|
In logic, proof by contradiction is a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction. Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and ''reductio ad impossibilem''. It is a particular kind of the more general form of argument known as ''reductio ad absurdum''. 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."〔G. H. Hardy, ''A Mathematician's Apology; Cambridge University Press, 1992. ISBN 9780521427067. ''(p. 94 ).〕 ==Principle== Proof by contradiction is based on the law of noncontradiction first stated by Aristotle. This states that an assertion or mathematical statement cannot be both true and false. That is, a proposition ''Q'' and its negation ~''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 absurdam''. If ''P'' is the proposition to be proved: #''P'' is assumed to be false, that is ~''P'' is true. #It is shown that ~''P'' implies two mutually contradictory assertions, ''Q'' and ~''Q''. #Since these cannot both be true, the assumption that ''P'' is false must be wrong, and ''P'' must be true. An alternate but slightly more confusing form derives a contradiction with the statement to be proved: #''P'' is assumed to be false. #It is shown that ~''P'' implies ''P''. #Since ''P'' and ~''P'' cannot both be true, the assumption must be wrong and ''P'' must be true. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Proof by contradiction」の詳細全文を読む スポンサード リンク
|