I'm currently coauthoring a book intended to teach first-year students basic proof techniques. One of the chapters, written by my coauthor, is about basic logic. In that chapter the negation of a statement is defined, and he says that he likes to think of $\neg P$ as the new sentence "$P$ is false".


Being familiar with Tarski's result on the undefinability of a truth predicate, I suggested that this new sentence doesn't properly belong to the original system, but instead belongs to a meta-system, as does any other statement which defines truth/falsity. (Moreover, those statements are not the negation, because they are also restricted to speaking about a specific model of the theory.) Admittedly, this is a subtle point (probably something we won't even tell our students about), but in talking with my coauthor I realized that I'm not entirely sure how we should then go about defining the negation, other than saying "not $P$".


So, here are my questions, which will hopefully shed light on this subject for me.


1. Unlike saying "$P$ is false", would it be okay to say "$P$ does not hold"? Or is that also, implicitly, defining a truth predicate? On the other hand, is it actually **okay** to define the negation as "$P$ is false" if we are speaking of falsity not as a predicate?
2. One of the problems with having a truth predicate would be the ability to create well-formed formulas of the form "I am a false statement." Once we have a negation operation, what prevents us from creating a sentence which says "I am the negation of myself"?



Defining negation
The only problem with defining negation is that there are at least two notions of "negation":


Negation (let's call it $\lnot\_1$) as a syntactic operation is quite clear: you prepend the symbol $\lnot$ to a formula. 


Negation (or $\lnot\_2$) is also a unary function on the two-element Boolean algebra $\{\top,\bot\}$ or $\{0,1\}$(but perhaps you want to use a different name for this operation, such as "complement"). 


If you want to give a "meaning" to a formula $P$, it is quite reasonable to say that the meaning is "$P$ is true"; then clearly the meaning of $\lnot P$ is "$P$ is false". 


Teaching (or at least mentioning) the distinction between syntax and semantics, and also between language ($P$) and metalanguage ("$P$ is true") can be a source of confusion, but sooner or later students will realize that this is an important point.


