In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical argument, or proof by contradiction, is the form of argument that attempts to establish a claim by showing that following the logic of a contrary proposition or argument would lead to absurdity or contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof. This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. In mathematics, the technique is called proof by contradiction. In formal logic, this technique is captured by an inference rule for reductio ad absurdum. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. G. H. Hardy described proof by contradiction as "one of a mathematician's finest weapons", saying "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 of a reductio ad absurdum argument can take a range of forms, as can be seen in the following examples of refutation by contradiction:
The Earth cannot be flat; otherwise, since the Earth is assumed to be finite in extent, we would find people falling off the edge. There is no smallest positive rational number; if q {\displaystyle q} were the smallest positive rational, then q / 2 {\displaystyle q/2} would be a positive rational number that is smaller than q {\displaystyle q} because it equals half of q {\displaystyle q} , and it would also not be smaller than q {\displaystyle q} because q {\displaystyle q} is assumed to be the smallest positive rational. The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses (empirical evidence). The second example is a mathematical proof by contradiction (also known as an indirect proof), which argues that the denial of the premise would result in a logical contradiction ( q / 2 {\displaystyle q/2} is both smaller and not smaller than q {\displaystyle q} ). A mathematical proof employing proof by contradiction usually proceeds as follows:
The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of noncontradiction. Since assuming P to be false leads to a contradiction, it is concluded that P is in fact true. An important special case is the existence proof by contradiction: in order to demonstrate that an object with a given property exists, we derive a contradiction from the assumption that all objects satisfy the negation of the property.
Greek philosophy Reductio ad absurdum was used throughout Greek philosophy. The earliest example of a reductio argument can be found in a satirical poem attributed to Xenophanes of Colophon (c. 570 – c. 475 BCE). Criticizing Homer's attribution of human faults to the gods, Xenophanes states that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and ox bodies. The gods cannot have both forms, so this is a contradiction. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false. Greek mathematicians proved fundamental propositions using reductio ad absurdum. Euclid of Alexandria (mid-4th – mid-3rd centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples. The earlier dialogues of Plato (424–348 BCE), relating the discourses of Socrates, raised the use of reductio arguments to a formal dialectical method (elenchus), also called the Socratic method. Typically, Socrates' opponent would make what would seem to be an innocuous assertion. In response, Socrates, via a step-by-step train of reasoning, bringing in other background assumptions, would make the person admit that the assertion resulted in an absurd or contradictory conclusion, forcing him to abandon his assertion and adopt a position of aporia. Elenctic refutation depends on a dichotomous thesis, one that may be divided into exactly two mutually exclusive parts, only one of which may be true. Then Socrates goes on to demonstrate the contrary of the commonly accepted part using the law of non-contradiction. According to Gregory Vlastos, the method has the following steps:
Socrates' interlocutor asserts a thesis, for example, "Courage is endurance of the soul", which Socrates considers false and targets for refutation. Socrates secures his interlocutor's agreement to further premises, for example, "Courage is a fine thing" and "Ignorant endurance is not a fine thing". Socrates then argues, and the interlocutor agrees, that these further premises imply the contrary of the original thesis, in this case, it leads to: "courage is not endurance of the soul". Socrates then claims that he has shown that his interlocutor's thesis is false and that its negation is true. The technique was also a focus of the work of Aristotle (384–322 BCE), particularly in his Prior Analytics where he referred to it as demonstration to the impossible (Ancient Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. 'demonstration to the impossible', 62b). Another example of this technique is found in the sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap.
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