We could have used the tilde, which has been used in formal logic & maths for negation in very many contexts for a long time.

It's used instead in C and many C-like languages for the far less useful bitwise negation. Of course, we could have had it work in the same way as bitwise vs logical and & or, by dialling up the symbol. Which would have massively improved its visibility compared to the bang.

But for some reason, no. They chose the bang instead.