It is impossible, or absurd, to say that one can both not exist and affirm one’s non-existence; the one affirming non-existence would have to exist in order to affirm one’s non-existence. Likewise, we may argue for logic by the impossibility of the contrary or absurdity of the opposite; in denying logic one is affirming it.
By “contrary” here we simply mean the denial of whatever is in view. Contrary is being used in an informal and conversational way, and not in its philosophical sense. In the philosophical or logical sense contraries cannot both be true but they can both be false, whereas here we want to say that if the contrary of a position is false or at any rate impossible, then the original position must be true or necessary.