In general, when you negate a statement $p$ to get a statement $\neg p$, two things should be true:
- Both $p$ and $\neg p$ cannot be true at the same time.
- At least one of $p$ or $\neg p$ will always be true: they cannot both be false at the same time.
These are the two characteristics of a negation.
Sometimes we can use these as a quick check of whether we took the negation correctly (even though this is not always easy). For example:
- "Every person likes logic" and "Some people do not like logic" can't both be true at the same time. If every person likes logic, there aren't any people who don't like logic.
- Maybe it's hard to see if "Every person likes logic" and "Some people do not like logic" can both be false at the same time, but they can't.
- However, I can give an example in which both "every person likes logic" and "every person does not like logic" are false, and so this is the wrong negation. Suppose there are two people; one of them likes logic, and the other doesn't.
If we look at "tomorrow will rain" and "all days other than tomorrow will not rain", then
- These can be true at the same time: imagine that tomorrow is the only day in history it rains, and it will never rain again before or after.
- These can also be false at the same time: imagine that it rains today, but not tomorrow.
So "all days other than tomorrow will not rain" definitely isn't the negation.
On the other hand, looking at "tomorrow will rain" and "tomorrow will not rain", it's clear that one of these must happen, but both can't happen at the same time: this is the correct negation.