Negating first order logic

Question

I am struggling to understand how to really negate in first order logic. Take the following examples:

"Somebody loves everybody" Negating this would be: "It is not the case that somebody loves everybody". I am having trouble simplifying this even further, as it is not enough to simply leave it like this.

or

"Socrates loves nobody" Negating this would be: "It is not the case that Socrates loves nobody". I think this simplifies to "Socrates loves somebody"?

Any advice as to how to approach these types of problems would be greatly appreciated.

Answer 1

Break down the first statement into a more mathematical notation first "Somebody loves everybody" means - there exists a person A such that A loves everybody. Now break down the everybody part some more - there exists a person A such that (for all people B, A loves B)

So now negate it. There does not exist a person A such that (for all people B, A loves B).

Since there does not exist a person A satisfying the statement in the parenthesis, that must mean the statement in the parenthesis is false for all people.

For all people A it is not true that (for all people B, A loves B)

How can the statement (for all people B, A loves B) be false? It must mean that there exists a person C such that A does not love C.

Put this back together.

For all people A there exists a person C such that A does not love C.

You see how some and every change roles when doing a negation. That is $\exists$ and $\forall$ in the more symbolic notation.

Answer 2

The first example should be, nobody loves everybody. Your treatment of the second example is fine. The trick is to parse each statement as saying a certain kind of thing either does or doesn't exist. Negating changes "all do it" to some don't, and "some do it" to none do. If you restate the original problem as negating "none get his love", the rest is trivial.