Reductio ad absurdum

In logic, reductio ad absurdum—Latin for "reduction to absurdity"—is also called argumentum ad absurdum, apagogical argument, or proof by contradiction. This argument form works by showing that if you follow the logic of an opposing claim, you end up with an absurdity or contradiction, thereby supporting the original claim. While mathematicians use it freely, not every school of mathematical thought accepts this kind of nonconstructive proof.

The technique dates back to ancient Greek philosophy and has appeared throughout history in formal math, philosophical reasoning, and debate. In mathematics, it is known as proof by contradiction.

In formal logic, it is captured by a specific inference rule. More broadly, proof by contradiction covers any argument that establishes a statement by reaching a contradiction, even if the initial assumption is not the direct negation of the statement being proved. This general sense is also called indirect proof, proof by assuming the opposite, or reductio ad impossibile.

Mathematician G. H. Hardy called proof by contradiction "one of a mathematician's finest weapons," adding, "It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."

Examples

The "absurd" conclusion in a reductio ad absurdum argument can take various forms. For example: The Earth cannot be flat; if it were, and it is finite, people would fall off the edge. Another example: There is no smallest positive rational number. If q were the smallest, then q/2 would be a positive rational smaller than q (since it is half of q), yet it would also not be smaller because q is supposedly the smallest—a contradiction.

A typical mathematical proof by contradiction proceeds as follows: The proposition to prove is P. Assume P is false (¬P). Then show that ¬P leads to a falsehood, often by deriving two contradictory statements, Q and ¬Q, invoking the law of noncontradiction. Since assuming ¬P leads to a contradiction, P must be true. A special case is the existence proof by contradiction: to show an object with a given property exists, derive a contradiction from assuming all objects lack that property.

Greek philosophy

In Greek philosophy, reductio ad absurdum was widely used. The earliest known example appears in a satirical poem by Xenophanes of Colophon (c. 570–475 BCE).

Criticizing Homer for giving gods human faults, Xenophanes notes humans also think gods have human bodies. But if horses and oxen could draw, they would depict gods with horse and ox bodies. Since gods cannot have both forms, this is a contradiction, so attributing other human traits—like faults—to gods is also false.

Greek mathematicians like Euclid (mid-4th to mid-3rd centuries BCE) and Archimedes (c. 287–212 BCE) used reductio ad absurdum to prove fundamental propositions. Plato’s earlier dialogues, featuring Socrates, elevated the technique into a formal dialectical method called elenchus (or the Socratic method). Typically, Socrates’ opponent made an innocent-sounding assertion. Socrates then, step by step, using other background assumptions, forced the opponent to admit that the assertion led to an absurd or contradictory conclusion, making him abandon it and fall into aporia.

Elenctic refutation depends on a dichotomous thesis—one divisible into two mutually exclusive parts, only one true. Socrates then demonstrates the contrary of the commonly accepted part using the law of noncontradiction. According to Gregory Vlastos, the method has these steps: (1) Socrates’ interlocutor asserts a thesis (e.g., "Courage is endurance of the soul") that Socrates considers false.

(2) Socrates gets agreement to further premises (e.g., "Courage is a fine thing" and "Ignorant endurance is not a fine thing"). (3) Socrates argues that these premises imply the opposite of the original thesis (here, "Courage is not endurance of the soul"). (4) Socrates claims the thesis is false and its negation true.

Aristotle (384–322 BCE) also focused on this technique, especially in his Prior Analytics, where he called it "demonstration to the impossible" (62b). Another example is the sorites paradox: if 1,000,000 grains of sand form a heap, and removing one grain still leaves a heap, then a single grain (or even none) would also be a heap.

Buddhist philosophy

Much of Madhyamaka Buddhist philosophy centers on showing how reductio ad absurdum arguments undermine all positions.