Transform arguments to logical propositions

Question

I have the following arguments that I need to transform to logical propositions and prove if they are true

Every person that goes to Hawaii walks beneath coconuts

There are persons that, walking beneath coconuts, have their head cracked

Therefore, who goes to Hawaii may have their heads cracked

So I started like this:

∀x( (P(x) ^ H(x)) -> W(x))

as the first argument, P means is that x is a person;

H means that x goes to Hawaii;

W is x walks beneath coconuts.

∃x( (P(x) -> (W(x) ^ C(x))

as my second argument, with C meaning that x has a cracked head.

But I have no idea how to tranform the last argument in a logical notation. My biggest problem is the "may have their heads cracked", beacuse I don't know how to express possibility in this case. I thought about using v (or) but it didn't feel right. The sorce material I'm using don't have examples like this, and so far I've found nothing similar in forums.

Any tips are appreciated, thanks!

Answer 1

For the second premise, you need $$\exists x( ((P(x) \land W(x)) \land C(x)) $$

For the last (conclusion): "Therefore, who goes to Hawaii may have their heads cracked" you might translate as follows: "For all persons who go to Hawaii, there are some who have their heads cracked."

$$\forall(x) \Big(\big(P(x)\land H(x)\big)\rightarrow \exists(x)\big(C(x)\big)\Big)$$