Proof by contradiction

In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile.[1]

The 3rd step is based on the following possible truth value cases of a valid argument p → q.

This negation is what proof by contradiction attempts to show cannot be true.

Since when p(F), the implication is mathematically true regardless of the truth value of q, the negation of implication-elimination employed when using proof by contradiction challenges the implication from a state where:

it is mathematically indistinguishable from when p(F) AND ( q(T) XOR q(F) ), stating effectively "if P is false, the implication p → q is true and no claim can be made"

That is, if a negated assumed statement is shown to be possible via valid logic, then the assumed statement (before it was negated) is false. This fact is why proof by contradiction works.

An existence proof by contradiction assumes that some 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 nonconstructive proof as universally valid.

Proof by contradiction also depends on the law of the excluded middle, also first formulated by Aristotle. This states that either an assertion or its negation must be true

Intuitionist mathematicians do not accept the law of the excluded middle, and thus reject arbitrary proof by contradiction as a viable proof technique. However, they do accept the following variation, called "proof of negation".

A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational.[3] If it were rational, it would be expressible as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. But if a/b = 2, then a2 = 2b2. Therefore, a2 must be even, and because the square of an odd number is odd, that in turn implies that a is itself even — which means that b must be odd because a/b is in lowest terms.

On the other hand, 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 must be even, which means that b must be even as well.

So b is both odd and even, a contradiction. Therefore, the initial assumption—that 2 can be expressed as a fraction—must be false.[4]

The method of proof by contradiction has also been used to show that for any non-degenerate right triangle, the length of the hypotenuse is less than the sum of the lengths of the two remaining sides.[5] By letting c be the length of the hypotenuse and a and b be the lengths of the legs, one can also express the claim more succinctly as a + b > c. In which case, a proof by contradiction can then be made by appealing to the Pythagorean theorem.

First, the claim is negated to assume that a + b ≤ c. In which case, squaring both sides would yield that (a + b)2 ≤ c2, or equivalently, a2 + 2ab + b2 ≤ c2. A triangle is non-degenerate if each of its edges has positive length, so it may be assumed that both a and b are greater than 0. Therefore, a2 + b2 < a2 + 2ab + b2 ≤ c2, and the transitive relation may be reduced further to a2 + b2 < c2.

On the other hand, it is also known from the Pythagorean theorem that a2 + b2 = c2. This would result in a contradiction since strict inequality and equality are mutually exclusive. The contradiction means that it is impossible for both to be true and it is known that the Pythagorean theorem holds. It follows from there that the assumption a + b ≤ c must be false and hence a + b > c, proving the claim.

Consider the proposition, P: "there is no smallest rational number greater than 0". In a proof by contradiction, we start by assuming the opposite, ¬P: that there is a smallest rational number, say, r.

Now, r/2 is a rational number greater than 0 and smaller than r. But that contradicts the assumption that r was the smallest rational number (if "r is the smallest rational number" were Q, then one can infer from "r/2 is a rational number smaller than r" that ¬Q.) This contradiction shows that the original proposition, P, must be true. That is, that "there is no smallest rational number greater than 0".

For other examples, see proof that the square root of 2 is not rational (where indirect proofs different from the one above can be found) and Cantor's diagonal argument.

A curious logical consequence of the principle of non-contradiction is that a contradiction implies any statement; if a contradiction is accepted as true, any proposition (including its negation) can be proved from it.[10] This is known as the principle of explosion (Latin: ex falso quodlibet, "from a falsehood, anything [follows]", or ex contradictione sequitur quodlibet, "from a contradiction, anything follows"), or the principle of pseudo-scotus.

Thus a contradiction in a formal axiomatic system is disastrous; since any theorem can be proven true, it destroys the conventional meaning of truth and falsity.

The discovery of contradictions at the foundations of mathematics at the beginning of the 20th century, such as Russell's paradox, threatened the entire structure of mathematics due to the principle of explosion. This motivated a great deal of work during the 20th century to create consistent axiomatic systems to provide a logical underpinning for mathematics. This has also led a few philosophers such as Newton da Costa, Walter Carnielli and Graham Priest to reject the principle of non-contradiction, giving rise to theories such as paraconsistent logic and dialethism, which accepts that there exist statements that are both true and false.[11]

G. H. Hardy described proof by contradiction as "one of a mathematician's finest weapons", saying "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."[12]