Ambiguous Sentence to Translate to Predicate Logic

Question

I have been given the following sentence to translate into predicate logic:

"Everyone can go home only if the work is done."

I have come up with two interpretations:

A) "Everyone can go home only if THEIR work is done." (But other people may still have their work to do, so they will be staying behind.)

Translated as:

FOR ALL x H(x) implies W(X)

Where H(x) is "x can go home" and W(X) is "X's work is done".

B) "Everyone goes home only if the work 'in general' is done." (Ie: The team stays behind until everybody is done, even if they have nothing left to do.

Translated as:

FOR ALL x H(X) implies w

Where w is "all work, across all x's, is done."

What do you think?

Answer 1

Let $H(x)$ be "$x$ can go home" and $W(x)$ be "$x$'s work is done". Then under your first interpretation $\forall x(H(x)\to W(x))$ as you wrote, and under the second $\forall x(H(x)\to \forall y\,W(y))$.