2
$\begingroup$

enter image description here

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?

  • 2
    I've chosen one of your five questions to answer. Take time to understand the answer, and ask any questions you may have about it. But it is *entirely inappropriate* to post five statements, each of which you want translated into logic, then its negation, and then translated back to natural language. – 2017-02-10
  • 1
    Since you've "analyzed the logic forms" of $a -e$, if you include the analysis of strictly **one** more statement above, I will work with you on negating it, and aid you in translating the negation back to English. If you're willing to do that, please post the statement, and your analysis (presenting it in logical form), below my answer below. – 2017-02-10
  • 1
    "There is a someone who has heard of a someone who does not like them." is the negation of "Anyone who has heard of everybody, will be liked by everybody." ... how delightful ... I am going to try to explain this to my wife ... ha ha ha – 2017-02-10

1 Answers 1

2

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."

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.