Negation
Negation, or denial, is the opposite of affirmation. It may be something that somebody does ("I deny what you have said") or the answer "No" to a question, but its full expression is generally a sentence. One sentence or statement may be the negation or denial of another, or we may call a statement simply a negation, or a negative statement, as opposed to an affirmative one, or affirmation. A negation in the last sense will contain some sign of negation, such as the "not" in "Grass is not pink" or "Not all leaves are green," the "no" in "No Christians are communists," or the phrase "it is not the case that" in "It is not the case that grass is pink." The negation of a sentence may simply be the same sentence with "it is not the case that" prefixed to it, or it may be some simpler form equivalent to this. For example, it might be said that "It is not the case that grass is pink" is negated or denied not only by "It is not the case that it is not the case that grass is pink" but also by the plain "Grass is pink" and that "If he has shut the door, it must have been open" is negated or denied by "He could have shut it even though it was already shut."
Contradictory negation, or contradiction, is the relation between statements that are exact opposites, in the sense that they can be neither true together nor false together—for example, "Some grass is brown" and "No grass is brown." Contrary negation, or contrariety, is the relation between extreme opposites (which may very well both be false)—for example, "No grass is brown" and "All grass is brown." Incompatibility is the relation between statements that cannot both be true, whether or not they stand at opposite ends of a scale ("This is black all over" is incompatible with "This is green all over" as well as with "This is white all over").
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