Translate from English a quantified logical statement, negate it, and translate back to english

Question

I have analyzed the logical form for each of these. However, I am not sure what to do when it says to negate the statement and translate back to english?

Answer 1

I'll given you an answer to (a) and (c).

Every likes somebody.

$$\forall x, \exists y(L(x, y))\tag{(a)}$$

$$\lnot \forall x \exists y\Big(L(x,y)\Big)\equiv \exists x \forall y\Big(\lnot L(x, y)\Big)$$

"There is someone who everyone dislikes."

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Anyone who has heard of everybody, will be liked by everybody.

$$\forall x \forall y(H(x, y) \rightarrow L(y, x))\tag{c}$$

Now, we will negate (c). $$\lnot\Big(\forall x \forall y(H(x, y)\rightarrow L(y, x))\Big) \equiv \exists x \exists y\Big(\lnot (H(x, y) \rightarrow L(y, x))\Big)$$

$$\equiv \exists x\exists y\Big(\lnot (\lnot H(x, y)\lor L(y, x))\Big)$$

$$\equiv \exists x \exists y((H(x, y) \land \lnot L(y,x)$$

Which can be translated: There is a someone who has heard of a person who does not like him/her.