Reductio ad Absurdum

A little more on this type of argumentation. Stephen F. Barker in The Elements of Logic points out that the type of argumentation which contains a conditional premise and which is known as reductio ad absurdum ("reduction to the absurd" or "reduction to absurdity") can be illustrated as follows. Assuming p is true,

If p, then not p (e.g., If there is a largest integer (a positive number), then there is not a largest integer.)

Conclusion: Therefore, not p (e.g., Therefore, it is not the case that there is a largest integer.)

The reasoning above is valid because the antecedent of the proposition "If p then not p" is false. By reducing the antecedent of the proposition "If p then not p" to absurdity, it shows that the antecedent is false. For if the antecedent were true, then the consequent of the proposition "If p, then not p" would be inconsistent with the propositional antecedent. So the antecedent must be false.