In logic, reductio ad absurdum (Latin for '"reduction to absurdity"'), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical arguments, negation introduction or the appeal to extremes, is a form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.[1][2] It can be used to disprove a statement by showing that it would inevitably lead to a ridiculous, absurd, or impractical conclusion,[3] or to prove a statement by showing that if it were false, then the result would be absurd or impossible.[4][5] Traced back to classical Greek philosophy in Aristotle's Prior Analytics[5] (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. 'demonstration to the impossible', 62b), this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate.[6]

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 (also known as an indirect proof[7]), 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).[8]

In reasoning to argue a claim, a fallacy is reasoning that is evaluated as logically incorrect and that undermines the logical validity of the argument and permits its recognition as unsound. Regardless of their soundness, all registers and manners of speech can demonstrate fallacies.

Because of their variety of structure and application, fallacies are challenging to classify so as to satisfy all practitioners. Fallacies can be classified strictly by either their structure or content, such as classifying them as formal fallacies or informal fallacies, respectively. The classification of informal fallacies may be subdivided into categories such as linguistic, relevance through omission, relevance through intrusion, and relevance through presumption.[1] On the other hand, fallacies may be classified by the process by which they occur, such as material fallacies (content), verbal fallacies (linguistic), and again formal fallacies (error in inference). In turn, material fallacies may be placed into the more general category of informal fallacies, while formal fallacies may be clearly placed into the more precise category of logical (deductive) fallacies.[clarification needed] Yet, verbal fallacies may be placed into either informal or deductive classifications; compare equivocation which is a word or phrase based ambiguity (e.g., "he is mad", which may refer to either him being angry or clinically insane) to the fallacy of composition which is premise and inference based ambiguity (e.g., "this must be a good basketball team because each of its members is an outstanding player").[2]

The conscious or habitual use of fallacies as rhetorical devices is prevalent in the desire to persuade when the focus is more on communication and eliciting common agreement rather than on the correctness of the reasoning. The effective use of a fallacy by an orator may be considered clever, but by the same token, the reasoning of that orator should be recognized as unsound, and thus the orator's claim, supported by an unsound argument, will be regarded as unfounded and dismissed.[3]

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).[9] 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 oxen 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 (mid-3rd – mid-4th centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples.[10]

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.[11] 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.[7] The technique was also a focus of the work of Aristotle (384–322 BCE). [5] 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 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 used reductio ad absurdum arguments 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).[12]

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.[13][14] That is, a proposition {\displaystyle Q}Q and its negation {\displaystyle \lnot Q}{\displaystyle \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,[7] has formed the basis of reductio ad absurdum arguments in formal fields such as logic and mathematics.

Logical fallacy

In common speech,[2] a reductio ad absurdum is also often deployed in a manner which results in a logical fallacy, in a similar manner to the slippery slope fallacy.[7] The fallacious nature of this deployment of the argument lies in the assumption that in order for something to be true, it must be true in all circumstances. For example, the assertion that if a reduction in working hours results in an increase in productivity[15] then working no hours would result in the highest productivity. The fallacious nature of this argument lies in the assumption that the relationship between working hours and productivity is entirely linear.

In rhetoric, antanaclasis (/æntəˈnækləsɪs, ˌæntænəˈklæsɪs/; from the Greek: ἀντανάκλασις, antanáklasis, meaning "reflection",[1] from ἀντί anti, "against", ἀνά ana, "up" and κλάσις klásis "breaking") is the literary trope in which a single word or phrase is repeated, but in two different senses.[2] Antanaclasis is a common type of pun, and like other kinds of pun, it is often found in slogans.[3]

>>8507835

>In rhetoric, antanaclasis (/æntəˈnækləsɪs, ˌæntænəˈklæsɪs/; from the Greek: ἀντανάκλασις, antanáklasis, meaning "reflection",[1] from ἀντί anti, "against", ἀνά ana, "up" and κλάσις klásis "breaking") is the literary trope in which a single word or phrase is repeated, but in two different senses.[2] Antanaclasis is a common type of pun, and like other kinds of pun, it is often found in slogans.[3]

Examples

Your argument is sound, nothing but sound. — Benjamin Franklin. The word sound in the first instance means "solid" or "reasonable". The second instance of sound means "noise".[4][5]

Although we're apart, you're still a part of me. — Lyrics from "Blueberry Hill" by Fats Domino.[6]

Time isn't wasted, when you're getting wasted. — Lyrics from "I Love College (song)" by Asher Roth.

“In Genua, someone set out to make dreams come true… Remember some of your dreams?” – Sir Terry Pratchett.[7] The first usage of dreams refers to aspirations or desires, while the second refers to literal dreams.

Shakespeare

Put out the light, then put out the light. — From Othello. Othello utters these words to himself as he enters Desdemona's chamber while she sleeps, intending to murder her. The first instance of put the light out means he will quench the candle, and the second instance means he will end the life of Desdemona.[8]

I will dissemble myself in't; and I would I were the first that ever dissembled in such a gown. — In Twelfth Night, the fool Feste, where dissemble changes from "disguise" to "act hypocritically".[9][10]

Whoever hath her wish, thou hast thy Will

And Will to boot, and Will in overplus… — Shakespeare's Sonnet 135. The speaker is named Will, but the woman he is addressing has another lover who is also named Will. In this sonnet, the word will is used thirteen times, meaning "William", "sexual desire", "penis", or "vagina", depending on the context (and it usually means more than one of these things at once).[11][12]

Witticisms

"Time flies like an arrow; fruit flies like a banana" is an example of a garden path sentence – the first half of the sentence misleads the reader into parsing the second half incorrectly. The exact origin of the phrase is unknown, but differing versions of it have appeared in print since the 1960s.

Benjamin Franklin, at the signing of the Declaration of Independence, is reported to have said: "We must, indeed, all hang together, or assuredly we shall all hang separately".[13] However, the phrase has also been attributed to Richard Penn in Alexander Graydon's Memoirs of a Life,[14] and appeared in Frederic Reynolds' play Life, first published in 1801.[15]

In an essay entitled "The Literati of New York City", Edgar Allan Poe wrote of George B. Cheever: "He is much better known, however, as the editor of The Commonplace Book of American Poetry, a work which has at least the merit of not belying its title, and is exceedingly commonplace".[16][17]

The American football coach Vince Lombardi once told his team: "If you aren't fired with enthusiasm, you will be fired, with enthusiasm".[18]

Advertising

The long cigarette that's long on flavor". — Pall Mall cigarettes[19]

We make the traveler's lot a lot easier. — Overseas National Airways[20]

Latin literature

The Roman poet Lucretius in De rerum natura Book 3 line 365 observes that we sometimes find ourselves temporarily blinded by bright objects because "lumina luminibus quia nobis praepediuntur" (because our eyes are impeded by the lights), taking advantage of the fact that in Latin the same word can mean both "eye" and "light".