Reductio ad absurdum
Argument form that proves a claim by showing its opposite leads to absurdity.
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Reductio ad absurdum, Latin for "reduction to absurdity," is a logical argument that proves a claim by showing that the opposite leads to nonsense or a contradiction. It is also called argumentum ad absurdum, apagogical argument, or proof by contradiction. While mathematicians use it freely, not all schools of mathematical thought accept this kind of nonconstructive proof.
The technique dates back to Ancient Greek philosophy and has been a staple of formal math, philosophy, and debate ever since. In mathematics, it is known as proof by contradiction, and in formal logic, it is captured by a specific inference rule. More broadly, proof by contradiction—also called indirect proof, proof by assuming the opposite, or reductio ad impossibile—covers any argument that establishes a statement by arriving at a contradiction, even if the initial assumption is not the exact negation of the statement being proved. Mathematician G. H. Hardy called it "one of a mathematician's finest weapons," comparing it favorably to a chess gambit: a chess player sacrifices a pawn or piece, but a mathematician offers the game itself.
The "absurd" conclusion can take various forms. For example, the Earth cannot be flat because, if it were finite and flat, 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 smaller positive rational—a contradiction, since *q* is both the smallest and not the smallest. The first example relies on empirical evidence; the second is a mathematical proof by contradiction, deriving a logical contradiction.
A typical mathematical proof by contradiction works as follows: the proposition to be proved is *P*. Assume *P* is false (i.e., assume ¬*P*). Show that ¬*P* leads to a falsehood, usually by deriving two mutually contradictory assertions, *Q* and ¬*Q*, appealing to the law of noncontradiction. Since assuming *P* false 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 the assumption that all objects lack that property.
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 attributing human faults to the gods, Xenophane
- field
- Logic, mathematics, philosophy
- known_for
- Proof by contradiction; establishing claims by deriving absurdity from contrary propositions
- earliest_known_use
- Satirical poem attributed to Xenophanes of Colophon (c. 570 – c. 475 BCE)
- notable_early_practitioners
- Euclid of Alexandria, Archimedes of Syracuse, Plato, Aristotle
- also_called
- Argumentum ad absurdum, apagogical argument, proof by contradiction, indirect proof, proof by assuming the opposite, reductio ad impossibile
Lore & Background
The earliest example of a reductio argument appears in a satirical poem by Xenophanes of Colophon, who criticized Homer's attribution of human faults to the gods. Xenophanes argued that if horses and oxen could draw, they would depict gods with horse and ox bodies, leading to a contradiction about divine forms, thus showing such attributions false. Greek mathematicians like Euclid and Archimedes used reductio ad absurdum to prove fundamental propositions. In Plato's dialogues, Socrates employed the method as a formal dialectical technique (elenchus), forcing opponents to abandon assertions by revealing they led to contradictions. Aristotle referred to it as 'demonstration to the impossible' in his Prior Analytics.
Reader's Guide
Reductio ad absurdum is a foundational argument form in logic and mathematics, valued for its power to establish truths by exposing contradictions in opposing views. In mathematics, it is called proof by contradiction, and G. H. Hardy described it as 'one of a mathematician's finest weapons,' noting that a mathematician offers the game itself, unlike a chess player who sacrifices a pawn. The method proceeds by assuming the negation of a proposition, deriving a contradiction (such as Q and not-Q), and concluding the original proposition is true. It has been used across cultures, including in Madhyamaka Buddhist philosophy by Nāgārjuna, who employed prasaṅga (consequence) arguments to show that essentialist theories lead to absurdities. Contemporary philosophers like Lewis White Beck and Robert L. Holmes have also utilized reductio arguments in their works. The technique relies on Aristotle's principle of non-contradiction, which states that a proposition and its negation cannot both be true. While widely accepted in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof.
Did You Know?
- The earliest known reductio ad absurdum argument is found in a satirical poem by Xenophanes of Colophon, criticizing Homer's portrayal of gods.
- In Buddhist Madhyamaka philosophy, reductio arguments (prasaṅga) were used by Nāgārjuna to show that theories of essential nature lead to absurd consequences.
- G. H. Hardy called proof by contradiction 'one of a mathematician's finest weapons,' saying it is a finer gambit than any chess gambit.
- The sorites paradox, which questions when a heap of sand ceases to be a heap, is an example of reductio ad absurdum reasoning.
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