# Translate to predicate logic “the one who dont love himself cannot love anyone else”

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?

• You probably either want to not specify that $x$ is a person (and instead assume that the domain of discourse is "people") or specify that $y$ is a person, at least. – Patrick Stevens Mar 11 '18 at 13:12

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

• Actually it's "...such that x loves x or x loves y" – Henrik supports the community Mar 11 '18 at 13:17
• This seems OK to me, but I am used to : symbol to signal "such that" for example $\forall x :$, $\nexists y :$ et.c. but I have not studied much logic so maybe I recall / learned it wrong. – mathreadler Mar 11 '18 at 13:27
• @Henrik Thanks; I misread the OP's post. – J.G. Mar 11 '18 at 14:05

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.