TITLE: How can I get the negation of $\exists!$ (unique existential quantification)?
QUESTION [8 upvotes]: How can I get the negation of $\exists!$ (unique existential quantification)? if it's $\forall$, So if I wanna re-negate the last one, I'll get $\exists$ but it's not the same as what we started with! Did I do something wrong here?
For example: $$P : \exists! x\in \mathbb{R} \text{ such that } x^2 = 0$$ it means $\exists x \in \mathbb{R}\text{ such that } x^2 = 0\wedge x $ is unique, So the negation is $\forall x \in \mathbb{R}\space x^2 \ne 0 \vee x \text{ isn't unique}$ is this a false statement?

REPLY [2 votes]: Using the simplest definition of $\exists!$ I know, which is $$\langle \exists! x :: P(x) \rangle \;\equiv\; \langle \exists y :: \langle \forall x :: P(x) \equiv x = y \rangle \rangle \;,$$ the negation you're looking for is $$\langle \forall y :: \langle \exists x :: P(x) \equiv x \not= y \rangle \rangle \;.$$