Please note: This is a companion version & not the original book. Sample Book Insights: #1 The law of the excluded middle requires that a thing must either possess a given attribute or must not possess it. A thing must be one way or the other; there is no middle ground. However, we frequently see this principle misused. #2 The law of noncontradiction states that a thing cannot be and not be at the same time. When you try to believe that a statement is both true and not true at the same time, you are guilty of self-contradiction. #3 The method of reductio ad impossibile, or argument by contradiction, was first used by Zeno of Elea in the fourth century B. C. It was a form of argument used to confound the other side, and it went like this: If P is true, then Q is true. But since it is impossible for Q to be true and not true at the same time, it is therefore impossible for P to be true. #4 The method of proof by contradiction was used by the Stoics to validate their rules of logic. Euclid employed this technique as well, and it was used to prove the theorem in geometry about the congruence of alternate interior angles formed by a straight line falling on parallel lines.