Negation:
Denial; statement to
the effect that something is not the case. Note that this
involves an interesting ability of the human mind:
Representing to itself that
suchandsuch 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.
