Negation of a true proposition It starts by someone asking an exercise question that whether negation of
2 is a rational number

is
2 is an irrational number

Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irrational. 
My argument is that because first proposition is always true because 2 could not be anything else but rational number, any false sentence could be negation of the first proposition, including the given sentence above. They said what I do is not "negation" in their sense.
So my question is, is it true that every false statements are negation of true statement? 
 A: If you are talking about the mathematical logic, the answer of your question is true. As this logic is two-valued (binary) and it means the negation of a false statement must be true and vice versa. A statement that all the time is false, is denoted by $\bot$ and the all the time true statement is called tautology and denoted by $\top$, in the mathematical logic.
In addition, you can find three-valued logic and more like temporal logic. However, I don't think you are seeking about them.
A: The answer to your question depends on which logic framework you think corresponds the most the everyday life. There are two such ones which answer differently to your question :


*

*Classical logic : the framework in which most people think naturally. As OmG said, you can think of it as 2-valued, that is, that statements can be interpreted as either true or false. In this logic, $ \neg \neg P $ is equivalent to $ P $ ($\neg$ is negation). So the answer to your question would be yes.

*Intuitionistic logic : here we have something that could be considered less intuitive, as we have all the classical axioms except the axiom of the excluded middle which is that for any proposition $ P $, $ P \lor \neg P $ ($\lor$ is logical or). The reasoning behind it is that it would be impossible to have a way construct a proof of either $ P $ or $\neg P$ for any possible $ P $, and in that way it is the correct framework for constructive logic. For example, reductio ad absurdum is not possible in this, because you have $ P \Rightarrow \neg \neg P $ but not $ \neg \neg P \Rightarrow P $. This logic is also not 2-valued, and this is of interest in model theory, with possible models being Heyting algebras, in which you could find "statements" that are never the negation of another one. As an example, take the Heyting algebra of open sets of $ \mathbb{R} $. Then $ \mathbb{R}^*$ is an open set that is never the negation of another one (that is, it is never equal to the interior of $  A^c $ for $ A $ an open set).
