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"?

• the negation is "you know at least one thing" Aug 5, 2016 at 14:15
• What's the logic symbol for 'Jon Snow'?
– Pedro
Aug 5, 2016 at 15:13
• @VatsalManot I note in your question that you mention sentential logic, which is usually understood to mean propositional calculus. Propositional calculus has no quantification, so the idea that "Jon Snow knows nothing" could be represented as $\neg \exists\,x.\mathit{knows}(\mathit{JonSnow},x)$ doesn't work so well. Without quantifiers, you're left with just negating the sentence: "It is not true that Jon Snow knows nothing." Aug 5, 2016 at 20:06
• @JoshuaTaylor: I see, thanks for the clarification! Aug 5, 2016 at 20:07
• You to are abusing the amiguity of "A is not B". You mean it to say "it is not the case that A = B". This is equivalent to saying "'I am a fish' is not 'cars use gas'". Those statements arent the same. Your friend means it to say "A = not(B)". This is equivalent to saying "'I am a fish' is not('I am not a fish')". In your friends case NOT(I know nothing) is "I know something" which is different than "I know everything". In your case "I know nothing" is not the same thing as "I know everything". So you are both right in your different uses of the term. Aug 5, 2016 at 20:30

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".

• My friend is still yammering on about "applying Boolean laws". How does one deal with such people? 🙄 Aug 5, 2016 at 14:21
• Ask your friend what is the negation of "You know something, Jon Snow”, perhaps.
– MJD
Aug 5, 2016 at 14:35
• @MJD: Good idea! Aug 5, 2016 at 14:38
• @VatsalManot: Tell him that by his logic either he eats nothing at all or he eats everything (including stones). Aug 5, 2016 at 15:21
• Perhaps they are a terrible sea lion? wondermark.com/1k62 Aug 5, 2016 at 19:04

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.

• +1 for simple explanation (simply add "do not") :) Maybe this is even better way to explain this logic issue to VatsalMascot's friend. Aug 6, 2016 at 6:31

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."