Exercise Quantifiers

Introduction to the topic

Input

Watch the following video on logical determiners:

<mediaplayer> http://youtu.be/5PRL23XcaFY</mediaplayer>

Exercises

After having watched the video, work on the following tasks.

Task 1 Identify the determiners in the following sentence.

(a) Juliet talked to some stranger at the party.

(b) Every Capulet is an enemy to some Montague.

(c) Many people in Verona are not happy about the Capulet-Montague feud.

Check your solutions here:

(a) some

(b) every, some

(c) many

Task 2 Identify the formula that corresponds to the translation of the sentence.

Some Montague who was at the party fell in love with Juliet.

	∃x (montague1(x) : (at-party1(x) ∧ fall-in-love-with2(x,juliet)))
	∃x ((montague1(x) ∧ at-party1(x)) : fall-in-love-with2(x,juliet))
	∃x (montague1(x) : (at-party1(x) ∧ fall-in-love-with2(x,juliet))
	∃x ((montague1(x) ∧ fall-in-love-with2(x,juliet)) : at-party1(x))

Task 3 The sentence: Some Tybalt loved some Montague. is translated into the formula
∃ y (montague₁(y) : love₂(tybalt,y).

Mark all the cells in the table that stand for a true statement.

montague1(y) zwisch	love2(tybalt,y)zwisch
		Romeo
		Mercutio
		Juliet
		Tybalt
		Laurence
		Paris

Given this table, is the overall formula true or false? (Give a reason for your answer.)

Check your solutions here:

The formula is false, because there is no individual in our model for which both the restrictor and the scope are true.

Task 4 Variable assignment function
Start with the following variable assigment function g: g(u) = Romeo, g(v) = Juliet, g(w) = Romeo, g(x) = Laurence, g(y) = Mercutio, g(z) = Juliet

Provide the changed variable assignment function g[v/Paris].

Check your solutions here:

g[v/Paris](u) = g(u) = Romeo
g[v/Paris](v) = Paris
g[v/Paris](w) = g(w) = Romeo
g[v/Paris](x) = g(x) = Laurence
g[v/Paris](y) = g(y) = Mercutio
g[v/Paris](z) = g(z) = Juliet

More exercises on quantifiers

The following material is an adapted form of material created by student participants of the project e-Learning Resources for Semantics (e-LRS).
Involved participants: AnKa, Katharina, Lara

Restricted Quantifiers

Find the right formula for the sentence below.

Some students who heard the concert were interviewed by Holmes.

	∃x (student(x) : (hear(x,concert) ∧ interview(holmes,x)))
	∃x ((student(x) ∧ hear(x,concert)) : interview(holmes,x))
	∃x (student(x) ∧ hear(x,concert) ∧ interview(holmes,x))
	∃x ((student(x) ∧ interview(holmes,x)) : hear(x,concert))

Different types of Quantifiers

Which type(s) of quantifiers does the sentence below have?

1 Ramon signs every sculpture he makes.

	existential
	universal

2 Some playwright also wrote famous sonnets.

	existential
	universal

3 Shakespeare wrote for King James.

	existential
	universal

4 All pupils read some plays by Shakespeare in school.

	existential
	universal

2. Write down the logical formula(e) that correspond to the sentence Ramon signs every sculpture he makes.

Check your solutions here

Sentence: Ramon signs every sculpture he makes.

Universal Quantifier

∀x ((sculpture(x) ∧ make(ramon, x)) ⊃ sign(ramon, x))

Paraphrse: "For every thing x, if x is a sculpture and x is made by Ramon then x is signed by Ramon."

We use the name constant ramon for both the name (Ramon) and the personal pronoun he that referes to Ramon.

In restricted quantifier notation

∀x ((sculpture(x) ∧ make(ramon, x)) : sign(ramon, x))

Here, the N' is "sculpture he makes" and therefore its translation appears in the part before the colon.

Scopal Ambiguity

1. In which way is the following sentence ambiguous?

Everyone loves someone.

The following pictures may help you:

• 

  One Reading

• 

  Another Reading

Check your solutions here:

In this sentence, the scopal ambiguity is created by the two quantifiers everyone and someone.

When looking at the two pictures that try to help you, you can see two possible readings:

1. For every person there is, there is at least one other person who loves him / her.

2. There is one person that is loved by everyone else.

2. Write down the two possible logical forms.

Check your solutions here:

1. For every person there is at least one person who loves him / her:

∀x (person(x) ⊃ ∃y (person(y) ∧ love(x,y)

Or, in restricted-quantifier notation: ∀x (person(x) : ∃y (person(y) : love(x,y)

2. There is one person that is loved by everyone:

∃y (person(y) ⊃ ∀x (person(x) ∧ love(x,y)

Or, in restricted-quantifier notation: ∀x (person(x) : ∃y (person(y) : love(x,y)

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