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Difference between revisions of "Reductio ad absurdum"

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A logical derivation rule that allows one to conclude that if a list <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804301.png" /> of statements and a statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804302.png" /> imply both a statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804303.png" /> and the statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804304.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804305.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804306.png" />. The rule of reductio ad absurdum can, e.g., be written in the form
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A logical derivation rule that allows one to conclude that if a list $\Gamma$ of statements and a statement $A$ imply both a statement $B$ and the statement $\neg B$, then $\Gamma$ implies $\neg A$. The rule of reductio ad absurdum can, e.g., be written in the form
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$$
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\frac{\Gamma, A \rightarrow B\,;\ \Gamma,A \rightarrow \neg B}{\Gamma \rightarrow \neg A}
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804307.png" /></td> </tr></table>
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Reductio ad absurdum is a [[sound rule]] in the majority of logico-mathematical calculi.
  
Reductio ad absurdum is a [[Sound rule|sound rule]] in the majority of logico-mathematical calculi.
 
  
  
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====Comments====
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Informally, the name  "reductio ad absurdum"  is also used for the rule that if $\Gamma$ together with $\neg A$ implies a contradiction, then $\Gamma$ implies $A$. This is of course equivalent to the above (and therefore sound) in classical logic, but it is not a sound rule of inference in [[intuitionistic logic]].
  
====Comments====
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Informally, the name  "reductio ad absurdum"  is also used for the rule that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804308.png" /> together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r0804309.png" /> implies a contradiction, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r08043010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080430/r08043011.png" />. This is of course equivalent to the above (and therefore sound) in classical logic, but it is not a sound rule of inference in intuitionistic logic.
 

Latest revision as of 18:28, 17 September 2017

A logical derivation rule that allows one to conclude that if a list $\Gamma$ of statements and a statement $A$ imply both a statement $B$ and the statement $\neg B$, then $\Gamma$ implies $\neg A$. The rule of reductio ad absurdum can, e.g., be written in the form $$ \frac{\Gamma, A \rightarrow B\,;\ \Gamma,A \rightarrow \neg B}{\Gamma \rightarrow \neg A} $$

Reductio ad absurdum is a sound rule in the majority of logico-mathematical calculi.


Comments

Informally, the name "reductio ad absurdum" is also used for the rule that if $\Gamma$ together with $\neg A$ implies a contradiction, then $\Gamma$ implies $A$. This is of course equivalent to the above (and therefore sound) in classical logic, but it is not a sound rule of inference in intuitionistic logic.

How to Cite This Entry:
Reductio ad absurdum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reductio_ad_absurdum&oldid=41884
This article was adapted from an original article by S.Yu. Maslov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article