Proof by contradiction

Reductio ad absurdum (Latin for "reduction to the absurd", traceable back to the Greek ἡ εις άτοπον απαγωγη (hi eis átopon apagogi), "reduction to the impossible", often used by Aristotle), also known as an apagogical argument or reductio ad impossibile, is a type of logical argument where one assumes a claim for the sake of argument, arrives at an absurd result, and then concludes that the original assumption must have been wrong, since it led to this absurd result. This is also known as proof by contradiction. 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 which cannot be false, must then be true.

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.

The term reductio ad absurdum is also often used for arguments where a conclusion is derived which it is thought that everyone will agree to be false or absurd, or which at least certain persons being argued against will agree is false or absurd. However this is a weak form of reductio, since the final decision to reject the original premise is contingent upon whether the conclusion derived from it really is agreed to be absurd. While a formal contradiction is by definition absurd (unacceptable), someone could reject a weak reductio ad absurdum argument by simply replying that they accept the purportedly absurd conclusion.

In philosophy and everyday discussion

The following dialogue is an example of reductio ad absurdum:

A — All beliefs are of equal validity and cannot be denied.

B — If that's the case, then C is correct in his belief, even though C believes something that is considered to be wrong by most people, such as that the Earth is flat.

A — True.

B — Then some beliefs can be denied.

The following is a trickier reduction, but one which is stronger from the philosophical point of view because it does not rely on A's accepting that C's opinion is wrong:

A — You should respect C's belief, for all beliefs are of equal validity and cannot be denied.

B —

1. I deny that belief of yours and believe it to be invalid.
2. According to your statement, this belief of mine (1) is valid, like all other beliefs.
3. However, your statement also contradicts and invalidates mine, being the exact opposite of it.
4. The conclusions of 2 and 3 are incompatible and contradictory, so your statement is logically absurd.

In each case, B has used a reduction to the absurd to argue his or her point. Note: the above example is also an example of the liar paradox.

A very familiar example:

Mother — Why did you start smoking?

Son — All my friends were doing it.

Mother — You're saying that if all your friends jumped off a cliff, you would do that too?

Here, the mother refutes the son's justification by showing the absurdity of its consequences.

Sometimes, people attempt to construct a valid reductio ad absurdum, but it degenerates into a fallacious straw man argument, which may involve elements of a slippery slope argument.

For example:

A — I don't think the police should arrest teenagers for soft drug possession.

B — So, you are basically arguing the police should not enforce the law and we should live in a society of violent chaos.

Among stupid people, there is a misconception that reductio ad absurdum just means "a silly argument".

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. Assuming p, one proves R and ~R, together they 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, together they 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 x = r₀/2. Then x is a rational number, and it's greater than 0; and x is smaller than r₀. But that is absurd — it 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.

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.

It is important to note that 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, 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).

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