Adjectives and Quantifiers

LIN 141: Semantics

Masoud Jasbi

Adjectives

Translation to L_λ: Adjectives

《Homer》= h

《Marge》= r

《Lisa》= l

《Bart》= b

《Maggie》= m

《Marvin》= v

《kind》= [λx.kind(x)]

《lazy》= [λx.lazy(x)]

《red》= [λx.red(x)]

《impressed》= [λx.impressed(x)]

《proud》= [λx.proud(x)]

《scared》= [λx.scared(x)]

Translating Predicative Adjectives

《Lisa is kind》=

S

NP

VP

Lisa

is

AP

kind

A

<<e,t>,<e,t>>

[λP.P]

e

l

<e,t>

[λx.kind(x)]

Rule: NN

Rule: FA

Prediction?

Our model suggests that a copula like “is” does not carry much meaning or information in the compositional structure of the sentence.

Therefore, the composition can also work without it!

All other things being equal, a language should be able to drop a copula and compose the sentence without such a verb.

Copula Construction in AAVE

In African American Vernacular English, the copula can often be dropped.

“He is tired.” → “He tired.”

“She is an expert.” → “She an expert.”

"I don't say stuff to people most of the time. Mostly I just look at them like they stupid."

But it’s more complicated! (always)

Wyatt (1991) found that AAE preschoolers were more likely to use zero copula: after pronoun subjects (56%) rather than noun subjects (21%); before locative predicates (35%) and adjective predicates (27%) rather than noun predicates (18%); and in second person singular and plural predicates (45%) rather than third person singular predicates (19%). In addition, the zero copula occurred less than 1% of the time in past tense, first person singular, and final clause contexts.

​

Toya A. Wyatt, "Children's Acquisition and Maintenance of AAE." Sociocultural and Historical Contexts of African American English, ed. by Sonja L. Lanehart. John Benjamins, 2001

Zero Copula Languages

https://wals.info/feature/120A#2/21.4/157.2

​

Wait a minute ...

《Lisa is proud of Maggie》=

《proud》= [λx.proud(x)]

《Lisa》= l

《Maggie》= m

​

Modifier Adjectives

《Lisa is a kind girl》=

S

NP

VP

Lisa

is

NP

kind

A

e

l

Rule: NN

Rule: FA

a

D

girl

NP

NP

?

<e,t>

[λx.girl(x)]

<e,t>

[λx.kind(x)]

Solution 1: Adding a composition rule

Rule 3 (Predication Modification):

Let 𝛾 be a syntactic node with daughters 𝛼 and 𝛽.

If 《𝛼》and 《𝛽》are both type <e,t>, then:

《𝛾》= [λv.《𝛼》(v) ⋀《𝛽》(v)]

where v is a variable of type e that does not occur free in either 《𝛼》and 《𝛽》.

​

PM Rule

《Lisa is a kind girl》=

S

NP

VP

Lisa

is

NP

kind

A

e

l

<<e,t>,<e,t>>

[λP.P]

Rule: NN

Rule: FA

a

D

girl

NP

NP

<<e,t>,<e,t>>

[λP.P]

<e,t>

[λx.girl(x)]

​

​

<e,t>

[λx.kind(x)]

​

​

<e,t>

[λx.kind(x)⋀ girl(x)]

​

​

Rule: PM

Solution 2: Type Shifting

Predicate-to-modifier shift (MOD)

If 《𝛼》is of category <e,t> then:

《MOD 𝛼》= [λP.[λv.《𝛼》(v) ⋀ P(v)]]

Example:

《kind》= [λx. kind(x)]

《MOD kind》= [λP. [λx. kind(x) ⋀ P(x)]

​

P-to-M Shift

《Lisa is a kind girl》=

S

NP

VP

Lisa

is

NP

kind

A

<<e,t>,<e,t>>

[λP.P]

a

D

girl

NP

NP

<<e,t>,<e,t>>

[λP.P]

<e,t>

[λx.girl(x)]

​

​

<<e,t>,<e,t>>

[λP.[λx.kind(x)⋀ P(x)]]

​

<e,t>

[λx.kind(x)]

Moment of Reflection

“Lisa is a kind girl” → Lisa is kind and Lisa is a girl.

“This is a red car” → This is red and this is a car.

Does this always work?

“Maggie is a tall 4-year-old” → Maggie is tall and Maggie is a 4-year-old.

“This is a fake gun” → This is fake and this is a gun!

Types of Adjective

1. Intersective
2. Subsective
3. Non-subsective
4. Privative

Intersective Adjectives

An adjective ADJ is intersective iff for all N, ⟦ADJ N⟧ = ⟦ADJ⟧∩⟦N⟧.

In other words when something is [ADJ N], it is both [ADJ] and [N].

Examples: red

Is a [red chair] red?

Is a [red chair] a chair?

Any other examples of intersective adjectives?

red things

chairs

Subsective Adjectives

An adjective ADJ is subsective iff, for all N, ⟦ADJ N⟧ ⊆ ⟦N⟧.

Example: skillful

A [skillful violinist] is only skillful as a violinist.

Not for example a surgeon!

Can you think of more examples?

Non-subsective Adjectives

An adjective ADJ is non-subsective if there is at least one N such that, ⟦ADJ N⟧ ⊈ ⟦N⟧

Example: alleged

Is an [alleged fraud] an alleged thing?

Is an [alleged fraud] a fraud?

Any other examples?

former?

ex?

Privative Adjectives

An ADJ is privative iff for all N, ⟦ADJ N⟧ ∩⟦N⟧ = ∅

Example: fake

Is a [fake gun] a fake thing?

Is a [fake gun] a gun?

Is a [fake ID] fake?

Is a [fake ID] an ID?

Can you think of other examples?

Quantifiers

Quantificational Nominals

We have dealt with two types of nominals:

Proper Names like “Lisa” that refer to entities in the model

Common Nouns like “simpson” or “therapist” that refer to sets of entities

Quantificational nominals are another type with more complex meanings:

Someone annoys Homer.

Everyone loves Maggie.

No-one likes Marvin.

...

Subject Q-words

Someone annoys Homer.

Everyone loves Maggie.

No-one likes Marvin.

...

annoys

loves

likes

...

Homer

Maggie

Marvin

...

Translation to L_λ: Lexical Entries

《Someone annoys Homer》= ∃x[annoy(x,h)]

《Homer》= h

《annoy》= [λy.[λx.annoy(x,y)]]

《someone》= ?

​

​

Translating Subject Q-words

《Someone annoys Homer》=

<e,<e,t>>

[λy.[λx.annoy(x,y)]]

?

?

S

VP

annoys

NP

N

Homer

Someone

NP

e

h

​

N

Translating Subject Q-words

《Someone annoys Homer》=

t

∃x[annoy(x,h)]

<<e,t>, t>

[λP.∃x[P(x)]

​

<e,<e,t>>

[λy.[λx.annoy(x,y)]]

e

h

e

h

S

VP

annoys

NP

N

Homer

Someone

NP

e

h

​

N

Translation to L_λ: Lexical Entries

《Someone annoys Homer》= ∃x[annoy(x,h)]

《Homer》= h

《annoy》= [λy.[λx.annoy(x,y)]]

《someone》= [λP.∃x[P(x)]]

Translation to L_λ: Lexical Entries

《Everyone loves Maggie》= ∀x[love(x,m)]

《Maggie》= m

《loves》= [λy.[λx.love(x,y)]]

《someone》= [λP.∃x[P(x)]]

《everyone》= ?

[λP.∀x[P(x)]]

​

​

《Everyone loves Maggie》=

<<e,t>, t>

[λP.∀x[P(x)]

​

<e,<e,t>>

[λy.[λx.loves(x,y)]]

S

VP

loves

NP

N

Maggie

everyone

NP

e

m

​

N

Translation to L_λ: Lexical Entries

《No-one likes Marvin》= ¬∃x[love(x,v)]

《Marvin》= v

《likes》= [λy.[λx.likes(x,y)]]

《someone》= [λP.∃x[P(x)]]

《everyone》= [λP.∀x[P(x)]]

《no-one》= ?

[λP.¬∃x[P(x)]]

​

​

《No-one likes Marvin》=

<<e,t>, t>

[λP.¬∃x[P(x)]

<e,<e,t>>

[λy.[λx.likes(x,y)]]

S

VP

likes

NP

N

Marvin

no-one

NP

e

v

​

N

Subject Determiner Phrases

A therapist annoys Homer.

Every Simpson loves Maggie.

No Simpson likes Marvin.

...

Translation to L_λ: Lexical Entries

《A therapist annoys Homer》= ∃x[therapist(x) ⋀ annoy(x,h)]

《Homer》= h

《therapist》= [λx.therapist(x)]

《annoys》= [λy.[λx.annoy(x,y)]]

《a》= ?

《A therapist annoys Homer》=

<e,t>

[λx.therapist(x)]

​

<e,<e,t>>

[λy.[λx.annoy(x,y)]]

S

VP

annoys

NP

N

Homer

therapist

NP

e

h

​

N

D

NP

a

?

?

《A therapist annoys Homer》=

t

∃x[therapist (x) ⋀ annoy(x,h)]

<e,t>

[λx.therapist(x)]

​

<e,t>

[λx.therapist(x)]

<e,<e,t>>

[λy.[λx.annoy(x,y)]]

e

h

e

h

S

VP

annoys

NP

N

Homer

therapist

NP

e

h

​

N

D

NP

a

<<e,t>, <<e,t>,t>>

[λPλQ.∃x[P(x)⋀Q(x)]]

Translation to L_λ: Lexical Entries

《Every Simpson loves Maggie》= ∀x[simpson(x) → love(x,m)]

《Maggie》= m

《Simpson》= [λx.simpson(x)]

《loves》= [λy.[λx.love(x,y)]]

《a》= [λPλQ.∃x[P(x)⋀Q(x)]]

《every》= ?

[λPλQ.∀x[P(x) → Q(x)]]

《Every Simpson loves Maggie》=

<e,t>

[λx.simpson(x)]

​

<e,<e,t>>

[λy.[λx.love(x,y)]]

S

VP

loves

NP

N

Maggie

Simpson

NP

e

m

​

N

D

NP

every

<<e,t>, <<e,t>,t>>

[λPλQ.∀x[P(x) → Q(x)]]

Translation to L_λ: Lexical Entries

《No Simpson likes Marvin》= ¬∃x[simpson(x) ⋀ love(x,m)]

《Marvin》= v

《Simpson》= [λx.simpson(x)]

《likes》= [λy.[λx.like(x,y)]]

《a》= [λPλQ.∃x[P(x)⋀Q(x)]]

《no》= ?

[λPλQ.¬∃x[P(x) ⋀ Q(x)]]

《No Simpson loves Maggie》=

<e,t>

[λx.simpson(x)]

​

<e,<e,t>>

[λy.[λx.likes(x,y)]]

S

VP

likes

NP

N

Marvin

Simpson

NP

e

v

​

N

D

NP

no

<<e,t>, <<e,t>,t>>

[λPλQ.¬∃x[P(x)⋀Q(x)]]

Generalized Quantifiers

English Lexical Determiners

every, each, all,

most,

two, three, … ten, ...

several, a few, a dozen, many, few

both, neither

some, a, the

no

How can we translate “most” into L_λ?

How about “several”, “a few”, …?

We can’t!

We need a more “general” theory of quantification!

The Role of Determiners

In our translations to L_λ, determiners were type <<e,t>,<<e,t>,t>>.

They take two “sets” and return truth values.

We can show this as D(A, B)

⟦adult⟧^M,g = { , , }

⟦child⟧^M,g = { , , }

⟦walk⟧^M,g = { , , , , }

⟦crawl⟧^M,g = { }

D_Simpsons = { , , , , , }

⟦simpson⟧^M,g = { , , , , }

⟦therapist⟧^M,g = { }

Universal Quantification

《Every adult walks》= ∀x[adult(x) → walk(x)]

We can represent the meaning as:

⟦adult⟧^M,g ⊆ ⟦walk⟧^M,g

⟦every⟧^M,g = {<A, B>: A ⊆ B}

⟦walk⟧^M,g = { , , , , }

⟦adult⟧^M,g = { , , }

Existential Quantification

《A therapist walks》= ∃x[therapist(x) ⋀ walk(x)]

We can represent the meaning as:

⟦therapist⟧^M,g ∩ ⟦walk⟧^M,g ≠ ∅

⟦a(n)⟧^M,g = {<A, B>: A ∩ B ≠ ∅}

⟦walk⟧^M,g = { , , , , }

⟦therapist⟧^M,g = { }

Negative Quantification

《No adult crawls》= ￢∃x[adult(x) ⋀ crawl(x)]

We can represent the meaning as:

⟦adult⟧^M,g ∩ ⟦crawl⟧^M,g = ∅

⟦no⟧^M,g = {<A, B>: A ∩ B = ∅}

⟦adult⟧^M,g = { , , }

⟦crawl⟧^M,g = { }

Non-Classical Quantifiers

《Most Simpsons walk》= ?

We can represent the meaning as:

|⟦simpson⟧^M,g ∩ ⟦walk⟧^M,g |/ |⟦simpson⟧^M,g |> 0.5

⟦most⟧^M,g = {<A, B>: |A ∩ B| / |A| > 0.5}

⟦walk⟧^M,g = { , , , , }

⟦simpson⟧^M,g = { , , , , }

Monotonicity

Monotonicity in Functions

A function f: D₁ → D₂ is monotonic (or upward monotone, or monotone increasing) when:

a ≤ b → f(a) < f(b)

A function f: D₁ → D₂ is antitonic (or downward monotone, or monotone decreasing) when:

a ≤ b → f(a) ≥ f(b)

Monotonicity in Determiners

A determiner D is monotonic (upward monotone, monotone increasing):

On its first argument iff when A ⊆ X: D(A)(B) ⇒ D(X)(B)

On its second argument iff when B ⊆ X: D(A)(B) ⇒ D(A)(X)

Example:

simpson ⊆ person: some (simpson)(walks) ⇒ some (person) (walks)

walk ⊆ move: some (simpson)(walks) ⇒ some (simpson)(moves)

Determiner “some” is monotonic (upward monotone) on both its arguments.

Antitonicity in Determiners

A determiner D is antitonic (downward monotone, monotone decreasing):

On its first argument iff when A ⊆ X: D(A)(B) ⇐ D(X)(B)

On its second argument iff when B ⊆ X: D(A)(B) ⇐ D(A)(X)

Example:

simpson ⊆ person: no (simpson)(walks) ⇐ no (person) (walks)

walk ⊆ move: no (simpson)(walks) ⇐ no (simpson)(moves)

Determiner “no” is antitonic (downward monotone) on both its arguments.

Non-monotonic Determiners

A determiner D is non-monotone on an argument iff D is neither monotonic or antitonic (neither upward nor downward monotone) on that argument.

​

Exercise

Determine the monotonicity of the following determiners:

every (A)(B)

at most ten (A)(B)

few (A)(B)

exactly three (A)(B)

most (A)(B)

A= student

A’ = LIN141 student

B = sing

B’ = sings opera

student

sings

LIN141

student

Sing opera

Monotonicity and Grammar

Negative Polarity Items

Some language expressions seem to be grammatical only in negative sentences.

Consider the case of ever or any in English:

1. Chrysler dealers don’t ever sell any cars anymore.
2. # Chrysler dealers ever sell any cars anymore.

Or with quantifiers:

1. # Some Chrysler dealer ever sells any cars to anyone.
2. No Chrysler dealer ever sells any cars to anyone.

Antitonic Items

However, soon it was discovered that the licensing factor is not negation, but rather the antitonic or downward entailing environment of the expression:

• # Some student who had ever come to any lecture passed.
• Every student who had ever come to any lecture passed.
• No student who had ever come to any lecture passed.

Second argument:

• # Some student passed any exam ever.
• # Every student passed any exam ever.
• No student passed any exam ever.

Take away

The logical properties of meaning and communication constrain and shape human language and grammar.

Precise formalization of language and its meaning allows us to discover these properties and discover why language is the way it is!