Talk:On the negation of modal verbs
It is correct that : "You must do A" Can be written as: ^A -> P But then, there is a problem with negating the whole thing. That is,
^A and ^P
is true if and only if A is false and P is false. This means that we know that you will not be punished, but also that you did not (will not) play. Hence, this is not how we would negative sentences.
The regular negation of a verb (as in German or in “You do not have to do A”) means that whether we do A or not, we will not get punished for it, as you point out. Using logic, it would be ^P.
So, my conclusion is that the rules of logic may not that easily explain the negation in a language.
Let me also point out that the negation of “must” is indeed "You must (not play in the street.)", but the negation of other modal verbs works differently. For example, the negation of “can” would be "You (can not) play in the street." (maybe that is why it is spelled “cannot”).