In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction. 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.
The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show:
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 (also known as an indirect proof), which argues that the denial of the premise would result in a logical contradiction (there is a "smallest" number and yet there is a number smaller than it).
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.
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 (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. "demonstration to the impossible", 62b).  The Pyrrhonists and the Academic Skeptics extensively used reductio ad absurdum arguments to refute the dogmas of the other schools of Hellenistic philosophy.
Much of Madhyamaka Buddhist philosophy centers on showing how various essentialist ideas have absurd conclusions through reductio ad absurdum arguments (known as prasanga in Sanskrit). In the Mūlamadhyamakakārikā, Nāgārjuna's reductio ad absurdum arguments are used to show that any theory of substance or essence was unsustainable and therefore, phenomena (dharmas) such as change, causality, and sense perception were empty (sunya) of any essential existence. Nāgārjuna's main goal is often seen by scholars as refuting the essentialism of certain Buddhist Abhidharma schools (mainly Vaibhasika) which posited theories of svabhava (essential nature) and also the Hindu Nyāya and Vaiśeṣika schools which posited a theory of ontological substances (dravyatas).
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 (not-Q) 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.