Translating an English statement to it's logical equivalent: "No student is friendly but not helpful"

Question

I am curious about the correct interpretation of the following English sentence in predicate logic. I suppose, I may also have to ask an English grammarian.

Let the following predicates be given. The domain consists of all people.

$F(x) = x$ is friendly

$H(x) = x$ is helpful

$S(x) = x$ is a student

Express the following English sentence in terms of $F(x)$, $H(x)$, $S(x)$, quantifiers, and logical connectives.

"No student is friendly but not helpful."

Is it:

A

$¬∃x(S(x) ∧ F(x) ∧ ¬H(x))$

There does not exist a person such that that person is a student, that person is friendly, and that person is not helpful.

or

B

$∀x( S(x) → (¬F(x)∧H(x))$

that person is not friendly and helpful. For all people if a person is a student then

FOLLOW UP

It may be useful to note the ambiguity in the English, which is clarified by the first comment on my posting to the English Grammar & Usage stack exchange, linked HERE

Answer 1

Notice that when you wrote an English sentence corresponding to A, you had (more or less) the same English sentence with which you began.

Also notice that statement B says that if a person is a student, that person is not friendly, which your original sentence did not imply (and which we hope is not true!).

(This would be more appropriate in a comment, but I haven't the reputation.)

Answer 2

Note: $\;\neg \big(F(x) \wedge \neg H(x)\big) \iff \big(\neg F(x)\vee H(x)\big)\;$ by DeMorgan's Laws.

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Now $\;\neg \exists x\, \big(S(x)\wedge F(x)\wedge \neg H(x)\big)\;$ parses as: "there is nothing that is a student and friendly and not helpful," or more naturally: "no student is friendly and/but not helpful."   Which is what you were required to express.

So applying dual negation, DeMorgan's law, and implication equivalence to this actually produces: $\;\forall x\, \Big(S(x) \;\to\; \big(\neg F(x)\vee H(x)\big)\Big)\;$, which parses as "if anything is a student then it is not friendly or it is helpful," or "any student either is not friendly or is helpful."

Alternatively, the equivalent, $\;\forall x\, \Big(S(x) \;\to\; \neg \big(F(x)\wedge \neg H(x)\big)\Big)\;$, reads, somewhat awkwardly as, "any student is not both friendly and not helpful".

I'd stay with the first form of the expression; as it say what you want in a way that is most compatible with natural language.