“John may go to the park” – this sentence expresses that it is possible for John to go to the park. The key word here is may. Another way to phrase it is then would be “It’s possible for John to go to the park”.
However, if somebody says “John may not go to the park”, we tend to translate that as “John cannot go to the park.” – but when we use inductive reasoning by applying the same principals in the first example, we find out that this is not the case. Since the word may is simply expressing possibility, we can say “It’s possible for John to not go to the park”, this means that it’s possible for John to not go the park…but if it’s possible that he will not go to the park, it must also be possible that he could. If not, then the statement is false and should read “It is impossible for John to go to the park”.
So, with that said...may and may not mean the exact same thing. Right? Another example...
"This may be the case"
"This may not be the case"
What's the difference? The two mean the exact same things from a mathematical and grammatical point of view. Right? If I'm wrong, then English is confusing to me because it doesn't follow consistent logic