How to translate "Who doesn't eat meat, is vegetarian" into a formula with predicate letters Lx meaning x is meat, Exy meaning x eats y, and Vx meaning x is vegetarian?
I tried, "for every x and for every y, if y is meat and x doesn't eat y, then x is vegetarian" but am not sure about this...
Basically, what you said is: $$(\forall x)(\forall y)(L_y \wedge \neg E_{xy} \implies V_x)$$ However, this isn't quite right. For example, let's say $x$ is me, a Muslim, and $y$ is pork. Then, $L_y$ since chicken is meat and $\neg E_{xy}$ since I don't eat pork. However, I'm not a vegetarian because I still eat some meat, so the above statement is false.
Therefore, we need to say that if someone does not eat ALL meat, then they are a vegetarian. Here's how we say $x$ does not eat any meat: $$(\forall y)(L_y \implies \neg E_{xy})$$ This means that for any thing, if that thing is meat, then $x$ does not eat it. Thus, $x$ does not eat any kind of meat. Now, we need to say that this statement implies that $x$ is a vegetarian, so we have: $$(\forall x)((\forall y)(L_y \implies \neg E_{xy}) \implies V_x)$$
