When to use which quantifier with predicate logic? 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?

The two answers are equivalent.

"$\lnot \forall$" is the same as "$\exists \lnot$".

If **not all** cats are black, there must be **some** cat **that is not** black.

Thus, we have that $$ ¬∀x \ (Gx \to Lxx) \iff ∃x \ ¬(Gx \to Lxx) \text{.} $$

Now we apply the tautological equivalence $$ \lnot (p \to q) \iff (p \land \lnot q) $$

(We can check it with a truth-table: $\lnot (p \to q)$ is TRUE just in case when $p$ is TRUE and $q$ is FALSE.) to get the final result, $$ ∃x \ (Gx \land \lnot Lxx) \text{.} $$