In logic, reductio ad absurdum, also known as argumentum ad absurdum, apagogical arguments, negation introduction or the appeal to extremes, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction. It can be used to disprove a statement by showing that it would inevitably lead to a ridiculous, absurd, or impractical conclusion, or to prove a statement by showing that if it were false, then the result would be absurd or impossible. Traced back to classical Greek philosophy in Aristotle's Prior Analytics, this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show:
The Earth cannot be flat; otherwise, we would find people falling off the edge.
There is no smallest positive rational number because, if there were, then it could be divided by two to get a smaller one.
The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. The second example is a mathematical proof by contradiction, which argues that the denial of the premise would result in a logical contradiction.
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. 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 utilizing reductio ad absurdum. Euclid of Alexandria and Archimedes of Syracuse are two very early examples. The earlier dialogues of Plato, relating the discourses of Socrates, raised the use of reductio arguments to a formal dialectical method, 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. The technique was also a focus of the work of Aristotle. The Pyrrhonists and the Academic Skeptics extensively used reductio ad absurdum arguments to refute the dogmas of the other schools of Hellenistic philosophy.
Buddhist Philosophy
Much of MadhyamakaBuddhist philosophy centers on showing how various essentialist ideas have absurd conclusions through reductio ad absurdum arguments. In the MūlamadhyamakakārikāNāgārjunareductio ad absurdum arguments are used to show that any theory ofsubstance or essence was unsustainable and therefore, phenomena such as change, causality, and sense perception were empty of any essential existence. Nāgārjuna's main goal is often seen by scholars as refuting the essentialism of certain Buddhist Abhidharma schools which posited theories of svabhava and also the HinduNyāya and Vaiśeṣika schools which posited a theory ofontological substances.
Principle of non-contradiction
Aristotle clarified the connection between contradiction and falsity in his principle of non-contradiction, which states that a proposition cannot be both true and false. That is, a proposition and its negation cannot both be true. Therefore, if a proposition and its negation can both be derived logically from a premise, it can be concluded that the premise is false. This technique, known as indirect proof or proof by contradiction, has formed the basis of reductio ad absurdum arguments in formal fields such as logic and mathematics.
