My friend and I are trying to settle a debate here. Someone joked that they're "not Jon Snow" when debating a yes/no question, and I retorted that the logical negation of "you know nothing, Jon Snow" is not "you know everything, Jon Snow", but my other friend claims that it is. The question, as reduced, stands:
Is "you know everything" the (using S.L.) logical negation of "you know nothing"?
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The negation of "You know nothing." is "You do not know nothing." Which implies that your knowledge is non-zero; not that it is necessarily absolute.
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Imagine someone would say to you "you know nothing!" Now you want to refute that. What would you say?
If you answer "I know everything" you would at best be laughed at. Maybe he'd reply with something like "oh, so you know what I dreamed this night? Let's hear!"
No, what you'd likely answer is: "It's not true that I know nothing." Or formulated more positively: "There are definitely some things that I know." Or in short: "I know something."
No, the negation of "you know nothing" is "you know something". "You know nothing" is of the form $(\forall x) \: \neg P(x)$, where $P(x)$ is "you know $x$". So its negation is $(\exists x) \: P(x)$, which is "you know something" or slightly more precisely "you know at least one thing".