Interpret the predicate Love($x,y,t$) as "$x$ loves $y$ at time $t$". Write the following statements with predicate logic.
Sometimes two people fall in love with each other forever.
I know that this is defined as $$(\forall x)(\forall y)(\forall t)\text{Love}(x,y,t)$$ but why not $$(\exists t)(\forall x) (\forall y)\text{Love}(x,y,t)?$$
Is it because forever (all the time) negates sometimes or is it a trick question?
On
I know that this is defined as $$(\forall x)(\forall y)(\forall t)\text{Love}(x,y,t)$$
It is not. That is: "Everyone loves everyone all the time."
but why not $$(\exists t)(\forall x) (\forall y)\text{Love}(x,y,t)?$$
No. That is, "There is a time everyone loves everyone."
You want to say: "Every time after some time there are two different people and they each love the other."
It's a matter of language first. "sometimes two people fall in love forever" should be analysed as: there are persons and some moment in time such that these persons love each other from that moment onwards. It's not $\forall x$ etc. because it holds for some people, not all of them. And if you start with $\exists t \forall x \forall y$ this means there is some moment that all people love each other, which is not what is asked for.
Personally I would use the translation:
$$\exists x \exists y \exists t \forall t': \lnot(x=y) \land (t ' > t \to \textrm{Love}(x,y,t') \land \textrm{Love}(y,x,t'))$$