Help with logical quantifiers

Question

Let $L(x,y)$ be "$x$ loves $y$". Then is the statement: "Nobody loves everybody" equivalent to $$∀x ∀y \overline{L(x,y)} $$

Answer 1

The first order sentence you wrote, $\forall x\,\forall y\, \neg\, L(x,y)$, means "nobody loves anybody", as Frentos observes. Notice that "nobody" means "there isn't somebody" — "nobody is such that $P$" means "there is no person $x$ such that $P(x)$", which is to say $\neg\exists x\, P(x)$.

In this case, $P(x)$ is "$x$ loves everybody". We can render that with the formula $\forall y\, L(x,y)$. Thus "nobody loves everybody" can be symbolized by $$\neg\exists x\,\forall y\, L(x,y).$$ Using the rules for interchanging negation and quantifiers, $\neg\forall\equiv\exists\neg$ and $\neg\exists\equiv\forall\neg$, this is equivalent to $$\forall x\,\exists y\, \neg\, L(x,y).$$

Answer 2

Your current answer, $∀x ∀y \overline{L(x,y)}$, says for every pair of people $x$ and $y$, $x$ does not love $y$. That is, nobody loves anybody.

Hint: Think about "nobody has property $P$" as "there does not exist a person $x$ such that $x$ has property $P$"

Hint: How would you write "somebody loves everybody"? I.e. "there is a person $x$ such that $x$ loves everybody"?

Answer 3

I think you want if L(x,y) means x loves y that there is no x such that for all y L(x,y). That is, "not (there exists x[ for all y L(x,y) ])." which can be symbolized from this English version.