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 $(\forall x) \: \
eg P(x)$, 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".