Can't see the intuition behind the validity of this formula: $\exists x(\exists yP(x,y) → \forall z \exists wP(z,w))$

Question

I know that

$$\vdash_{\mathcal G}\exists x(\exists yP(x,y) → \forall z \exists wP(z,w))$$

(I have read and done a syntactic proof of this.)
And therefore also

$$\models \exists x(\exists yP(x,y) → \forall z \exists wP(z,w))$$

By soundness of FOL.

But I can't make sense of how this formula can be valid. It looks wrong to me. The way I read it is: if there exists some $x$ and $y$ such that $P(x,y)$, then forall $z$ and some $w$, $P(z,w)$. But how can that be valid?

The order of quantifiers is significant in FOL, so it might be simpler to consider a logically equivalent formula on a simpler form. This formula can be rewritten to a formula in Prenex Normal Form, for example like this:

$$\exists x \forall y \forall z \exists w: P(x,y) → P(z,w)$$

But this formula does not look like it is valid, either. Here is an attempt at a counter-example in the form of an interpretation of the formula (a structure for it): Let $P = \{0\} \times \mathbb{N}$. Then $P(0,x)$ for all $x \in \mathbb{N}$, but there is no $w \in \mathbb N$ s.t. $P(z,w)$ for all $z \in \mathbb N$. In fact, there is only one $z \in \mathbb N$ s.t. $P(z,w)$ holds for some $w$, namely $z = 0$.

Answer 1

This is just the Drinker Paradox in disguise.

Note the following:

$$\exists x(\exists yP(x,y) → \forall z \exists xP(z,x))\iff \exists x(\exists yP(x,y) → \forall y \exists xP(y,x)).$$

Denoting the predicate $\exists x_2P(x_1,x_2)$ by $D(x_1)$ turns the subject statement into

$$\exists x(D(x)\to \forall yD(y)).$$

If you allow me to rewrite this as $\exists x(\neg D(x)\lor \forall yD(y))$, then I think it is intuitive to think of this is as saying "either someone doesn't drink, or everyone does".

Note that this isn't what this formula conveys immediately. The statement "either someone doesn't drink, or everyone does" should be formalized as $\exists x\neg D(x)\lor \forall yD(y)$, but they are equivalent and since this is a matter of intuition, I think this close enough.

If it is any help, see my comment here.

Answer 2

Assuming the universe is non-empty, consider two cases:

1. $\forall x\ \exists y\ P(x,y)$.

   Here your formula is true, because the conclusion of implication is trivially true.

2. $\exists x\ \forall y\ \neg P(x,y)$.

   Here your formula is true, because for that $x$ we have $\neg P(x,x)$ and so the premise of implication is false.

In fact your formula is very similar to reordering quantifiers, namely

$$\exists x\ \forall y\ P(x,y)\ \to\ \forall y\ \exists x\ P(x,y).$$

I hope this helps $\ddot\smile$

Answer 3

The intuition is clear, take $x$ to be an element with no corresponding $y$ such that $P(x,y)$, if that is possible. Then the hypothesis will be false, and the statement true. On the other have if for every $x$ there is a $y$, $P(x,y)$ then the conclusion is correct and you can take any element for $x$.

Answer 4

But the reading :

if there exists some $x$ and $y$ such that $P(x,y)$, then for all $z$ and some $w$, $P(z,w)$

is not correct; the first $\exists$ is "outside" the parentheses.

Try with an "informal" argument, starting rewriting the formula as :

$∃x(\lnot ∃yP(x,y) \lor ∀z∃wP(z,w))$

you immediately see that you cannot "move inside" the first $\exists$ without changing the meaning of the formula.

If we remember that we can "distribute" $\exists$ over $\lor$, because $\exists$ "is like" an "inifinite" disjunction", we have that our formula is "equivalent" to :

$∃x(\lnot ∃yP(x,y)) \lor ∃x∀z∃wP(z,w)$

which is :

$\lnot ∀x∃yP(x,y) \lor ∃x∀z∃wP(z,w)$.

In the second disjunct, the quantifier $\exists x$ does not do any job, because $x$ in not free in it; thus, forget it.

Finally, we have :

$\lnot ∀x∃yP(x,y) \lor ∀z∃wP(z,w)$

which is "clearly" valid.