Truth of a sentence in predicate logic I'm working through the exercises in this Logic problem set and I'm struggling to understand ex. 7.5:

Determine whether the following sentence is logically true in predicate logic:

There is someone such that, if he or she is asleep, everyone is asleep.


The solutions section suggests this is True and gives a formal 'Natural Deduction' proof that I don't really understand (it probably doesn't help that I don't have a copy of the book the solutions keep referring to).
Is there another way to approach this?

This is a classic example of the mismatch between the meaning of if in classical logic and its ordinary intuitive meaning. It is called the drinker's paradox.
Let's use the fact that $\varphi \to \psi$ is equivalent to $\lnot \varphi \lor \psi$ in classical logic to paraphrase the sentence in natural language and avoid the troublesome if.
There exists a person w such that w is not asleep or everyone is asleep.

Let $c$ be an arbitrary person in our group of people.
Suppose everyone is asleep, then we pick $w$ to be $c$. Since everyone is asleep, this value of $w$ satisfies the condition.
Suppose at least one person is awake, call them $a$. Then we pick $a$ as $w$.
I think the reason why this is counterintuitive is that the choice of $w$ depends on who happens to be asleep right now or not, but the phrasing of the question in natural language implies that a fixed $w$ has some special property that someone causally determines which configurations of awake and asleep people are possible.
The paradox goes away completely if we give the conjunction or wide scope over exists. We can do this unambiguously in natural language by reordering the dijsuncts. Reordering the dijsuncts, and changing the order of the disjunction and existential quantifier in this specific case are both allowed in classical logic.
Everyone is asleep or there's a person who is not asleep.

