If we have a function $f(x)\,:\,[0,1]\mapsto[0,1]$, which is continuous and montonically decreasing, satisfies $f(0)=1, f(1)=0$ (to agree with classical logic) and is an involution, i.e.,
$$f(f(x))=x,\quad\forall x\in[0,1],$$
do these conditions uniquely determine that the logic "not" function must be $f(x)=1-x$? If yes, please give a proof. If no, please give an alternative $f(x)$ that also has these properties.
Background: In fuzzy logic, truth and falsehood are a matter of degree, which can be between $0$ and $1$. This question asks about whether the definition of the fuzzy "not" operation is unique in order to preserve certain properties of the classical logic.
I believe this works on $[0, 1]$?
$$f(x) = \sqrt{1 - x^2}$$