When to use which quantifier with predicate logic?

Question

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?

Answer 1

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, $$ ∃x \ (Gx \land \lnot Lxx) \text{.} $$

Answer 2

Both are correct. They are equivalent.

$\neg \forall x~(Gx\to Lxx)$ "Not all girls love themselves."

$\exists x~(Gx\wedge \neg Lxx)$ "Some girls don't love themselves."

Answer 3

I would say that the author of the textbook took a wrong approach. The task of translation is to communicate meaning as close as possible to the original. This is especially important in mathematics. Your answer follows English text verbatim, and the author’s answer is transformed. Albeit the transformation is legal, I bet that it baffles readers because transformation of logical formulas does not belong to the translation process. I wholeheartedly agree that your answer is more precise.

If you are picky, then you may want to know that there are circumstances where your answer and the author’s answer are not equivalent. Their equivalence is impossible to prove in intuitionistic logic. Intuitionistic logic is slightly weaker than classical logic that your textbook teaches. Intuitionistic logic does not contain the law of excluded middle, and truth tables are inapplicable to it; statements are proved by inference.