Is "you know everything" the (using sentential logic) logical negation of "you know nothing"? 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"?

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".
