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)

LRS combinatorics for quantifiers

Eventuality quantification

Which meaning contributions stem from which words in the sentence?

1 Sentence: Pat called.
Logical form: ∃e ((call • e) • pat)

e ¦	pat ¦	call ¦	call • e ¦	(call • e) • pat ¦	∃e (...)
						Pat
						called

2 Sentence: Pat didn't call.
Logical form: ¬∃e ((call • e) • pat)

e ¦	pat ¦	call ¦	call • e ¦	(call • e) • pat ¦	∃e (...) ¦	¬(...)
							Pat
							didn't
							call

Sentences with quantifiers

Which meaning contributions stem from which words in the sentence?

1 Sentence: Some students called.
Logical form: Some x (student • x: ∃e ((call • e) • x)

e ¦	x ¦	call ¦	student ¦	Some ¦	call • e ¦	(call • e) • x ¦	∃e (...) ¦	student • x ¦	Some x (...:...)
										Some
										students
										called

2 Sentence: Many farmers grow few vegetables.
Logical form for reading 1: Many x (farmer • x: Few y (vegetable • y : ∃e (((grow • e) • x) • y)))
Logical form for reading 2: Few y (vegetable • y: Many x (farmer • x : ∃e (((grow • e) • x) • y)))

e ¦	x ¦	y ¦	farmer ¦	vegetable ¦	grow ¦	Many ¦	Few ¦	farmer • x ¦	vegetable • y ¦	grow • e ¦	(grow • e) • y ¦	((grow • e) • y) • x ¦	∃e (...) ¦	Many x (...:...) ¦	Few y (...:...)
																Many
																farmers
																grow
																few
																vegetables

INTERNAL-CONTENT and MAIN value

Indicate the MAIN and the INTERNAL-CONTENT value of the words in the sentences.
(Hint: Copy the relevant parts from the indicated logical form of the sentences)

1 Sentence: Pat read many books.
Logical form: Many x (BOOK*x : Exists e (((READ*e)*x)*PAT)) Many x (book • x : ∃e (((read • e) • x) • pat))

Pat: MAIN IN-CONT

read: MAIN IN-CONT

many: MAIN IN-CONT

books: MAIN IN-CONT

2 Sentence: Alex might call.
Logical form: MIGHT (Exists e ((CALL*e)*ALEX))

Alex: MAIN IN-CONT

might: MAIN IN-CONT

call: MAIN IN-CONT