There are at least two people who love each other and no one else
If I were to negate this, this is my thinking:
It is not the case that there are at least two people who love each other and no one else
If there is not at least two people who love each other and no one else, then everybody must love everybody?
Am I doing this right?
On
Well an easy negation is "there aren't at least two people who love only each other", but assuming you want to know how to get a statement that doesn't have a negation on the whole thing, the trick for negated quantifiers like "for all" and "there exists" is that if you want to move the negation past the quantifier, you switch quantifiers and negate what's quantified.
For example:
$$ \neg \exists c (\forall p \in c(Love(p, x) \implies x \in c)) \\ \forall c (\neg \forall p \in c(Love(p, x) \implies x \in c)) \\ \forall c (\exists p \in c \ \neg (Love(p, x) \implies x \in c)) \\ \forall c (\exists p \in c(Love(p, x) \wedge \neg (x \in c))) \\ \forall c (\exists p \in c(Love(p, x) \wedge x \notin c)) $$
Every couple contains a partner that loves someone and that someone is not in the couple. Or, better put: a partner in every couple loves someone other than their partner.
No, the negation of "love noone else" is "love someone else".
Also, the negation of the statement includes cases where no two people love each other.