Proof by reductio ad absurdum is also known as proof by contradiction. It is one of the most powerful tools of reasoning at the disposal of logicians and mathematicians. Reductio proofs are used for conditionals, propositions of the form "if p then q." Most mathematical theorems take this form, for instance the Pythagorean Theorem: "If one of the angles in a triangle is a right angle, then the sum of the squares of the adjacent sides is equal to the square of the opposite side."

There are two key ideas in the reductio line of reasoning. First, a proposition must either be true or false, but not both. Therefore, if it cannot possibly be false, then it must be true. This is as simple as it is essential. Second, if a formally correct derivation from a set of assumptions leads...