I'm following my first logic course as part of my pre-masters programme. Currently I'm working on predicate logic.
I know that $∀$ is the universal quantifier, which stands for "all" or "every", and $∃$ is the existential quantifier, which stands for "some" or "there is".
In my textbook I tried the following question:
Translate the following sentences into predicate logical formulas. Assume the domain of discourse is human beings. Not all girls love themselves
I used $G$ for "Girl" and $L$ "Loves". My translation was as follows: $$ ¬∀x(Gx \to Lxx) $$ But, the solution the textbook gives is: $$ ∃x(Gx ∧ ¬Lxx) $$
I'm really wondering if both solutions are correct. Actually, I think mine is more precise considering that "Not All" is $¬∀x$ and $∃x$ is "Some".
But I guess I am missing something, or is this just a style thing and are both correct?