Is somebody no ones cousin or is somebody not a cousin to everybody?

Question

I am given the following first order sentence:

$\exists x \forall y(\neg Cousin(x, y))$

Note that the predicate Cousin(x, y) means "x and y are cousins".

I am asked to provide an English translation to this statement. My approach is to first go about interpreting the statement without the negation, that is: $\exists x \forall y(Cousin(x, y))$ which would give me:

There exists someone that is a cousin to everyone.

I then add the negation as per the beginning of the question which gives me:

There exists someone that is not a cousin to everyone.

However, upon looking at the solution I see that the correct interpretation is:

Somebody is no ones cousin

Could someone explain to me in an intuitive way why and where my interpretation fell short of the answer?

Answer 1

As I've said in the comments, your interpretation actually corresponds to:

$$\exists x \lnot \forall y (\text{Cousin}(x,y))$$

Instead of what you are asked to translate

$$\exists x \forall y (\lnot \text{Cousin}(x,y))$$

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I will make a feeble attempt at providing "intuition":

Before trying to translate first-order logic into colloquial English, it might be better to use "basic" language first.

Let $F(x,y)$ be some statement

$\exists$ stands for "exists"

$\forall$ stands for "for all"

$\exists x \forall y (F(x,y))$

Translates into: Exists at least one $x$ such that for all $y$ $F(x,y)$ is true.

$\exists x \forall y (\lnot F(x,y))$

Translates into: Exists at least one $x$ such that for all $y$ $F(x,y)$ is false.

$\exists x \lnot \forall y (F(x,y))$

Translates into: Exists at least one $x$ such that not for all $y$ $F(x,y)$ is true.

Clearly, "for all $y$ $F(x,y)$ is false" and "not for all $y$ $F(x,y)$ is true" are two different statements.

In your case you have

$$\exists x \forall y(\neg Cousin(x, y))$$

So again, using basic language first:

Exists $x$ such that for all $y$ statement $\text{Cousin}(x,y)$ is False.

To put it roughly:

There is somebody ($\exists x$) such that everybody ($\forall y$) is not his cousin ($\lnot \text{Cousin}(x,y)$).

Which in proper English means

Somebody is no ones cousin

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Answer 2

My approach is to first go about interpreting the statement without the negation ... I then add the negation as per the beginning of the question

But $\forall y (\lnot K)$ is not at all the same thing as $\lnot \forall y(K)$!

You CAN'T take out a negative in the center and put in front. You should not have thought that you could.

Consider. Then statement "Every bird cannot fly" ($\forall b(\lnot fly(b))$) and "Not every bird can fly" ($\lnot (\forall b(fly(b))$. The first is obviously false (doves can fly) and the second is obviously true (penguins can't).

Instead $\forall y(\lnot K)$ is $\not \exists y(K)$. "For all things this doesn't happen" is "There is no thing where this does happen".

So in this case $\exists x \forall y(\lnot Cousins(x,y))$ is the same as $\exists x\not\exists y(Cousins (x,y)$.

Or "there exists a guy for whom everybody is not his cousin" is the same as "there exists a guy for whom no-one is his cousin" which is the same as "there is a guy who has no cousins".

Then statement $\exists x \lnot (\forall y(cousins(x,y))$ is saying there is a guy who is not everybody's cousin. (That could be anybody! Nobody is everybody's cousin because not every body shares cousins!)

$\lnot (\forall y K)$ is not the sames as $\forall y (\lnot K)$. ("Not all birds fly" $\ne$ "All birds don't fly") $\lnot (\forall y K)$ is $\exists y (\lnot K)$. ("Not all birds fly" $=$ "there is a bird that doesn't fly").

So had you begun with $\exists x \exists y(\lnot Cousins(x,y))$ then your interpretation would be correct $\exists x\lnot(\forall y (Cousins(x,Y))$.

"There exists a person who has a person who is not his cousin" is the same as "there exists a person who is not everybody's cousin".

Answer 3

There exists someone that is not a cousin to everyone.

However, upon looking at the solution I see that the correct interpretation is:

Somebody is no ones cousin

You are not wrong, because those are equivalent claims.   The second is just abbreviated better.

They both say: "There is someone who is not-anyone's cousin." $$\def\Cousin{\operatorname{Cousin}}\exists x~\forall y~\neg\Cousin(x,y)\quad\iff\quad\exists x~\neg\exists y~\Cousin(x,y)$$

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It can also be restated as: "Not everyone has a cousin."

$$\neg \forall x~\exists y~\Cousin(x,y)$$