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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?

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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:

$$\exists x \exists y \exists t \forall t': \lnot(x=y) \land (t ' > t \to \textrm{Love}(x,y,t') \land \textrm{Love}(y,x,t'))$$

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  • $\begingroup$ The consequent should be a conjunction. As well as ensuring that they are different people, they are each to love the other. $\endgroup$ – Graham Kemp Feb 13 '18 at 8:25
  • $\begingroup$ @GrahamKemp True, I'll edit. also added the condition that $x \neq y$, or we could have self-love. $\endgroup$ – Henno Brandsma Feb 13 '18 at 8:42
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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."

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