Predicate Logic translation from English to formula

Question

I am studying for an exam and I have these exercises I practiced.

Let: Monkey(x) = "x is a monkey" Intelligent(x) = "x is an intelligent being" Likes(x, y) = "x likes y"

1. No monkey likes a tiger.

let Tigers(x) = "x is a tiger"

∀x(Tiger(x) => ∀y¬Likes(y, x)

2. There exists an intelligent monkey.

∃x(Monkey(x) => Intelligent(x))

3. Every monkey likes another monkey

∀x(Monkey(x) => ∀yLikes(y, x))

4. There is only one smart being who likes all other monkeys

∃!x(Monkey(x) => ∀yLikes(y, x))

5. Everybody likes my monkey, but my monkey doesn't like nobody else but be.

∀x(Monkey(x) ∧ ∃(y)(Monkey(y) ∧ x ̸= y ⇒ Likes(y, x)))

I could use some feedback, please? how would you write this?

Answer 1

1. Make sure that $y$ is a Monkey. Also, it is a little unusual to your sentence starts with introducing the Tigers, rather than the Monkeys, given that the subject term of the sentence is the Monkeys. Now, as it turns out, you can make this work, because both Tigers and Monkeys are universally quantified, but you're lucky it does. I would urge you to start by introducting the Monkeys, and then the Tigers.

2. Use a conjunction here instead of a conditional

3. $y$ needs to be a Monkey too. You may also have to explicitly specific that $y$ is not the same as $x$, since it says that $x$ likes some other monkey. And finally, it is some other monkey, so you need an existential for the $y$

4. Yet again, $y$ needs to be a Monkey.... and different from $x$. And I suppose that for the 'smart' you should use the Intelligent predicate. Finally, the English sentence is ambiguous: is there exactly one monkey that is Intelligent .. and that one intelligent monkey happens to like all other monkeys? Or is there exactly one monkey that has both properties of being intelligent and liking all other monkeys?

5. This should not be a universal at the start: it's a bout one monkey. In fact, it's about a specific monkey ... 'my monkey' ... there is not something in the key that lets you refer to 'my monkey'?