Note that this involves an interesting ability of the human mind: Representing to itself that such-and-such is not so, and saying and communicating this in words.

And evidently statements to the effect that something is not the case are as intuitively true or not as statements to the effect that something is the case, and indeed from the latter one can obtain the former by putting a "not" at a  grammatically appropriate place.

There are several subtle problems involved in the concept of negation, and indeed it is also involved in several subtle problems, like Russell's Paradox and other paradoxes.

First, how to treat double negation.

It seems to many people that it makes intuitive sense that if "p is not true" is not true, that then it follows that "p is true", but to intuitionists it has seemed otherwise.

Stated in terms of propositional logic, it concerns the question whether the formula (~~P --> P) is valid. In standard propositional logic it is valid, and is inferred from the valid formula (~P V P), but then intuitionists also reject its validity. Part of their reason for doing so is that of some propositions, including mathematical ones, one just does not know that it is true and one just does not know that it is false.

Second, the effects of the place of negation.

In one analysis, the problem just mentioned of double negation is an instance of this, or related to it. The problem is whether e.g. "it is not true that x is P" and "it is true that x is not P" are true in just the same conditions. In the former case quoted, the negation i.e. the term "not" is said to be external (to the formula "x is P") and in the latter case, the negation is said to be internal (idem).

And it would seem that in general if the latter is true i.e. if it is true that x is not P, that then it is not true that x is P, but not always conversely, as in the previous paragraph.

Thus - as Aristotle already noted, about future contingents - it would seem as if it is not true that tomorrow there is a seabattle, it need not necessarily be the case that it is true that tomorrow there is not a seabattle.

One way to keep external and internal negations apart terminologically, is to call the former denial and the latter negation.