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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.

(a) Everybody loves somebody at some point. $$(\forall x)(\exists y)(\exists t)\text{Love}(x, y, t)$$

(b) Nobody is in love all the time. $$(\forall x)(\exists y)(\forall t)\lnot\text{Love}(x, y, t)$$

(c) Some people never fall in love. $$(\exists x)(\exists y)(\forall t)\lnot\text{Love}(x, y, t)$$

(d) At some point everyone falls in love with someone. $$(\forall x)(\exists y)(\exists t)\text{Love}(x, y, t)$$

(e) Sometimes two people fall in love with each other forever. $$(\exists x,y)(\forall x)(\forall y)(\forall t)\text{Love}(x, y, t)$$

I am most certain about a through c, d and e were more tricky for mee.

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A) correct

B) should be $\forall x \exists t \forall y \neg Loves(x,y,t)$

C) should be $\exists x \forall y \forall t \neg Loves(x,y,t)$

D) move the $\exists t$ to the front

e) get rid of the universal x and universal y

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    Ah, so order matters, in predicate logic? That is nice to know! Thank you. – 2017-02-03
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    @WensworthKoohli Yes, order matters. For example: $\forall x \exists y Loves(x,y)$ means 'everyone loves someone', but $\exists y \forall x Loves(x,y)$ means 'someone is loved by everyone'. That is, in the second case you have (at least) one very special person that is loved by everyone! This is not necessarily the case for the first one. So the two mean different things. – 2017-02-03