Reductio ad absurdum

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'''Reductio ad absurdum''' is a type of [[logic]]al [[argument]] where one assumes a claim for the sake of argument, arrives at an "absurd" result (often a [[contradiction]]), and then concludes that the original assumption must have been wrong, since it led to this absurd result.
 
'''Reductio ad absurdum''' is a type of [[logic]]al [[argument]] where one assumes a claim for the sake of argument, arrives at an "absurd" result (often a [[contradiction]]), and then concludes that the original assumption must have been wrong, since it led to this absurd result.
  
Note that this is a [[Validity vs. soundness|logically valid]] technique that usually takes one of two forms:
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This is a [[Validity vs. soundness|logically valid]] technique that usually takes the form:
 
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1. Self contradiction:
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* Assume that '''P''' is true.
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* Deduce that '''P''' must also be false.
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* Therefore, '''P''' is false.
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2. Deducing two mutually contradictory statements:
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* Assume '''P''' is true.
 
* Assume '''P''' is true.
 
* From this assumption, deduce that '''Q''' is true.
 
* From this assumption, deduce that '''Q''' is true.
 
* Also deduce that '''Q''' is false.
 
* Also deduce that '''Q''' is false.
* Thus, '''P''' implies '''Q''' and '''not Q''' or, more simply, '''P''' implies a falsehood.
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* Thus, '''P''' implies both '''Q''' and '''not Q''' (a contradiction, which is necessarily false).
 
* Use ''[[wikipedia:Modus_tollens|modus tollens]]'' to conclude that '''P''' itself must be false.
 
* Use ''[[wikipedia:Modus_tollens|modus tollens]]'' to conclude that '''P''' itself must be false.
  
An example (from [[wikipedia:number theory|number theory]]) of the first form of this type of argument is:
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An example of this type of argument (from [[wikipedia:number theory|number theory]]) follows. It proves that there are infinitely many [[wikipedia:prime number|prime numbers]] by first assuming the opposite. Note that the term ''number'' is being used here to refer to [[Wikipedia:Natural number|positive integers]] only.
* Assume there are finitely many [[wikipedia:prime number|prime numbers]] and call this number ''N''.
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* Assume there are finitely many prime numbers; in particular, say there are ''N'' of them.
* We can therefore list all ''N'' prime numbers: 2,&nbsp;3,&nbsp;5,&nbsp;...,&nbsp;''p''<sub>''N''</sub>.
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* We can therefore list all ''N'' prime numbers: 2, 3, 5, 7,&nbsp;..., ''p''<sub>''N''</sub>. (Note that the proof doesn't rely on knowing the actual value of any of the primes. The list given here is merely to illustrate the argument.)
* Consider the product of all ''N'' prime numbers: 2&nbsp;&times;&nbsp;3&nbsp;&times;&nbsp;5&nbsp;&times;&nbsp;7&nbsp;&times;&nbsp;11&nbsp;&times;&nbsp;&middot;&middot;&middot;&nbsp;&times;&nbsp;''p''<sub>''N''</sub><sub>''</sub>.
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* Consider a number ''q'' that is one greater than the product of all these prime numbers: ''q''&nbsp;=&nbsp;(2&nbsp;&times;&nbsp;3&nbsp;&times;&nbsp;5&nbsp;&times;&nbsp;7&nbsp;&times;&nbsp;&middot;&middot;&middot;&nbsp;&times;&nbsp;''p''<sub>''N''</sub>)&nbsp;+&nbsp;1.
* Add one: (2&nbsp;&times;&nbsp;3&nbsp;&times;&nbsp;5&nbsp;&times;&nbsp;7&nbsp;&times;&nbsp;11&nbsp;&times;&nbsp;&middot;&middot;&middot;&nbsp;&times;&nbsp;''p''<sub>''N''</sub>)&nbsp;+&nbsp;1. Call this number ''q''.
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* Clearly ''q'' is greater than ''p''<sub>''N''</sub> and thus cannot be prime (since ''p''<sub>''N''</sub> was the largest prime).
* Notice that ''q'' is not divisible by any of these primes up to ''p''<sub>''N''</sub>, since the remainder, when ''q'' is divided by any the primes, will always be 1. (For example, ''q'' divided by 2 gives the [[wikipedia:quotient|quotient]] 3&nbsp;&times;&nbsp;5&nbsp;&times;&nbsp;7&nbsp;&times;&nbsp;11&nbsp;&times;&nbsp;&middot;&middot;&middot;&nbsp;&times;&nbsp;''p''<sub>''N''</sub>, with remainder 1.)
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* This means that ''q'' must be divisible by one of our listed primes. (By definition, if ''q'' is greater than 1 and not prime, it must be divisible by some number other than 1 and itself. But by a separate argument not reproduced here, every non-prime number greater than 1 must be divisible by at least one prime. Thus it suffices to check whether ''q'' is divisible by any of our prime numbers.)
* So ''q'' is either prime (''e''.''g''., 2&times;3&times;5&nbsp;+&nbsp;1&nbsp;=&nbsp;31) or it is divisible by a prime number which has not yet been listed (''e''.''g''., 2&times;3&times;5&times;7&times;11&times;13&nbsp;+&nbsp;1&nbsp;=&nbsp;30031&nbsp;=&nbsp;51&times;509). In either case, either ''q'' or one of its [[wikipedia:Prime_factor|prime factors]] does not appear in our list of ''N'' prime numbers.
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* Note that ''q'' is not divisible by any of the listed primes, since the remainder, when ''q'' is divided by any the primes, will always be 1. (For example, ''q'' divided by 2 gives the [[wikipedia:quotient|quotient]] 3&nbsp;&times;&nbsp;5&nbsp;&times;&nbsp;7&nbsp;&times;&nbsp;&middot;&middot;&middot;&nbsp;&times;&nbsp;''p''<sub>''N''</sub>, with remainder 1.)
* Thus we have found a prime number not appearing in our list of ''N'' prime numbers and thus there are at least ''N''&nbsp;+&nbsp;1 prime numbers. This is a contradiction.
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* Thus assuming finitely many primes we have deduced that there is a number ''q'' that is both divisible by one of our primes and not divisible by any of our primes. This is a contradiction.
 
* Therefore, we have to reject our original assumption. There must be infinitely many prime numbers.
 
* Therefore, we have to reject our original assumption. There must be infinitely many prime numbers.
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(This is essentially [[wikipedia:Euclid|Euclid's]] proof of the same result from ''[[wikipedia:Euclid's Elements|The Elements]]''.)
  
An example of the second form of this type of argument comes from [[wikipedia:Euclid|Euclid]]'s ''[[wikipedia:Euclid's_Elements|Elements]]'' which states that if two circles cut each other, they must have different centers:
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In less formal settings, making a ''reductio ad absurdum'' argument may be called taking an assumption to its [[wikipedia:logical extreme|logical extreme]] (or limit).
* Assume that if two circles cut each other, they have the same center ''Z''.
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* The length of a ray ''ZC'' to one of the cuts ''C'' (a point lying on both circles) must therefore equal the length of a ray ''ZA'' to a point ''A'' on the first circle that is not also on the second circle.
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* The line forming this ray, however, must also cut the second circle at a point ''B'' and the length of ''ZB'' therefore be different from the length of ''ZA''.
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* But, be definition, ''ZB'' equals the length of the ray ''ZC''.
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* Things which are equal to a common thing must therefore be equal to each other.  Thus, the length of ''ZA'' must equal the length of ''ZB''.
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* We have therefore deduced that the lengths of ''ZA'' and ''ZB'' must be both equal and unequal.
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* Therefore, the original assumption is false:  the circles must have different centers.
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==Counter-apologetics==
 
==Counter-apologetics==
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See [[Can God create a rock so heavy that he can't lift it?]] for an example in the context of [[counter-apologetics]] (the claim being assumed is that God is all-powerful).
 
See [[Can God create a rock so heavy that he can't lift it?]] for an example in the context of [[counter-apologetics]] (the claim being assumed is that God is all-powerful).
  
The problem with this type of argument is that the "absurdity" one reaches must actually be a logical contradiction in order for the argument to be valid. If the conclusion is simply unlikely, then the argument doesn't necessarily work. For example:
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The problem with this type of argument is that the "absurdity" one reaches must actually be a logical contradiction in order for the argument to be valid. If the consequence is simply unlikely, then the argument doesn't necessarily work. For example:
  
 
* If [[God]] doesn't exist, then life arose by purely natural means.
 
* If [[God]] doesn't exist, then life arose by purely natural means.
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* [[Therefore, God exists.]]
 
* [[Therefore, God exists.]]
  
Well... no. Ignoring the fact that the premise is faulty (as its possible, however unlikely that life arose by something other than purely natural means) the absurdity is not shown.  The assertion that its absurd is only used to come to the conclusion that [[God did it]].  The person that would be making this argument does not know how life arose, so to assume anything other than a deity seems absurd. (See: [[God of the gaps]])
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Well... no. Ignoring the fact that the premise is faulty (as it is possible, however unlikely, that life arose by something other than purely natural means) the absurdity is not shown.  The assertion that it is absurd is only used to come to the conclusion that [[God did it]].  The person that would be making this argument does not know how life arose, so to assume anything other than a deity seems absurd. (See: [[God of the gaps]])
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However, if the goal is to demonstrate that an assertion is ''untenable'' (i.e., not [[belief|believable]], as opposed to not [[truth|true]]), then deducing a merely unlikely statement may lead to the desired conclusion. This is, in fact, how arguments based on [[Statistical significance|statistical inference]] work.
  
 
[[Category:Argumentation]]
 
[[Category:Argumentation]]
 
[[Category:Logic]]
 
[[Category:Logic]]

Revision as of 10:33, 11 January 2011

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For more information, see the Wikipedia article:

Reductio ad absurdum is a type of logical argument where one assumes a claim for the sake of argument, arrives at an "absurd" result (often a contradiction), and then concludes that the original assumption must have been wrong, since it led to this absurd result.

This is a logically valid technique that usually takes the form:

  • Assume P is true.
  • From this assumption, deduce that Q is true.
  • Also deduce that Q is false.
  • Thus, P implies both Q and not Q (a contradiction, which is necessarily false).
  • Use modus tollens to conclude that P itself must be false.

An example of this type of argument (from number theory) follows. It proves that there are infinitely many prime numbers by first assuming the opposite. Note that the term number is being used here to refer to positive integers only.

  • Assume there are finitely many prime numbers; in particular, say there are N of them.
  • We can therefore list all N prime numbers: 2, 3, 5, 7, ..., pN. (Note that the proof doesn't rely on knowing the actual value of any of the primes. The list given here is merely to illustrate the argument.)
  • Consider a number q that is one greater than the product of all these prime numbers: q = (2 × 3 × 5 × 7 × ··· × pN) + 1.
  • Clearly q is greater than pN and thus cannot be prime (since pN was the largest prime).
  • This means that q must be divisible by one of our listed primes. (By definition, if q is greater than 1 and not prime, it must be divisible by some number other than 1 and itself. But by a separate argument not reproduced here, every non-prime number greater than 1 must be divisible by at least one prime. Thus it suffices to check whether q is divisible by any of our prime numbers.)
  • Note that q is not divisible by any of the listed primes, since the remainder, when q is divided by any the primes, will always be 1. (For example, q divided by 2 gives the quotient 3 × 5 × 7 × ··· × pN, with remainder 1.)
  • Thus assuming finitely many primes we have deduced that there is a number q that is both divisible by one of our primes and not divisible by any of our primes. This is a contradiction.
  • Therefore, we have to reject our original assumption. There must be infinitely many prime numbers.

(This is essentially Euclid's proof of the same result from The Elements.)

In less formal settings, making a reductio ad absurdum argument may be called taking an assumption to its logical extreme (or limit).

Counter-apologetics

See Can God create a rock so heavy that he can't lift it? for an example in the context of counter-apologetics (the claim being assumed is that God is all-powerful).

The problem with this type of argument is that the "absurdity" one reaches must actually be a logical contradiction in order for the argument to be valid. If the consequence is simply unlikely, then the argument doesn't necessarily work. For example:

  • If God doesn't exist, then life arose by purely natural means.
  • This is absurd (read: very, very unlikely).
  • Therefore, God exists.

Well... no. Ignoring the fact that the premise is faulty (as it is possible, however unlikely, that life arose by something other than purely natural means) the absurdity is not shown. The assertion that it is absurd is only used to come to the conclusion that God did it. The person that would be making this argument does not know how life arose, so to assume anything other than a deity seems absurd. (See: God of the gaps)

However, if the goal is to demonstrate that an assertion is untenable (i.e., not believable, as opposed to not true), then deducing a merely unlikely statement may lead to the desired conclusion. This is, in fact, how arguments based on statistical inference work.

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