Translate to predicate logic "the one who dont love himself cannot love anyone else"

Question

I want to translate the following sentence to predicate logic Sentence:

The one who dont love himself cannot love anyone else

Translated into predicate logic my attempt is

$\forall x(Person(x)\implies \lnot\exists y(Love(x,x)\lor Love(x,y)))$

Something tells me this is not correct. Would anyone care to show me how to do this correctly?

Answer 1

It should be $\forall x (\mathrm{Person}(x) \land \neg \mathrm{Love}(x, x) \to \neg\exists y(x\ne y\land\mathrm{Person}(y)\land \mathrm{Love}(x, y))$. What you wrote was, "Given any person $x$, no $y$ exists such that $x$ loves both $x$ or $y$".

Answer 2

This is probably not what the exercise wants, but I would make a distinction between does love and can love. That is, a person may be capable of loving someone, but simply doesn't. Let's write this as $Does(x,y)$ if $x$ does love $y$ and $Can(x,y)$ if $x$ can love $y$. Then the sentence becomes

$$\forall x(\lnot Does(x,x)\implies \forall y(\lnot(y=x)\implies\lnot Can(x,y)))$$

or, if you need to explicitly limit yourself to the universe of people,

$$\forall x(Person(x)\implies(\lnot Does(x,x)\implies\forall y(Person(y)\land\lnot(y=x)\implies\lnot Can(x,y))))$$

Note, the "$\lnot(y=x)$" here stems from the "else" in "anyone else." The sentence as stated does not assert that a person who doesn't love himself cannot love himself; if that were true, psychotherapists would have to admit defeat with a lot of their patients.