2. HW ANSWERS
Part A:
1. Amos, Bouncer and Cleo live at the Zoo.
Za&Zb&Zc
2. Bouncer is a reptile, but not an alligator.
Rb& ¬Ab
3. If Cleo loves Bouncer, then Bouncer is a monkey.
Lcb →Mb
4. If both Bouncer and Cleo are alligators, then Amos loves them both.
(Ab&Ac) → (Lab&Lac)
5. Some reptile lives at the zoo.
∃x (Rx&Zx)
3. Part 1: CONTINUATION
6. Every alligator is a reptile.
∀x (Ax → Rx)
7. Every animal that lives at the zoo is either a monkey or an alligator.
∀x (Zx→(Mx v Ax))
8. There are reptiles which are not alligators.
∃x (Rx & ¬Ax)
9. Cleo loves a reptile.
∃x (Rx & Lcx)
4. 10. Bouncer loves all monkeys that live at the zoo.
- We have two space predicate: between a name and “all monkeys”
- Since we are talking about “all” animals which have property of being “a
monkey” we will need a quantifier
∀x [(Mx&Zx) →Lbx]
5. PART C
1. Bertie is a dog who likes Samurai movies.
Bertie is a dog and Bertie likes to watch Samurai movies.
Db & Sb
2. Bertie, Emerson and Fergis are all dogs.
Berie is a dog, Emerson is a dog and Fergis is a dog.
Db & De & Df
3. Emerson is larger than Bertie, and Fergis is larger than Emerson.
Leb & Lfe
4. All dogs like samurai movies.
∀x (Dx →Sx)
6. 5. Only dogs like samurai movies.
∀x (Sx → Dx)
Doesn’t say that all dogs like samurai movies (which means that to be a dog
would be sufficient condition for liking samurai movies)
Rather it says, that it is necessary that everyone who like samurai movies
are dogs.
∀x (¬ Dx → ¬ Sx)
7. 6. There is a dog that is larger than Emerson.
∃x (Dx & Lxe)
7. If there is a dog larger than Fergis, then there is a dog larger than Emerson.
∃x (Dx & Lf) → ∃x (Dx & Le)
8. TEAM WORK
PART A of exercise sheet
1. Sue is easygoing
Es
2. Michael is taller than Sue and Sue is taller than Henry.
Tms & Tsh
3. Sue likes Henry and Michael likes Rita.
Lsh & Lmr
4. If Rita likes Henry, then Rita is taller than Henry.
Lrh → Trh
5. If Michael is easygoing, then Rita is not easygoing.
Em → ¬ Er
9. 6. Henry likes Rita but Rita doesn’t like Henry.
Lhr & ¬ Lrh
7. Rita is taller than Henry and Rita is not taller than Sue.
Trh & ¬Trs
10. PART 2
1. Everyone is easygoing.
∀xEx
2. No one likes Michael.
¬ ∃xLxm, ∀x ¬ Lxm
3. Michael likes everyone.
∀xLmx
4. Michael doesn’t like anyone.
There is no person that Michael likes.
¬ ∃xLmx, ∀x ¬ Lmx
5. Michael doesn’t like everyone.
It’s not the case that Michael likes all people (in his office).
¬∀xLmx
11. 6. Someone likes Sue.
There is a person (at least one) who likes Sue.
∃x Lxs
7. No one is taller than herself or himself.
¬ ∃xTxx,
∀x ¬ Txx
12. PART 3
1. Some mathematicians are famous.
UD: people
Mx: x is mathematician
Fx: x is famous
Gx: x is German
∃x(Mx&Fx)
2. Some mathematicians are not famous.
∃x(Mx& ¬ Fx)
13. 3. There is no mathematician who is famous.
¬ ∃x(Mx&Fx)
4. Some Germans are famous mathematicians.
∃x(Gx&Fx&Mx)
5. All donkeys are stubborn.
UD: donkeys
Sx: x is stubborn
∀xSx
14. 5. Only private universities are expensive.
UD: universities
Px: x is private
Ex: x is expensive
∀x (Ex → Px)
6. Whales are mammals.
UD: animals
Wx: x is a whale
Mx: x is a mammal
∀x (Wx → Mx)
15. HW ANSWERS
PART A
11. All the monkeys that Amos loves love him back.
∀x [(Mx & Lax) → Lxa]
14. Every monkey that Cleo loves is also loved by Amos.
∀x [(Mx&Lcx) → Lax]
15. There is a monkey that loves Bouncer, but sadly Bouncer doesn’t
reciprocate this love.
∃x(Mx& Lxb& ¬ Lbx)
16. PART R 1-7
1. Boris has never tried any candy.
¬ ∃xTx
2. Marzipan is always made with sugar.
∀x(Mx →Sx)
3. Some candy is sugar-free.
∃x ¬Sx
¬ ∀x Sx
4. The very best candy is chocolate.
∃x (Cx & ¬ ∃yByx)
17. 5. No candy is better than itself.
¬ ∃xBxx
6. Boris has never tried sugar-free chocolate.
¬ ∃x(Cx&¬ Sx&Tx)
7. Boris has tried marzipan and chocolate, but never together.
∃x(Mx&Tx) & ∃x(Cx&Tx)& ¬ ∃x(Cx&Mx&Tx)
18. TEAM WORK: Multiple Quantifiers
1. Everyone loves someone.
∀x∃yLxy
2. No one loves everyone.
¬ ∃x∀yLxy
3. There is no one who is unloved.
∀x∃yLyx
