Discrete Math Question with "all" and "except"

Question

I came across this question on an exam and am not sure why I am wrong. The question:

Using the following predicates over the domain of people:

$$A(x) = \text{x is an American.}$$

$$B(x) = \text{x likes Burgers.}$$

$$H(x) = \text{x likes Hot Dogs.}$$

Correctly represent the statement, "All Americans like Burgers except those who like Hot Dogs."

So my answer was:

$$\forall x,[A(x)\longrightarrow (B(x)\iff \neg H(x))]$$

The correct answer was:

$$\forall x,[A(x) \land B(x) \longrightarrow \neg H(x)]$$

My logic for my answer is that the statement essentially says all Americans either like burgers or hot dogs. My professor justifies his answer by saying that the statement is still true even if the left hand side is false. Now, where I'm confused is I think both statements are true, and mine gives more information. I think that the 'correct' answer misrepresents the statement in the case where $A(x)$ is true, $B(x)$ is false, and $H(x)$ is false. This results in F -> T, which makes the conditional true. However, by the provided statement, this should never happen. Because, assume $H(x)$ is empty. Then we have all Americans like burgers. So $B(x)$ must be true. Then, if $H(x)$ gains people, $B(x)$ is only false when $H(x)$ is true, so this is a contradiction for the 'correct' statement. However, my answer correctly accounts for this.

If someone could clarify, that would be much appreciated, thank you.

Answer 1

The English statement is ambiguous. One construction is, as you say, that all Americans like either burgers or hot dogs but not both. Another is that all Americans like burgers except for those who like hot dogs, who may or may not also like burgers.

If the first meaning is intended, I agree that your answer is better than the professor's for the reasons you stated. If the second meaning is intended, the correct answer is:

$$\forall x ~((A(x) \land \lnot H(x)) \Rightarrow B(x)).$$

This sentence says that any American who doesn't like hot dogs does like burgers. That is not equivalent to either of the solutions proposed in the question.

Answer 2

I am totally with you! While English is notoriously ambiguous, I think the most reasonable interpretation of a statement like: "All P's, with the exception of Q's, are R's" is to say that the P's that are not Q's are R's, but the P's that are Q's are not R's. When we say that the Q's are the exception, I think it is pretty clear that it is meant that being R does not hold for them.

And even if your professor wants to interpret the statement as saying that those P that are not Q's are R's, while for those P's that are Q's we don't know whether they are R's, you would get:

$\forall x (A(x) \to (\neg H(x) \to B(x))$

or, equivalently:

$\forall x (A(x) \to (\neg B(x) \to H(x))$

or:

$\forall x ((A(x) \land \neg H(x)) \to B(x))$

or:

$\forall x ((A(x) \land \neg B(x)) \to H(x))$

but certainly not your professor's:

$\forall x ((A(x) \land B(x)) \to \neg H(x))$

because that would be true if no Americans like either hot dogs or burgers ... meaning that even those who don;t like hot dogs still don;t burgers either. No reasonable interpretation would say that!