# How do you read this logical statement aloud, and how do you notate it in symbols?

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?

• I know that sometimes time becomes one of the scarcest of resources. I still don't see the problem with "several hours of study". – Asaf Karagila Jul 19 '11 at 6:15
• Alas, it's simply a matter of priorities. This is likely to be the only time in my life that the ability to read and construct logic notation will be useful to me. And as you might guess, the answer to a question pertaining to a work of Harry Potter fanfic is on the outskirts of any reasonable definition of "useful" to begin with. In any case, I doubt there are many questions on the Stack Exchange network that could not be answered after several hours of the askers' time; yet curiously, there are still plenty of people who are happy to answer questions here. – 75th Trombone Jul 20 '11 at 5:16

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.

• I think what's really going on is that the time of people's death is relevant but is suppressed in what is said. Dumbledore only means he doesn't want everyone to die now or soon, not that he doesn't want them to ever die. So the intended meanings of the two statements are not logically equivalent. – Tara B Mar 2 '13 at 19:26
• Sure, but Harry doesn't see a strong distinction between asking that people die soon and asking that people die later. They still die either way. See yudkowsky.net/singularity/simplified for an elaboration. – Qiaochu Yuan Mar 2 '13 at 19:34

It would seem that the author made an error. Like Qiaochu says,

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

But Dumbledore says, "I don't want everyone to die, Harry!"

Therefore,

"Dumbledore doesn't want everyone to die" is equivalent to "Dumbledore wants at least one person to be be immortal."

Therefore, in the quotation,

"You just don't want anyone to be immortal," Harry said with considerable irony.

Harry is making a mistake; actually, the author is making a mistake.

• Nevertheless, in the spirit of fanfiction: Actions do speak louder than words, and Dumbledore's machinations did lead to the deaths of Nicolas Flame and Tom Riddle, his two contemporaries who each had some mechanism for immortality. – Rolanda Hooch Mar 2 '13 at 19:17
• Neither Harry nor the author is making a mistake. I think the point is that Dumbledore would agree that he doesn't want anyone to be immortal, hence 'with considerable irony'. – Tara B Mar 2 '13 at 19:24