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. Principle. Proof by contradiction is based on the law of noncontradiction as first formalized as a metaphysical principle by Aristotle. Noncontradiction is also a theorem in propositional logic. This states that an assertion or mathematical statement cannot be both true and false. That  is, a proposition "Q" and its negation formula_1"Q" ("not-"Q"") cannot both be true. In a proof by contradiction, it is shown that the denial of the statement being proved results in such a contradiction. It has the form of a "reductio ad absurdum" argument, and usually proceeds as follows: The 3rd step is based on the following possible truth value cases of a valid argument p → q. It tells that if a false statement is reached via a valid logic from an assumed statement, then the assumed statement is a false statement. This fact is used in proof  by contradiction. Proof by contradiction is formulated as formula_6, where formula_7 is a logical contradiction or a "false" statement (a statement which truth value is "false"). If formula_7 is reached from formula_1"P" via a valid logic, then formula_10 is proved as true so p is proved as true. An alternate form of proof by contradiction derives a contradiction with the statement to be proved by showing that formula_1"P" implies "P". This is a contradiction so the assumption formula_1"P" must be false, equivalently "P" as true. This is formulated as formula_13. An existence proof by contradiction assumes that some object does n'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. Law of the excluded middle. 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 That is, there is no other truth value besides "true" and "false" that a proposition can take. Combined with the principle of noncontradiction, this  means that exactly one of formula_15 and formula_16 is true. In proof by contradiction, this permits the conclusion that since the possibility of formula_16 has been excluded, formula_15 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". If the proposition to be proved has itself the form of a negation formula_16, a proof by contradiction can start by assuming that formula_15 is true and derive a contradiction from that assumption. It then  follows that the assumption was wrong, so formula_15 is false. In such cases, the proof does not need to appeal to the law of the excluded middle. An example is the proof of irrationality of the square root of 2 given below. Relationship with other proof techniques. Proof by contradiction is closely related to proof by contrapositive, and the two are sometimes confused, though they are distinct methods. The main distinction is that a proof by contrapositive applies only to statements formula_15 that can be written in the form formula_23 (i.e., implications), whereas the technique of proof by contradiction applies  to statements formula_15 of any form: If the statement to be proven is an implication formula_28, then the differences between direct proof, proof by contrapositive, and proof by contradiction can be outlined as follows: Examples. Irrationality of the square root of 2. A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. 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" = , then "a" = 2"b". Therefore, "a" 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 "a" is a multiple of 4. If "a" is a multiple of 4 and "a" = 2"b", then 2"b" is a multiple of 4, and therefore "b" 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 can be expressed as  a fraction—must be false. The length of the hypotenuse. 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. 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") ≤ "c", or equivalently, "a" + 2"ab" + "b" ≤ "c". 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, "a" + "b" < "a" + 2"ab" + "b" ≤ "c", and the transitive relation may be reduced further to "a" + "b" < "c". On the other hand, it is also known from the Pythagorean theorem that "a  + b" = "c". 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. No least positive rational number. 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". Other. 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. Notation. Proofs by contradiction sometimes end with the word "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. Others sometimes used include a pair of opposing arrows (as formula_37 or formula_38), struck-out arrows (formula_39), a stylized form of hash (such as U+2A33: ⨳), or the "reference mark" (U+203B: ※), or formula_40. Principle of  explosion. 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. This is known as the principle of explosion (, "from a falsehood, anything [follows]", or "", "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. Hardy's view. 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."