 4), relative clauses (Chapter 5) and
quantifier scope (Chapter 7).
Main results #
interp_of_denotes,Partial.interp_of_denotes: where the book's rules assign a value, the engines compute it.the_student_fails,the_pizza,the_john_uninterpretable: presupposition failure and uninterpretability with the Fregean article (§4.4).interp_studentWhoSleeps,the_student_who_sleeps: a restrictive relative clause is an intersective modifier, so the student who sleeps is defined although the student is not (§5.1).det_noun_modifier_uninterpretable: the bracketing [[the noun] modifier] is uninterpretable (§4.5).scope_ambiguity_computed,surfaceScopeProp_mem_readings_inSitu: the two Quantifier Raising derivations of a doubly quantified sentence compute its two scope readings (§7.3), and the in-situ alternative of §7.2.1 derives the surface one.
Implementation notes #
The toy fragment's words stand in for the book's: student who sleeps for house which is
empty, every person sees some person for "Some publisher offended every linguist". With
interpTy .t = Prop the engines produce truth conditions as propositions.
References #
Model and lexicon #
A reading as a composition terminal on the entity domain E, at the determiner type.
Equations
- HeimKratzer1998.toDenotation d E W = ⟨Montague.Ty.det, d E⟩
Instances For
The terminals of a set of readings on a domain.
Equations
- HeimKratzer1998.terminals s E W = (fun (x : Quantifier.GQ.Family) => HeimKratzer1998.toDenotation x E W) '' s
Instances For
objectShift D is the object-position entry of the determiner D, which the book's lexical
rule (§7.2.1) derives from the basic one. It takes an object-first two-place predicate and the
subject and quantifies over the object.
Equations
- HeimKratzer1998.objectShift D P R x = D P fun (y : E) => R y x
Instances For
The object-position entry of a terminal of the determiner type, the book's lexical rule
applied to a lexical item, and none at any other type.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The object-position entries the lexical rule derives from a set of readings.
Equations
- HeimKratzer1998.objectShifts s = {d : Montague.Denotation E W | ∃ d₁ ∈ s, HeimKratzer1998.objectShift? d₁ = some d}
Instances For
The study's stand on the quantifier words, the one reading each of every and some makes available, as a terminal on the toy domain.
Equations
- HeimKratzer1998.quantifierReading English.Determiners.QuantityWord.every = some (HeimKratzer1998.toDenotation Quantifier.GQ.Family.every HeimKratzer.ToyEntity Unit)
- HeimKratzer1998.quantifierReading English.Determiners.QuantityWord.some_ = some (HeimKratzer1998.toDenotation Quantifier.GQ.Family.some HeimKratzer.ToyEntity Unit)
- HeimKratzer1998.quantifierReading x✝ = none
Instances For
Each chosen terminal is among the terminals of the word's available readings.
The leaf interpretation reads the quantifier words through their readings and the toy fragment's nouns and verbs through the toy lexicon.
Equations
Instances For
The book's rules #
The extensional rules of Chapters 3 to 5, one constructor per rule. The engine at M = Id
extends them.
The book's interpretation relation, relative to a leaf interpretation and an assignment.
- tn
{C L E W : Type}
{lex : L → Option (Montague.Denotation E W)}
{g : Assignment E}
{c : C}
{w : L}
{d : Montague.Denotation E W}
(h : lex w = some d)
: Denotes lex g (PhraseStructure.Tree.terminal c w) d
By Terminal Nodes, a leaf denotes what the lexicon gives it.
- nn
{C L E W : Type}
{lex : L → Option (Montague.Denotation E W)}
{g : Assignment E}
{c : C}
{t : PhraseStructure.Tree C L}
{d : Montague.Denotation E W}
(h : Denotes lex g t d)
: Denotes lex g (PhraseStructure.Tree.node c [t]) d
By Non-Branching Nodes, a node denotes what its only daughter does.
- faLeft
{C L E W : Type}
{lex : L → Option (Montague.Denotation E W)}
{g : Assignment E}
{c : C}
{t₁ t₂ : PhraseStructure.Tree C L}
{σ τ : Montague.Ty}
{f : Montague.Ty.Domain E W (σ ⇒ τ)}
{a : Montague.Ty.Domain E W σ}
(h₁ : Denotes lex g t₁ ⟨σ ⇒ τ, f⟩)
(h₂ : Denotes lex g t₂ ⟨σ, a⟩)
: Denotes lex g (PhraseStructure.Tree.node c [t₁, t₂]) ⟨τ, f a⟩
By Functional Application, a node whose left daughter is the function denotes its value.
- faRight
{C L E W : Type}
{lex : L → Option (Montague.Denotation E W)}
{g : Assignment E}
{c : C}
{t₁ t₂ : PhraseStructure.Tree C L}
{σ τ : Montague.Ty}
{a : Montague.Ty.Domain E W σ}
{f : Montague.Ty.Domain E W (σ ⇒ τ)}
(h₁ : Denotes lex g t₁ ⟨σ, a⟩)
(h₂ : Denotes lex g t₂ ⟨σ ⇒ τ, f⟩)
: Denotes lex g (PhraseStructure.Tree.node c [t₁, t₂]) ⟨τ, f a⟩
By Functional Application, a node whose right daughter is the function denotes its value.
- pm
{C L E W : Type}
{lex : L → Option (Montague.Denotation E W)}
{g : Assignment E}
{c : C}
{t₁ t₂ : PhraseStructure.Tree C L}
{P Q : Montague.Ty.Domain E W (Montague.Ty.e ⇒ Montague.Ty.t)}
(h₁ : Denotes lex g t₁ ⟨Montague.Ty.e ⇒ Montague.Ty.t, P⟩)
(h₂ : Denotes lex g t₂ ⟨Montague.Ty.e ⇒ Montague.Ty.t, Q⟩)
: Denotes lex g (PhraseStructure.Tree.node c [t₁, t₂])
⟨Montague.Ty.e ⇒ Montague.Ty.t, fun (x : Montague.Ty.Domain E W Montague.Ty.e) => P x ∧ Q x⟩
By Predicate Modification, two sister predicates conjoin.
- trace
{C L E W : Type}
{lex : L → Option (Montague.Denotation E W)}
{g : Assignment E}
{n : ℕ}
{c : C}
: Denotes lex g (PhraseStructure.Tree.trace n c) ⟨Montague.Ty.e, g n⟩
By the Traces and Pronouns Rule, a trace denotes the value of its index.
- pa
{C L E W : Type}
{lex : L → Option (Montague.Denotation E W)}
{g : Assignment E}
{n : ℕ}
{c : C}
{body : PhraseStructure.Tree C L}
{τ : Montague.Ty}
{F : E → Montague.Ty.Domain E W τ}
(h : ∀ (x : E), Denotes lex (Function.update g n x) body ⟨τ, F x⟩)
: Denotes lex g (PhraseStructure.Tree.bind n c body) ⟨Montague.Ty.e ⇒ τ, F⟩
By Predicate Abstraction, a binder abstracts over its index in the body.
Instances For
Where the book assigns a denotation, the engine at M = Id computes it.
The book's rules are deterministic.
Definite descriptions #
The Fregean article Partial.the is defined on predicates true of exactly one individual
(§4.4). In the toy model, with two students and one pizza, the student is undefined and the
pizza is not.
The toy lexicon for the partial engine, which pairs the definite article with the fragment's nouns, sleeps and names lifted to partial denotations.
Equations
- HeimKratzer1998.partialLex "the" = some ⟨(Montague.Ty.e ⇒ Montague.Ty.t) ⇒ Montague.Ty.e, Part.some HeimKratzer.Partial.the⟩
- HeimKratzer1998.partialLex "student" = some ⟨Montague.Ty.e ⇒ Montague.Ty.t, Part.some ↑HeimKratzer.Toy.student⟩
- HeimKratzer1998.partialLex "pizza" = some ⟨Montague.Ty.e ⇒ Montague.Ty.t, Part.some ↑HeimKratzer.Toy.pizza⟩
- HeimKratzer1998.partialLex "sleeps" = some ⟨Montague.Ty.e ⇒ Montague.Ty.t, Part.some ↑HeimKratzer.Toy.sleeps⟩
- HeimKratzer1998.partialLex "John" = some ⟨Montague.Ty.e, Part.some HeimKratzer.ToyEntity.john⟩
- HeimKratzer1998.partialLex x✝ = none
Instances For
The student is a presupposition failure in the toy model, which has two students.
The pizza denotes the pizza, the toy model's unique one.
The John, the article applied to an individual rather than a predicate, is uninterpretable, and the types alone decide it (§4.4.4).
The book's partial rules #
Chapter 4's revision of the rules for partial denotations, which the partial engine extends.
The book's revised interpretation relation, relative to a leaf interpretation and an assignment.
- tn
{C L E W : Type}
{lex : L → Option (Montague.PDenotation E W)}
{g : Assignment E}
{c : C}
{w : L}
{d : Montague.PDenotation E W}
(h : lex w = some d)
: Denotes lex g (PhraseStructure.Tree.terminal c w) d
By Terminal Nodes, a leaf denotes what the lexicon gives it.
- nn
{C L E W : Type}
{lex : L → Option (Montague.PDenotation E W)}
{g : Assignment E}
{c : C}
{t : PhraseStructure.Tree C L}
{d : Montague.PDenotation E W}
(h : Denotes lex g t d)
: Denotes lex g (PhraseStructure.Tree.node c [t]) d
By Non-Branching Nodes, a node denotes what its only daughter does.
- faLeft
{C L E W : Type}
{lex : L → Option (Montague.PDenotation E W)}
{g : Assignment E}
{c : C}
{t₁ t₂ : PhraseStructure.Tree C L}
{σ τ : Montague.Ty}
{f : Montague.Ty.PDomain E W (σ ⇒ τ)}
{a : Montague.Ty.PDomain E W σ}
(h₁ : Denotes lex g t₁ ⟨σ ⇒ τ, Part.some f⟩)
(h₂ : Denotes lex g t₂ ⟨σ, Part.some a⟩)
: Denotes lex g (PhraseStructure.Tree.node c [t₁, t₂]) ⟨τ, f a⟩
By Functional Application, a node whose left daughter is the function denotes its value, defined or not.
- faRight
{C L E W : Type}
{lex : L → Option (Montague.PDenotation E W)}
{g : Assignment E}
{c : C}
{t₁ t₂ : PhraseStructure.Tree C L}
{σ τ : Montague.Ty}
{a : Montague.Ty.PDomain E W σ}
{f : Montague.Ty.PDomain E W (σ ⇒ τ)}
(h₁ : Denotes lex g t₁ ⟨σ, Part.some a⟩)
(h₂ : Denotes lex g t₂ ⟨σ ⇒ τ, Part.some f⟩)
: Denotes lex g (PhraseStructure.Tree.node c [t₁, t₂]) ⟨τ, f a⟩
By Functional Application, a node whose right daughter is the function denotes its value.
- pm
{C L E W : Type}
{lex : L → Option (Montague.PDenotation E W)}
{g : Assignment E}
{c : C}
{t₁ t₂ : PhraseStructure.Tree C L}
{P Q : Montague.Ty.PDomain E W (Montague.Ty.e ⇒ Montague.Ty.t)}
(h₁ : Denotes lex g t₁ ⟨Montague.Ty.e ⇒ Montague.Ty.t, Part.some P⟩)
(h₂ : Denotes lex g t₂ ⟨Montague.Ty.e ⇒ Montague.Ty.t, Part.some Q⟩)
: Denotes lex g (PhraseStructure.Tree.node c [t₁, t₂])
⟨Montague.Ty.e ⇒ Montague.Ty.t,
Part.some fun (x : Montague.Ty.PDomain E W Montague.Ty.e) =>
(P x).bind fun (a : Montague.Ty.PDomain E W Montague.Ty.t) => Part.map (fun (b : Prop) => a ∧ b) (Q x)⟩
By Predicate Modification, two sister predicates conjoin where both are defined.
- trace
{C L E W : Type}
{lex : L → Option (Montague.PDenotation E W)}
{g : Assignment E}
{n : ℕ}
{c : C}
: Denotes lex g (PhraseStructure.Tree.trace n c) ⟨Montague.Ty.e, Part.some (g n)⟩
By the Traces and Pronouns Rule, a trace denotes the value of its index.
- pa
{C L E W : Type}
{lex : L → Option (Montague.PDenotation E W)}
{g : Assignment E}
{n : ℕ}
{c : C}
{body : PhraseStructure.Tree C L}
{τ : Montague.Ty}
{F : E → Part (Montague.Ty.PDomain E W τ)}
(h : ∀ (x : E), Denotes lex (Function.update g n x) body ⟨τ, F x⟩)
: Denotes lex g (PhraseStructure.Tree.bind n c body) ⟨Montague.Ty.e ⇒ τ, Part.some F⟩
By Predicate Abstraction, a binder abstracts over its index in the body, the abstract defined at an individual where the body has a defined value under the modified assignment.
Instances For
Where the book's revised rules assign a value, the partial engine computes it.
The book's revised rules are deterministic.
The pizza denotes the pizza by the book's own rules.
Whatever value the book's rules assign the student is undefined.
The toy lexicon lifted entrywise never fails, since its entries are first order.
Relative clauses #
A relative clause gets no rule of its own (p. 87): Predicate Abstraction over its trace makes it a predicate, which Predicate Modification intersects with the head noun (p. 88). The vacuous complementizer is a non-branching node, and the engine's abstraction is the indexed rule of §5.3.4.
The subject relative [CP who₁ [C̄ [S t₁ sleeps]]] (p. 101).
Equations
Instances For
The noun phrase [NP student [CP who₁ [C̄ [S t₁ sleeps]]]], standing in for the book's
house which is empty (p. 88).
Equations
Instances For
The open sentence [S John [VP sees t₁]].
Equations
- HeimKratzer1998.tree_johnSeesT = (PhraseStructure.Tree.leaf "John").bin ((PhraseStructure.Tree.leaf "sees").bin (PhraseStructure.Tree.tr 1))
Instances For
The object relative [CP which₁ [C̄ [S John [VP sees t₁]]]], standing in for the book's
which John abandoned t (p. 96).
Equations
Instances For
The subject relative denotes what sleeps denotes, at every assignment (p. 101).
The relative clause modifies its head intersectively, restrictive relatives being "just another kind of intersective modifier" (p. 88).
The noun phrase is a derivation in the book's rules, Predicate Modification over the noun and the Predicate Abstraction of the clause.
The open sentence has a value only relative to an assignment (definition (9), p. 94).
The open sentence has no assignment-independent value, since John sees Mary but not himself.
The object relative denotes the property of being seen by John at every assignment, the value of none of its subtrees.
The book's proof for which John abandoned t (pp. 96-98) is a derivation in its rules, by Predicate Abstraction, the vacuous complementizer, Functional Application twice and the Traces Rule.
The student who sleeps denotes John by the book's revised rules, the article applying to the noun phrase that the relative clause restricts.
The student who sleeps denotes John, though the student is a presupposition failure
(the_student_fails), since the restrictive relative narrows the article's domain to one student,
where a nonrestrictive one would leave the article applying to student alone (p. 88).
Under the bracketing [[the student] [who₁ [t₁ sleeps]]] that §4.5 rejects, the definite
denotes a truth value, so a sentence with it as subject is uninterpretable (p. 83).
Quantifier scope #
Quantifier Raising (§7.3) leaves a trace and a binder index, so a raised quantifier takes the abstracted predicate as its scope. The two QR trees of "every person sees some person" compute its two scope readings.
The surface-scope tree [S [DP every person] [1 [S [DP some person] [2 [S t₁ [VP sees t₂]]]]]]
has the reading ∀x[person(x) → ∃y[person(y) ∧ sees(x,y)]].
Equations
- One or more equations did not get rendered due to their size.
Instances For
The inverse-scope tree [S [DP some person] [2 [S [DP every person] [1 [S t₁ [VP sees t₂]]]]]]
has the reading ∃y[person(y) ∧ ∀x[person(x) → sees(x,y)]].
Equations
- One or more equations did not get rendered due to their size.
Instances For
The surface-scope reading ∀ > ∃ puts every over some, with x sees y.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The inverse-scope reading, ∃ > ∀.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Surface scope is true in the toy model, where John and Mary see each other.
Inverse scope is false in the toy model, where no one sees themselves.
The inverse reading entails the surface one, so a model can separate them only in the direction the toy model does.
On the surface-scope tree the engine computes the surface reading.
On the inverse-scope tree the engine computes the inverse reading.
The two QR derivations interpret to different meanings.
The surface-scope reading is a derivation in the book's rules, Functional Application around two Predicate Abstractions with the quantifier words as terminals.
Readings of the ambiguous lexicon #
Tree.readings composes the words' available readings, one resolution per occurrence; the
study's leaf interpretation is one such resolution.
The words' available readings, the quantifier words through the terminals of theirs and the toy fragment's words through the toy lexicon.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The study's leaf interpretation chooses among the available readings.
The surface-scope reading is among the readings of the surface tree.
Quantifiers in situ #
In §7.2.1's alternative to movement the book's lexical rule gives each determiner an
object-position entry (objectShift?), and the flat tree composes by Functional Application
alone.
The words' readings closed under the lexical rule, each quantifier word making its object-position entry available beside its basic one.
Equations
Instances For
A resolution of the quantifier words, the words in object taking their object-position
entry and the others their basic one.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Every resolution chooses among the readings the lexical rule makes available.
The book's in-situ tree on the toy fragment, [S [DP every person] [VP sees [DP some person]]], with no movement.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Under the book's subscripts, the object-position entry for some alone, the in-situ tree composes by Functional Application alone to the surface-scope reading.
The in-situ and QR derivations compute the same reading.
The surface-scope reading is among the readings of the flat tree under the lexical rule.
An object-position entry in subject position leaves its mother uninterpretable, [S [DP some person] [VP sleeps]] with some resolved to its object-position entry.
A basic entry in object position is the type mismatch of §7.1 the rule repairs, [S John [VP sees [DP some person]]] with every word resolved to its basic entry.