Exercise-ch6

Exercises for chapter 6: Quantifiers

Quantifier denotation

Complete the paraphrase for the following sentences with quantifiers.

1 Sentence: Pat read many books.

For x such that is a , Pat x.

2 Sentence: Most students work hard.

For such x is a , works .

3 Sentence: Pat knows some good movies.

For x , .

Complete the paraphrases for the following ambiguous sentences with quantifiers.

1 Sentence: Every linguist knows at least two languages.

Reading 1: For least x such that is a , for every y such that a , knows .

Reading 2: y that , at least x is , .

2 Sentence: Few actors star in many movies.

Reading 1: x is an actor, y is a , in .

Reading 2: y , is , x in .

Logical forms for sentences with quantifiers

Provide the logical form of the following sentences. (Ignore the eventuality quantification.)
(Use: (i) lower case letters for variables (e,x,y,z, ...), (ii) upper case letters for predicates and name constant (LIKE, PAT, ...), (iii) capitalized words for quantifiers (Every, Most, Atleast, ..., (iv) use "*" for the bullet operator.) Note that the brackets and the "bullet operator" also count as symbols.
Example:
Sentence: Alex met some student.
Logical form: Some x (STUDENT * x : ((MEET * e) * x) * ALEX)

1 Sentence: Chris read many book.

Logical form: x ( * ((READ e) * ) * )

2 Sentence: Few students know Pat.

Logical form:

Provide the logical form of the following sentences. For the solutions, mark the -seemingly emtpy- line below the sentence with your mouse.

1. Sentence: Every student asked Alex.
   Logical form: ∀ x (student • x : ((help • e) • alex) • x)
2. Sentence: Alex talked to Chris about many movies.
   Logical form: Many x (movie • x : (((talk-to-about • e) • chris) • x) • alex)
3. Sentence: Chomsky wrote at least four books that Alex has read.
   Logical form: AtLeast-4 x ((book • x ∧ ((read • e) • x) • alex) : ((write • u) • x) • chomsky)

Eventuality quantification

Add the eventualiy quantification in the right position in the logical forms of the following sentences. Write "_" if nothing needs to be put in a particular slot and "Exists" for the existential quantifier and add the appropriate variable.

1 Sentence: Pat was asleep.

Logical form: ((be-asleep • s) • pat)

2 Sentence: Pat wasn't asleep.

Logical form: ¬ ((be-asleep • s) • pat)

3 Sentence: Alex studied in Frankfurt and lived in Berlin.

Logical form: ( ((study-in • u) • frankfurt) • alex ∧ ((live-in • v) • berlin) • alex)

4 Sentence: Alex walked and Chris ran.

Logical form: ( (walk • e) • alex ∧ (run • u) • chris)

5 Sentence: Two students worked in the library.

Logical form for the collective reading: Two x ( student • x: (work-in-library • e) • x)

Logical form for the distributive reading: Two x ( student • x: (work-in-library • e) • x)