object with the desired property exists, thereby deriving a contradiction. Such a proof is again a refutation by contradiction. A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that is even smaller than q and still positive. Russell's paradox Russell's paradox, stated set-theoretically as "there is no set whose elements are precisely those sets that do not contain themselves", is a negated statement whose usual proof is a refutation by contradiction. 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 or ), struck-out arrows (), a stylized form of hash (such as U+2A33: ⨳), or the "reference mark" (U+203B: ※), or . 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." See also Law of excluded middle Law of noncontradiction Proof by exhaustion Proof by infinite descent Modus tollens References Further reading and external links Proof by Contradiction from Larry W. Cusick's How To Write Proofs Reductio ad Absurdum Internet Encyclopedia of Philosophy; ISSN 2161-0002 Mathematical proofs Methods of proof Theorems in propositional logic