If a language $L$ is context-sensitive, then every grammar $G$ that generates $L$ is also context-sensitive.

Question

I am not quite sure, whether the question in the title is true, but I would argue that it is not. If a language $L$ is context-sensitive, then there exists a context-sensitive grammar $G$ that generates $L$. However, this does not say anything about non-context-sensitive grammars, which in theory could also generate $L$. Would this be enough to argue that the statement in the title is wrong?

Answer 1

Then lines of your argument are fine, only the "non-context-sensitive grammars, which in theory could also generate $L$" should be filled with life, i.e., an example.

Formally, it is enough to add a rule that uses non-terminals, which do not occur in the original grammar, and which is not context-sensitive. For example, a grammar with the sole rule $ S \rightarrow a$ generates the finite language $\{a\}$.

We can add the rule $BCDEFG \rightarrow DEBC$. The grammar still generates the same language, but it is not regular, not context-free, and not even context-sensitive.

Of course, this is a pathological example, but we could also design a long derivation that finally derives simply $a$ like in a grammar with the rules $S\rightarrow AB, B \rightarrow CD, D\rightarrow EF, ACEF \rightarrow a$. Here no rule would be completely useless as above.