View source for Reductio Ad Absurdum

'''''Reducto ad absurdum''''' is [[Latin]] for "reduction to the absurd."  It is a form of argument that seeks to disprove a proposition by showing that that proposition inevitably leads to an absurdity, or of proving a proposition by assuming its negation and then showing that that assumption inevitably leads to an absurdity.<ref>[http://www.thefreedictionary.com/reductio+ad+absurdum Reductio ad absurdum]</ref>  One use is in [[mathematics]], as will be described in greater detail below.  ''Reductio ad absurdum'' should not be confused with the [[straw man fallacy]] or the [[slippery slope]] fallacy.

==In Mathematics==
''Reductio ad absurdum,'' also called "proof by contradiction" or "proof by assuming the opposite," is a method of [[mathematical proof]].  It involves assuming the opposite of what one is trying to [[prove]], and showing that this would lead to a [[contradiction]]. It works by the [[law of the excluded middle]]. The proof typically follows this structure:

#Create an initial assumption
#Follow a series of axiomatically valid steps
#Reach a contradiction
#Therefore, the initial assumption is incorrect

===Proof of Proposition ''P'' by Contradiction===
#Suppose ~P.
#...
#Therefore, Q.
#...
#Therefore, ~Q
#Hence, Q and ~Q, a contradiction
#Thus, P

===Example===
An example of this is [[Euclid]]'s proof of the [[infinitude]] of the [[Prime number|primes]]:

#Assume there are finitely many primes
#Take the product of all primes and call it ''N''. Since ''N''+1 is not in our finite set of primes, it must be composite
#By the [[fundamental theorem of arithmetic]], ''N''+1 has a prime factorization. But ''N+1'' is not divisible by any of the previous primes
#Since ''N''+1 is composite, there must be a prime missing from our set of primes. But this set contains all primes
#Therefore, our initial assumption ("there are finitely many primes") is invalid

==See also==
*[[Stolen concept]], a fallacy that can be exposed using a ''reductio ad absurdum''

==References==
<references/>

[[Category:Logic]]