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" – palio Aug 5 '16 at 14:15
• What's the logic symbol for 'Jon Snow'? – Pedro Tamaroff Aug 5 '16 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." – Joshua Taylor Aug 5 '16 at 20:06
• @JoshuaTaylor: I see, thanks for the clarification! – Vatsal Manot Aug 5 '16 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. – fleablood Aug 5 '16 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? 🙄 – Vatsal Manot Aug 5 '16 at 14:21
• Ask your friend what is the negation of "You know something, Jon Snow”, perhaps. – MJD Aug 5 '16 at 14:35
• @MJD: Good idea! – Vatsal Manot Aug 5 '16 at 14:38
• @VatsalManot: Tell him that by his logic either he eats nothing at all or he eats everything (including stones). – user21820 Aug 5 '16 at 15:21
• Perhaps they are a terrible sea lion? wondermark.com/1k62 – Alejandro Aug 5 '16 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. – Spook Aug 6 '16 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."