Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”) a) All dogs have fleas. b) There is a horse that can add. c) Every koala can climb. d) No monkey can speak French. e) There exists a pig that can swim and catch fish.

Solution:

By using a different domain for each question

We need to define the following predicates:

hasFleas(x): x has fleas.

canAdd(x): x can add.

canClimb(x): x can climb.

canSpeakF rench(x): x can speak French.

canSwim(x): x can swim.

canCatchF ish(x): x can catch fish.

a) All dogs have fleas.

Domain = dogs

∀x hasF leas(x)

Negation: ¬∀x hasF leas(x) ≡ ∃x ¬hasF leas(x)

In English: There is a dog which doesn’t have any fleas.

b) There is a horse that can add.

Domain = horses

∃x canAdd(x)

Negation: ¬∃x canAdd(x) ≡ ∀x ¬canAdd(x)

No horse can add. (There is no horse that can add)

c) Every koala can climb.

Domain = koalas

∀x canClimb(x)

Negation: ¬∀x canClimb(x) ≡ ∃x ¬canClimb(x)

In English: There is a koala which can’t climb.

d) No monkey can speak French.

Domain = monkeys

∀x ¬canSpeakF rench(x)

Negation: ¬∀x ¬canSpeakF rench(x) ≡ ∃x canSpeakF rench(x)

In English: There is a monkey which can speak French.

e) There exists a pig that can swim and catch fish.

Domain = pigs

∃x (canSwim(x) ∧ canCatchF ish(x))

Negation: ¬∃x (canSwim(x) ∧ canCatchF ish(x))

≡ ∀x ¬(canSwim(x) ∧ canCatchF ish(x))

≡ ∀x (¬canSwim(x) ∨ ¬canCatchF ish(x))

No pig can swim and catch fish. (There is no pig that can swim and catch fish)