How to read this in simple English?

Question

$∃x.P(x) \Rightarrow ∀x.P(x) $

How can I read this in simple English? I translated it as: There exists an element x for which P(x) implies that for all elements x, P(x) is true - but I feel like this doesn't make much sense. What am I doing wrong here?

Answer 1

$\exists x P(x)$ is read as

there exists $x$ such that $P(x)$ holds

Likewise $\forall x P(x)$ is read as

for every $x$, $P(x)$ holds.

$A \implies B$ is read as

If $A$ then $B$.

Thus, bringing all of this together, $\exists x P(x) \implies \forall x P(x)$ is read as

If there exists $x$ such that $P(x)$, then $P(x)$ holds for every $x$.

Which is the same as what you have.

Answer 2

Your translation is right: suppose that there is an element $x$ that makes $P(x)$ true. Then all elements make $P(x)$ true.

This does sound a little weird, but just because it is weird, that doesn't make the translation incorrect. (Logic is pretty weird.) There are predicates $P(x)$ that make $\exists x. P(x) \Rightarrow \forall x. P(x)$ true. One possible $P(x)$ is a predicate that doesn't actually depend on $x$, such as the statement '2+2=4'. Do you see how $\exists x. P(x) \Rightarrow \forall x. P(x)$ is true for this $P(x)$?