Reductio ad absurdum (Latin for "reduction to the absurd") also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument and derives an absurd or ridiculous outcome, and then concludes that the original claim must have been wrong as it led to an absurd result. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along

It makes use of the law of non-contradiction — a statement cannot be both true and false. In some cases it may also make use of the law of excluded middle — a statement must be either true or false. This article uses forms of logical notation For a concise description of the symbols used in this notation see Table of logic symbols. The phrase is traceable back to the Greek ἡ εἰς ἄτοπον ἀπαγωγή (hē eis átopon apagōgḗ), meaning "reduction to the absurd", often used by Aristotle. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great.

In mathematics and formal logic, this refers specifically to an argument where a contradiction is derived from some assumption (thus showing that the assumption must be false). However, Reductio ad absurdum is also often used to describe any argument where a conclusion is derived in the belief that everyone (or at least those being argued against) will accept that it is false or absurd. This is a comparatively weak form of reductio, as the decision to reject the premise requires that the conclusion is accepted as being absurd. Although a formal contradiction is by definition absurd (unacceptable), a weak reductio ad absurdum argument can be rejected simply by accepting the purportedly absurd conclusion. Such arguments also risk degenerating into strawman arguments, an informal fallacy caused when an argument or theory is twisted by the opposing side to appear ridiculous. A straw man argument is an Informal fallacy based on misrepresentation of an opponent's position An Informal fallacy is an argument whose stated premises fail to support their proposed conclusion

Contents 1 Explanation 2 Examples 2.1 Cubing-the-cube puzzle 2.2 In mathematics 2.3 In mathematical logic 2.4 Humour 3 Notation 4 Quotations 5 References 6 Further reading

Explanation

In formal logic, reductio ad absurdum is used when a formal contradiction can be derived from a premise, allowing one to conclude that the premise is false. If a contradiction is derived from a set of premises, this shows that at least one of the premises is false; if there are several, other means must be used to determine which ones. Mathematical proofs are sometimes constructed using reductio ad absurdum, by first assuming the opposite of the theorem the presenter wishes to prove, then reasoning logically from that assumption until presented with a contradiction. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Upon reaching the contradiction, the assumption is disproved and therefore its opposite, due to the law of excluded middle, must be true. This article uses forms of logical notation For a concise description of the symbols used in this notation see Table of logic symbols. Such proofs in mathematics are sometimes called informal proofs, but are no less valid than a "formal" mathematical proof arrived at through reduction to equality.

There is a fairly common misconception that reductio ad absurdum simply denotes "a silly argument" and is itself a formal fallacy. In philosophy, a formal fallacy or a logical fallacy is a pattern of reasoning which is always wrong However, this is not correct; a properly constructed reductio constitutes a correct argument. When reductio ad absurdum is in error, it is because of a fallacy in the reasoning used to arrive at the contradiction, not the act of reduction itself.

Examples

A classic reductio proof from Greek mathematics is the proof that the square root of 2 is irrational. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 If it were rational, it could be expressed as a fraction a/b in lowest terms, where a and b are integers. But if a/b = √2, then a² = 2b². That implies a² is even. Since the square of an odd number is odd, that in turn implies that a is even. If a is even, then a² is a multiple of 4. If a² is a multiple of 4 and a² = 2b², then 2b² is a multiple of 4, and therefore b² is even. If b² is even then b is even. But now a and b are both even. Therefore the fraction a/b is not in lowest terms. That is a contradiction. Therefore the initial assumption—that √2 is rational—must be false.

Cubing-the-cube puzzle

A more recent use of a reductio argument is the proof that a cube cannot be cut into a finite number of smaller cubes with no two the same size. Consider the smallest cube on the bottom face; on each of its four sides, either a neighbouring cube or the border of the main cube is rising above it. This means that any larger cube will not fit on top of it (the "footprint" of such a cube is too large). Since different cubes aren't permitted to have the same sizes, only smaller cubes can be placed directly on top of it. But then the smallest of these would likewise be surrounded by larger cubes, so could only have smaller cubes directly on top of it. . . and so on, in an infinite regress, requiring an infinite number of cubes, which violates our conditions. An infinite regress in a series of propositions arises if the truth of proposition P 1 requires the support of proposition P 2 and for any proposition (This gives rise to a proof by induction that the cubing-the-cube puzzle is also unsolvable in dimensions higher than three. )

In mathematics

Say we wish to disprove proposition p. In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence The procedure is to show that assuming p leads to a logical contradiction. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield Thus, according to the law of non-contradiction, p must be false.

Say instead we wish to prove proposition p. We can proceed by assuming "not p" (i. e. that p is false), and show that it leads to a logical contradiction. Thus, according to the law of non-contradiction, "not p" must be false, and so, according to the law of the excluded middle, p is true. This article uses forms of logical notation For a concise description of the symbols used in this notation see Table of logic symbols.

In symbols:

To disprove p: one uses the tautology (p → (R ∧ ¬R)) → ¬p, where R is any proposition and the ∧ symbol is taken to mean "and". In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation Assuming p, one proves R and ¬R, and concludes from this that p → (R ∧ ¬R). This and the tautology together imply ¬p.

To prove p: one uses the tautology (¬p → (R ∧ ¬R)) → p where R is any proposition. Assuming ¬p, one proves R and ¬R, and concludes from this that ¬p → (R ∧ ¬R). This and the tautology together imply p.

For a simple example of the first kind, consider the proposition "there is no smallest rational number greater than 0". In a reductio ad absurdum argument, we would start by assuming the opposite: that there is a smallest rational number, say, r₀.

Now let . Then x is a rational number, and it's greater than 0; and x is smaller than r₀. (In the above symbolic argument, "x is the smallest rational number" would be R and "r (which is different from x) is the smallest rational number" would be ¬R. ) But that contradicts our initial assumption that r₀ was the smallest rational number. So we can conclude that the original proposition must be true — "there is no smallest rational number greater than 0".

It is not uncommon to use this first type of argument with propositions such as the one above, concerning the non-existence of some mathematical object. One assumes that such an object exists, and then proves that this would lead to a contradiction; thus, such an object does not exist. For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891

On the other hand, it is also common to use arguments of the second type concerning the existence of some mathematical object. One assumes that the object doesn't exist, and then proves that this would lead to a contradiction; thus, such an object must exist. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of argument as universally valid. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. See further Nonconstructive proof.

In mathematical logic

In mathematical logic, the reductio ad absurdum is represented as:

if

then

or

if

then

In the above, p is the proposition we wish to prove or disprove; and S is a set of statements which are given as true — these could be, for example, the axioms of the theory we are working in, or earlier theorems we can build upon. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. We consider p, or the negation of p, in addition to S; if this leads to a logical contradiction F, then we can conclude that the statements in S lead to the negation of p, or p itself, respectively.

Note that the set-theoretic union, in some contexts closely related to logical disjunction (or), is used here for sets of statements in such a way that it is more related to logical conjunction (and). In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of

Humour

The often humorous outcome of extending the simplification of a flawed statement to ridiculous proportions with the aim of criticising the result is frequently utilised in forms of humour. Humour or humor (see spelling differences) is the tendency of particular cognitive experiences to provoke Laughter and provide Amusement In fiction, seemingly simple and innocuous actions that are extended beyond reasonable circumstance to chaotic outcomes, typically by use of stereotype and literal interpretation, can also be categorised as reductio ad absurdum^[1]. See farce. A farce is a Comedy written for the stage or film which aims to Entertain the audience by means of unlikely extravagant and improbable situations disguise and mistaken For example, this technique is used and then subsequently analysed in The Big Bang Theory^[2]. The Big Bang Theory is an American Situation comedy created and executive produced by Chuck Lorre and Bill Prady, which premiered Wherein, Sheldon's identification of the technique reduces Leonard's joke to its fundamental nature, and his subsequent indepth explanation of it stretches this simplicity to the degree by which it is no longer funny. This effectively applies reductio ad absurdum to the topic of reductio ad absurdum, which in itself becomes the next joke.

Notation

Proof by reductio ad absurdum often end "Contradiction!", or "Which is a contradiction. ". Isaac Barrow and Baermann used the notation Q. Isaac Barrow (October 1630 &ndash May 4, 1677) was an English scholar and Mathematician who is generally given credit for his early role E. A. , for "quod est absurdum" ("which is absurd"), along the lines of Q.E.D., but this notation is rarely used today^[3]. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley^[4]

Quotations

In the words of G. H. Hardy (A Mathematician's Apology), "Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 A Mathematician's Apology is a 1940 essay by British mathematician G 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. Chess is a recreational and competitive Game played between two players. A gambit is a Chess opening in which the first player risks or sacrifices material usually a pawn, with the hope of achieving a resulting advantageous "

In the first paragraph of the Quentin Section (Part 2: June Second, 1910) of William Faulkner's The Sound and the Fury, Quentin's Father, Mr. William Faulkner (born William Cuthbert Falkner) ( September 25, 1897 – July 6, 1962) was an American Author The Sound and the Fury is one of the most celebrated novels of the Twentieth century, written by American author William Faulkner, which makes use Compson, gives his son a watch that has been in the family for many generations. His father explains, "It [The Watch] was Grandfather's and when Father gave it to me he said I give you the mausoleum of all hope and desire; it's rather excruciating-ly apt that you will use it to gain the reducto absurdum of all human experience which can fit your individual needs no better that it fitted his or his father's". This example represents a corruption of the Latin phrase Reductio ad absurdum.

References

Further reading

J. Franklin and A. Daoud, Proof in Mathematics: An Introduction, Quakers Hill Press, 1996, ch. 6

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