@MiNiPanda: You're slightly incorrect. The example you've shown is that of a statement which leads to a contradiction no matter what truth value you try, leading to a paradox and the statement is neither true, nor false. The paradox isn't that it takes multiple truth values, the paradox is that it takes $0$ truth values!
The only way for a statement to be both true and false is to have an inconsistent set of axioms. A set of axioms is said to be inconsistent if it leads to both proving and disproving a statement, that is, if any statement S can be shown to be both true and false under the given set of axioms. The thing with inconsistent axioms is that you can prove anything in them! No matter what the statement P is, you can show that P is both true and false. This makes inconsistent axiomatic systems utterly useless.
Sidenote: We don't know if maths is consistent or not! There are several axiomatic systems that are used across maths. Different axioms lead us to different styles of maths. We've proved some axiomatic systems (let's say $X$, $Y$) to be consistent, but for the proof, we need to be doing our proofs under some different axiomatic system (let's say $Z$). An example would be: the consistency of "Peano Arithmetic axiomatic system" can be proven under the "ZFC axiomatic system". But now we need to prove the consistency of the $Z$, because if $Z$ is inconsistent, our proofs for the consistency of $X, Y$ are useless as you can prove anything in $Z$. One way of resolving this would be if some axiomatic system $B$ could prove its own consistency! Sadly, that's impossible. Godel's incompleteness theorems show that given an axiomatic system $B$, if $B$ is infact consistent, then it cannot prove its own consistency. If it can, then it would mean that $B$ is inconsistent! (Some constraints on $B$ are applicable.)