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Suppose claim $P$ is "I know that I don't know you". My gut feeling says "I don't know that I don't know you". One set of lecture notes I have says I need to negate everything in the sentence in a nested fashion, which yields "I don't know that I know you". Any clarification is very much appreciated.

user64066
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    I think that the advise "to negate everything in the sentence in a nested fashion" refers to the way to "move inside" the negation sign of a formula... The negation of a sentence $P$ is $\lnot P$, full stop. But if $P$ is a "complex" formula, like $R \land \lnot Q$ you can start with $\lnot (R \land \lnot Q)$ and "move inside" $\lnot$ using De Morgan's laws to get : $\lnot R \lor Q$. – Mauro ALLEGRANZA Dec 14 '14 at 18:59
  • As usual when coming from natural language, the very question is how you formalize the sentence, I would say. And it seems quite hard in this case, since "I know that X" looks different from "I know X", where the last might be a relation, but what should be the first? Edit: So to give an answer I would need more information about your lecture. – aphorisme Dec 14 '14 at 19:26

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I would formalize the proposition "I know that I don't know you" in the following way: First I would introduce the relation $K(x)$ via $$K(x) :\Leftrightarrow \text{I know that x}$$ whereby $x$ is a proposition. Let $A$ be the proposition "I don't know you". So your given proposition "I know that I don't know you" is $K(A)$. The negation of $K(A)$ is $\neg K(A)$ or "I don't know that I don't know you" (as you suggested first).

The proposition "I don't know that I know you" is $\neg K (\neg A)$. Because the negation of $R(x)$ for any relation $R$ is $\neg R(x)$ not $\neg R(\neg x)$, this proposition is not the right negation. Note that $\neg x$ only make sense if $x$ itself is a proposition which is not the case for every relation $R$. So $\neg R(\neg x)$ cannot be the right negation of $R(x)$.

Stephan Kulla
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