It doesn’t have to be a totally new language. Javascript is an example where there are multiple front-end languages. You can go as far as formally prove the implementation is correct, too. Language theory is an abstraction for mathematical logic and proofs, so it touches anything logical at all.
Of course, for example, if you're in need of an automated theorem prover, your only options are Coq, Agda, Isabelle/HOL etc for now all of which come with very serious trade-offs which leads to the development of newer languages such as Lean etc.
The language is supposed to tell you if your thinking is broken. If a program is a logical statement, and your logic is impossible, the program should not compile. Reading dead memory or having race conditions should not be possible, yet look at security vulnerabilities today.