Exercise 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.

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

2. Write down the corresponding logical formula(s).

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.

1. In which way is the following sentence ambiguous?

Everyone loves someone.

The following pictures may help you:

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)

	∃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))

	existential
	universal

	existential
	universal