[Math] How to read this logical statement aloud, and how do you notate it in symbols

Question

Harry Potter and the Methods of Rationality is a wonderful work of fan fiction by AI researcher and decision theorist Eliezer Yudkowsky. In Chapter 39, this exchange takes place between Dumbledore and Harry:

"…I don't want everyone to die, Harry!"

"You just don't want anyone to be immortal," Harry said with considerable irony. It seemed that elementary logical tautologies like All x: Die(x) = Not Exist x: Not Die(x) were beyond the reasoning abilities of the world's most powerful wizard.

(Harry is a bit more intelligent in this work, in case you failed to notice.)

My question pertains to that tautology. He seems to have converted individual logic notation symbols or groups of symbols to single English words. I can't read logical notation in the first place, so even if I could find examples of similar notation I probably wouldn't be able to parse them without several hours of study. And it may be that you can't notate this particular statement without getting creative with the syntax in ways only a proper math/logic expert could do intelligently.

So I ask you wonderful people to take it in both directions for me: How would you notate that statement properly? And then how would you read it aloud in spoken English?

Answer 1

Let $\text{Die}(x)$ denote "Dumbledore wants $x$ to die." Then Harry claims that

$(\forall x) \text{Die}(x) \Leftrightarrow (\neg \exists x) \neg \text{Die}(x)$

which follows from standard properties of how quantifiers and negation behave in logic. $\forall$ is "for all" (the universal quantifier), $\neg$ is "not" (negation), and $\exists$ is "exists" (the existential quantifier). The above should be read (depending on how closely you like to adhere to notation)

"For all $x$, Dumbledore wants $x$ to die" is equivalent to "There does not exist $x$ such that Dumbledore does not want $x$ to die."

or, as the text says,

"Dumbledore wants everyone to die" is equivalent to "Dumbledore does not want anyone to be immortal."

Of course what is really going on here is a framing effect: it is a well-documented psychological phenomenon that people can have very different reactions to two logically equivalent rephrasings of the same situation.