# Predicate logic translation $\exists y \exists x \neg P(x,y)$

Let P be the predicate P(x,y) "x owns y" where x represents people; y represents objects.

$$\exists y \exists x \neg P(x,y)$$

I am trying to convert the above statement into plain english, for some reason this one is giving me trouble but it should be very easy... I know that you cannot directly negate objects or people.

I read; "There exists an object and there exists person such that some person does not own some object."

"Some people do not own some objects"

The correct solution is given as; "Some things are not owned by everyone"

What is the correct thought process here? I know this is trivial but if you could be very detailed it would be helpful, I am trying to help a friend understand why this solution is correct; Although it seems I need the practice too.

EDIT: Thank you all very much for your help!

• In my opinion both answers are either incorrect or correct. semantically yours transmits the same idea as $\exists y \exists x \neg P(x,y)$ and as "Some things are not owned by everyone", but you inverted the order of the quantification. Of course they mean the same, but it's not the same statement. This why I think your answer might be deemed incorrect, depending on the criteria. But if someone decides to be this picky, then he can't accept the given solution either, because it doesn't use the predicate $P(x,y)$ it uses a new one $Q(x,y)$ that stands for '$y$ is owned by $x$'. – Git Gud Oct 25 '14 at 1:46
• The given solution is also translating $\exists y\neg\forall xQ(x,y)$, so I think the solution you propose is actually closer to $\exists y \exists x \neg P(x,y)$ than the given one. – Git Gud Oct 25 '14 at 1:48
• @GitGud I agree, thank you for you input. – mar10 Oct 25 '14 at 2:09

A straightforward translation is

there are an object $y$ and a person $x$ such that $x$ does not own $y$.

This still sounds more like mathematics than like everyday English, so we try to improve it. That there is someone who does not own a particular object means precisely then it’s not the case that everyone owns that object, so we can further translate:

there is an object that isn’t owned by everyone.

This is a perfectly fine answer. One could also focus on the person. That there is an object that a certain person does not own means precisely that this person does not own every object, so we could equally well go with:

there is a person who does not own everything.

(The reason that we can focus on either the person or the object is that when two adjacent quantifiers are of the same kind — both $\exists$ or both $\forall$ — they can be interchanged without affecting the meaning of the expression.)

• Thank you; This is exactly what I was looking for. I appreciate the assistance Brian! – mar10 Oct 25 '14 at 2:04
• @mar10: You’re very welcome! – Brian M. Scott Oct 25 '14 at 2:04

I believe your first translation is an accurate, direct translation, except that I would phrase it as, "There exists an object and there exists a person such that that person does not own that object."

Your second translation is equivalent to the given statement (and therefore as correct as the "accepted" solution).

Since the "accepted" solution involves manipulation of the quantifiers (replacing an existential with a universal), we might as well further manipulate the statement mathematically. I get $$\neg \forall y \forall x P(x,y)$$ or, "Not everything is owned by everyone."

• Thank you David for this perspective as well. – mar10 Oct 25 '14 at 2:37

The following are equivalent statements: $$\begin{array}{rl} \exists y\exists x\,\neg P(x,y) & \text{"There is something that someone does not own."} \\ \exists x\exists y\,\neg P(x,y) & \text{"There is someone that does not own something."} \\ \exists y\neg \forall x\, P(x,y) & \text{"There is something that not everyone owns."} \\ \exists x\neg \forall y\, P(x,y) & \text{"There is someone that does not own everything."} \\ \neg\forall y\forall x\,P(x,y) & \text{"Not all things are owned by everyone."} \\ \neg\forall x\forall y\,P(x,y) & \text{"Not everyone owns everything."} \end{array}$$

In summary: "Ownership is not universal."