TheInfoList

OR:

In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...
, (
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
for "reduction to absurdity"), also known as (
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
for "argument to absurdity") or ''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 Ancient Greek philosophy arose in the 6th century BC, marking the end of the Greek Dark Ages. Greek philosophy continued throughout the Hellenistic period and the period in which Greece and most Greek-inhabited lands were part of the Roman Empire ...
and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate.

# Examples

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, since Earth assumed to be finite in extent, we would find people falling off the edge. * There is no smallest positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
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 (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).

# Greek philosophy

''Reductio ad absurdum'' was used throughout
Greek philosophy Ancient Greek philosophy arose in the 6th century BC, marking the end of the Greek Dark Ages. Greek philosophy continued throughout the Hellenistic period and the period in which Greece and most Greek-inhabited lands were part of the Roman Empire ...
. The earliest example of a argument can be found in a satirical poem attributed to
Xenophanes of Colophon Xenophanes of Colophon (; grc, Ξενοφάνης ὁ Κολοφώνιος ; c. 570 – c. 478 BC) was a Greek philosopher, theologian, poet, and critic of Homer from Ionia who travelled throughout the Greek-speaking world in early Classical ...
(c. 570 – c. 475 BCE). Criticizing
Homer Homer (; grc, Ὅμηρος , ''Hómēros'') (born ) was a Greek poet who is credited as the author of the '' Iliad'' and the '' Odyssey'', two epic poems that are foundational works of ancient Greek literature. Homer is considered one of ...
'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 Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
(mid-4th – mid-3rd centuries BCE) and
Archimedes of Syracuse Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...
(c. 287 – c. 212 BCE) are two very early examples. The earlier dialogues of
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
(424–348 BCE), relating the discourses of
Socrates Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no ...
, raised the use of arguments to a formal dialectical method (), also called the
Socratic method The Socratic method (also known as method of Elenchus, elenctic method, or Socratic debate) is a form of cooperative argumentative dialogue between individuals, based on asking and answering questions to stimulate critical thinking and to draw o ...
. 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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
(384–322 BCE), particularly in his '' Prior Analytics'' where he referred to it as ( grc-gre, ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, , demonstration to the impossible, 62b). The Pyrrhonists and the Academic Skeptics extensively used ''reductio ad absurdum'' arguments to refute the
dogma Dogma is a belief or set of beliefs that is accepted by the members of a group without being questioned or doubted. It may be in the form of an official system of principles or doctrines of a religion, such as Roman Catholicism, Judaism, Islam o ...
s of the other schools of
Hellenistic philosophy Hellenistic philosophy is a time-frame for Western philosophy and Ancient Greek philosophy corresponding to the Hellenistic period. It is purely external and encompasses disparate intellectual content. There is no single philosophical school or ...
.

# Buddhist philosophy

Buddhist philosophy Buddhist philosophy refers to the philosophical investigations and systems of inquiry that developed among various schools of Buddhism in India following the parinirvana of The Buddha and later spread throughout Asia. The Buddhist path com ...
centers on showing how various
essentialist Essentialism is the view that objects have a set of attributes that are necessary to their identity. In early Western thought, Plato's idealism held that all things have such an "essence"—an "idea" or "form". In ''Categories'', Aristotle si ...
ideas have absurd conclusions through ''reductio ad absurdum'' arguments (known as ''prasaṅga'' - "consequence" - in Sanskrit). In the
Mūlamadhyamakakārikā The ''Mūlamadhyamakakārikā'' ( sa, मूलमध्यमककारिका, ''Root Verses on the Middle Way''), abbreviated as ''MMK'', is the foundational text of the Madhyamaka school of Mahāyāna Buddhist philosophy. It was compose ...
, 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 The Abhidharma are ancient (third century BCE and later) Buddhist texts which contain detailed scholastic presentations of doctrinal material appearing in the Buddhist ''sutras''. It also refers to the scholastic method itself as well as the f ...
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'').Wasler, Joseph. ''Nagarjuna in Context.'' New York: Columibia University Press. 2005, pgs. 225-263.

Aristotle clarified the connection between contradiction and falsity in his
principle of non-contradiction In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...
, which states that a proposition cannot be both true and false. That is, a proposition $Q$ and its negation $\lnot Q$ (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 arguments in formal fields such as
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...
and mathematics.

*
Appeal to ridicule Appeal to ridicule (also called appeal to mockery, ''ad absurdo'', or the horse laugh) is an informal fallacy which presents an opponent's argument as absurd, ridiculous, or humorous, and therefore not worthy of serious consideration. Appeal to ...
* Argument from fallacy *
Contraposition In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a stateme ...
*
List of Latin phrases __NOTOC__ This is a list of Wikipedia articles of Latin phrases and their translation into English. ''To view all phrases on a single, lengthy document, see: List of Latin phrases (full)'' The list also is divided alphabetically into twenty page ...
* Mathematical proof * Prasangika *
Slippery slope A slippery slope argument (SSA), in logic, critical thinking, political rhetoric, and caselaw, is an argument in which a party asserts that a relatively small first step leads to a chain of related events culminating in some significant (usuall ...
* Strawman

# Sources

* Pasti, Mary. Reductio Ad Absurdum: An Exercise in the Study of Population Change. United States, Cornell University, Jan., 1977. * Daigle, Robert W.. The Reductio Ad Absurdum Argument Prior to Aristotle. N.p., San Jose State University, 1991.