Scope Ambiguity

Scope and Anaphora

antecedent vs. postcedent

anaphor vs. cataphor

Predicate logic is suited to capture natural language meaning

allow recursion = recursivity

two sources of recursion

coordination: john likes cats, dogs, beer
embedding: John said that Mary knws that Bill thinks that Jane claims that

some boy kissed every girl.

Every girl was kissed by some boy.

Someone mentioned tehy called everyone.

\forall x: Px\forall y(M(x, Cxy))

linear order: negative polarity item

graph TD
  DS -.Transformation.-> SS
    SS -.send off.-> PF
    SS -.send off.-> LF
    PF -.acoustic representation.-> SS
    LF -.semantic interpretation.-> SS

Transformation:

Jane laughed at Bill
Bill was laughed at by Jane

CALLOUT: annotation, connotation and denotation

annotation
- a critical or explanatory commentary or analysis
- a comment added to a text
- the process of writing such comment or commentary
- (computing) metadata added to a document or program
- (genetics) information relating to the genetic structure of sequences of bases
connotation: intension.
- A meaning of a word or phrase that is suggested or implied, as opposed to a denotation, or literal meaning. A characteristic of words or phrases, or of the contexts that words and phrases are used in.
  
  The connotations of the phrase "you are a dog" are that you are physically unattractive or morally reprehensible, not that you are a canine.
- A technical term in logic used by J. S. Mill and later logicians to refer to the attribute or aggregate of attributes connoted by a term, and contrasted with denotation .
  
  The two expressions "the morning star" and "the evening star" have different connotations but the same denotation (i.e. the planet Venus).
denotation
- its explicit or direct meaning, as distinguished from the ideas or meanings associated with it or suggested by it. Simply put, a word’s denotation is what that word means or directly represents.

Solution for scope ambiguity

Quantifier-raising - NC RM - syntactic structure comes before the semantic structure - The movement we make in SS to remove ambiguity in DS is called quantifier-raising. - take the quantifier to the higher position to show the scope

Quantifier-in - Montague grammar - The derivational illustration is called quantifier-in. - each predicate take an argument once a time

Quantifier storage - Cooper storage - semantic ambiguity not represented in syntactic structure - semantic representation in which scope ambiguities are obtained without special syntactic rules

Quantifier-in

interrogative: asking a question

which woman does every man love?

which scopes over every.

Scope ambiguity

e.g. some boy did not laugh.

\exist x (Boy(x) and ~Laugh(x))
~\exist x (Boy(x) and Laugh(x))

some boy kissed no girl.

\exist x (Boy(x) and ~\exist y (Girl(y) and Kiss(x, y)))
~\exist y (Girl(y) and \exist x (Boy(x) and Kiss(x, y))): there was no girl kissed by a boy

every boy kissed no girl.

\forall x (Boy(x) and ~\forall y(Girl(y) and Kiss(x, y)))

Deictic

No boy said he was hungry.

No boy was present. He was outside instead.: “he” is trying to refer to ”no boy” but outside the scope.

pronoun \(\sub\) anaphora

Discourse Anaphora

e.g.

Every student was present and she was interested.

every: scopes over “Every student was present”

every: an indefinite quantifier. “she”’s antecedent is not clear

“she” is hardly bound by the antecedent. “she” is free * ungrammatical: 不合语法的, syntactic

infelicitous: 不合适的, semantic, fit the context

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sentence pronoun: within the same clause
discourse pronoun: in separate clauses

sentence quantifier:

Some boy said he was hungry.
No boy said he was hungry.

discourse quantifier:

Some boy was present; he was hungry.
#No boy was present; he was hungry.   // he is free

coreference individual constants
```
Fred thought he was the the best
```

binding individual variables

Every student thinks he/she is the best

So we may conclude the following rules for e-type anaphora. BUT this part has NOT been verified with any authority. Do NOT take them as given truths during exams.

F: universal quantifier + singular pronoun
T: universal quantifier + plural pronoun
F: negated quantifier + singular pronoun
T: negated quantifier + plural pronoun, semantically plural but grammatically singular
T: existential quantifier + singular pronoun

e.g. No boy thinks that he has a chance.

~\exist x(Boy(x) and Think(x, Has-a-chance(x)))

A particular boy said he wanted to kiss every girl. He then did it.

\exist !x(Bx and  W(x, K(x, \forall y(Gy -> K(x, y))))) and K(x, y)

Donkey anaphora

if a farmer owns a donkey, he beats it.

* \exist x (Fx and \exist y (Dy and O(x, y))) -> B(x, y)
\forall x \forall y (Fx and Dy and O(x, y) -> B(x, y))

= every farmer who owns a donkey beats it.

\exist x(Fx and \exist y (Dy and O(x, y)) -> B(x, y))  // y is free

❗❗❗

A donkey sentence is such that an expected existential is interpreted as universal taking wide scope.

donkey pronoun can be: it, him, they (can also be plural forms)

“a”: generic indefinite

A woman is a difficult thing to please.

[Every farmer [who owns a donkey] beats it.]

universal wide scope: it scopes more over the relative clause

The problem - Existential with narrow scope - interpreted as universal with wide scope - in conditional clauses - in restriction of every

Conclusion - the machinery of predicate logic is broken - cannot capture meaning of natural language

If a student tries, she passes the exam.

(\exist x(Sx and Tx)) -> Py   ; y is free
\exist x((Sx and Tx)) -> Py)

interpretation

\forall x((Sx and Tx) -> px)

Solutions for donkey anaphora:

E-type anaphora

Gareth Evans, 1970s, philosopher, university of Oxford, logic, philosophy of mind
pronoun outside the scope of their binder

initial examples

A student came in. She had a question about the exam.
she = the student came in

Bill owns some sheep and Max vaccinates them.
them = the sheep Bill owns. E-type pronoun, some sheep scopes over the first half

If a student likes Copenhagen, she is happy.

she = for every case we examine, the student is

every student who reads a semantic paper likes it.

Bill owns a cat. Max takes care of it.
Bill is a cat-owner. #Max takes care of it.

DRT (Dynamic binding theory)
- discourse anaphora and donkey pronouns
- intermediate level
- Irene Heim (1982) and Hans Kamp (1981)
- Discourse Representation Theory (DRT)
  - embedding conditions
  - language of boxes
  - boxes constantly updated
  - embedded boxes in accessible
Unselective binding example of the subject is ‘unselectively bound’ by a special ‘generic operator.
```
Dogs bark.
A dog barks.
```

Reference: Unselective Binding

Chapter 6 in short: Discourse/Donkey Anaphora

(加粗的是Donkey anaphora和E-type anaphora的区别)

Discourse: basic unit of interpretation

Dynamic theory of meaning: look beyond the meaning of individual sentences and determine the way they are pieced together to make discourse
discourse anaphora: anaphora in sequence of sentences (instead of single sentence)
Donkey sentence: is such that an expected existential is interpreted as universal taking wide scope. / sentences that contain a pronoun with clear meaning but whose syntactical role in the sentence poses challenges to grammarians (wikipedia)
- universal: each, every, all, not all, not every
- existential: a, one, some, no
- e.g.
```
every farmer who owns a donkey beats *it*.
it: (corresponding to the) existential ("a") but interpreted as universal
```
```
every police officer who arrested a murder insulted *him*.
him
```
```
every farmer who owns some sheep cleans *them*.
them
```
donkey anaphora = donkey pronoun: it, him, they (can also be plural forms)

Analysis of donkey anaphora: Montague grammar

e.g.

\forall x(Farmer(x) and \exist y (Donkey(y) and Owns(x,y)) -> Beat(x,y))

Anaphoric relations in sentence and discourse - E-type anaphora: pronoun outside the scope of binder, not bound, content of pronoun reconstructed, reconstruction based on context - in separate sentences

    ```
    A student came in. *She*(the student came in) had a question about the exam.
    ```

- in the same sentence but outside the scope
    ```
    If a student likes Copenhagen, *she*(for every case we examine, the student in question who likes Copenhagen) is happy.
    ```

- problem of compound: antecedent must appear as a noun?
    ```
    Bill owns a cat. Max takes care of it.
    Bill is a cat-owner. # Max takes care of it.
    ```

unselective binding + Discourse Representation Theory (DRT)
- embedded conditions
- language of boxes, boxes constantly updated
- Accessibility: the antecedent to be in a box ‘as high as’(same box or left box) or ‘higher than’ the discourse referent for the pronoun
- ⇒ binds all left variables. unselective quantification over all the variables. unselective binding
dynamic binding
- TODO

Anaphora resolution - TODO