Proof by contradiction

Reductio ad absurdum (Latin: "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, derives an absurd or ridiculous outcome, and then concludes that the original assumption must have been wrong as it led to an absurd result.

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. 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.

Reductio ad absurdum is also often used to describe an 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. However, this is a 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 can also commonly incorporate the appeal to ridicule, an informal fallacy caused when an argument or theory is twisted by the opposing side to appear ridiculous.

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, but other means must be used to determine which one. 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. Upon reaching the contradiction, the assumption is disproved and therefore its opposite, due to the law of excluded middle, must be true. 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. 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. It it were rational, it could be expressed as a fraction a/b in lowest terms, where a and b are positive 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. (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. The procedure is to show that assuming p leads to a logical contradiction. 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.

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". 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, $r_{0}$.

Now let $x={\frac {r_{0}}{2}}$. Then x is a rational number, and it's greater than 0; and x is smaller than $r_{0}$. (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 $r_{0}$ 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.

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. In schools such as intuitionism, the law of the excluded middle is not taken as true. From this way of thinking, there is a very significant difference between proving that something exists by showing that it would be absurd if it did not; and proving that something exists by constructing an actual example of such an object. These schools will still, however, accept arguments of the first kind concerning non-existence. A famous example of the second kind is Brouwer's own proof of his fixed point theorem, which shows that it is impossible for certain fixed points not to exist, without being able to show how to obtain one in the general case.

To form a valid proof, it must be demonstrated that the assumption being made for the sake of argument implies a property that is actually false in the mathematical system being used. The danger here is the logical fallacy of argument from lack of imagination, where it is proven that the assumption implies a property which looks false, but is not really proven to be false. Traditional (but incorrect!) examples of this fallacy include false proofs of Euclid's fifth postulate (a.k.a. the parallel postulate) from the other postulates.

The reason these examples are not really examples of this fallacy is that the notion of proof was different in the 19th century; (Euclidean) geometry was seen as being a 'true' reflection of physical reality, and so deducing a contradiction by concluding something physically implausible (like the angles of a triangle not being 180 degrees) was acceptable. Doubts about the nature of the geometry of the universe led mathematicians such as Bolyai, Gauss, Lobachevsky, Riemann, among others, to question and clarify what actually constituted 'geometry'. Out of these men's work, resulted Non-Euclidean geometry. For a further exposition of these misunderstandings see Morris Kline, Mathematical Thought: from Ancient to Modern Times.

In mathematical logic

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

if

$S\cup \{p\}\vdash F$

then

$S\vdash \neg p.$

or

if

$S\cup \{\neg p\}\vdash F$

then

$S\vdash p.$

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. 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).

Notation

Proof by reductio ad absurdum often end "Contradiction!", or "Which is a contradiction.". Isaac Barrow and Baermann used the notation Q.E.A., for "quod est absurdum" ("which is absurd"), along the lines of Q.E.D., but this notation is rarely used today. A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley

Quotes

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. 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."

See also

• Reductio ad Hitlerum