Translation of predicate logic into english confusion

Question

For this question: Interpret the predicate $\mathrm{Love}(x,y,t)$ as "$x$ loves $y$ at time $t$". Write the following statements with predicate logic:

• Nobody is in love all the time

I do not understand why the answer is:

$$∀x∃t∀y¬\mathrm{Love}(x,y,t)$$ and not $$(∀x)(∃y)(∀t)¬\mathrm{Love}(x,y,t)$$ because the answer would translate to: For all $x$, at some time $t$, for all $y$, it is not the case that $x$ loves $y$ at time $t$ which doesn't really make sense to me.

Whereas, the answer I came up with (which isn't correct) would translate to: For all $x$, there exists some $y$, at time $t$, such that it is not the case that $x$ loves $y$ at time $t$ and I don't understand why this is incorrect

Answer 1

Nobody is in love all the time.

$\iff$ Nobody is all the time in love.

$\iff$ There does not exist a person who all the time is in love.

$\iff$ Every person at some time is not in love.

$\iff$ For each person, there exists a time when it is not that they are in love.

$\iff$ For each person, there exists a time when, with each person, it is not that they are in love.

$\iff \forall x \,\exists t \,\forall y \,\lnot \mathrm{Love}(x,y,t).$

Answer 2

I think that the initial sentence means that there is no such person that can be in love his/her whole life. Equivalently, for each person there exists at least one moment of time when he/she is not in love (with anyone). That translated into predicate language gives the "correct" answer.

Your answer at the same time says: for each person there is someone, who that person never loves. Or: nobody experiences love for every single person at some points of time.

Answer 3

Hint: you need to capture the concept of being in love at a certain time. Person $x$ is in love at time $t$ if there is some person $y$ whom $x$ loves at time $t$: $(\exists y)\mathrm{Love}(x, y, t)$.

"Nobody is in love all the time" is equivalent to saying that for each person $x$ there is a time $t$ at which $x$ is not in love, i.e., $(\forall x)(\exists t)\lnot((\exists y)\mathrm{Love}(x, y, t))$, which is equivalent to $(\forall x)(\exists t)(\forall y)\lnot\mathrm{Love}(x, y, t)$, which is the supplied answer.

Your answer $(∀x)(∃y)(∀t)¬\mathrm{Love}(x,y,t)$, says that for every person $x$, there is some person $y$ whom $x$ is not in love with at any time. But that does not preclude the possibility that everybody is in love all the time. E.g., if there are three people, Anne, Bob and Charlie, and if Bob and Charlie both love Anne (and nobody else) and Anne loves Bob (and nobody else), then everybody is in love, but nobody loves Charlie.

Answer 4

The correct answer says: For everyone there is some time where they loves noone (= everyone not).

Your answer says: For everyone there is someone they never (= at all times not) loves.

Your answer is not strong enough because $\exists y$ only requires that there exists some unloved person, which does not exclude the possibility that each $x$ is in love with some other $y$ at all times. At the same time it is too strong in the sense that it denies the love for a particular individual at all times, whereas the English sentence would permit for everyone to be loved by anyone at some point as long as noone is in love with someone all the time.

Answer 5

Nobody is in love all the time

That is: "There is not someone who, for any time, there is someone they love then."

$$\lnot\exists x~\forall t~\exists y~L(x,y,t)$$

So this is also: "Everybody has some time where anybody is not someone loved then."

$$\forall x~\exists t~\forall y~\lnot L(x,y,t)$$

Note: This accounts for the possibility that people may change whom they love over different times.)

(∀x)(∃y)(∀t)¬Love(x,y,t)

This is: "Anybody has somebody that they do not love all the time."

This does not account for the possibility that there are people who love more than one person. Kate who loves Tom, Dick, and Harry, might not love each of them all the time, but may still love at least one among them at any time.