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Difference between revisions of "Law of the excluded middle"

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The law in classical logic stating that one of the two statements  "A"  or  "not A"  is true. The law of the excluded middle is expressed in mathematical logic by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057730/l0577301.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057730/l0577302.png" /> denotes [[Disjunction|disjunction]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057730/l0577303.png" /> denotes [[Negation|negation]]. From the intuitionistic (constructive) point of view, establishing the truth of a statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057730/l0577304.png" /> means establishing the truth of either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057730/l0577305.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057730/l0577306.png" />. Since there is no general method for establishing in a finite number of steps the truth of an arbitrary statement, or of that of its negation, the law of the excluded middle was subjected to criticism by representatives of the intuitionistic and constructive directions in the foundations of mathematics (cf. [[Intuitionism|Intuitionism]]; [[Constructive mathematics|Constructive mathematics]]).
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The law in classical logic stating that one of the two statements  "A"  or  "not A"  is true. The law of the excluded middle is expressed in mathematical logic by the formula $A\lor\neg A$, where $\lor$ denotes [[Disjunction|disjunction]] and $\neg$ denotes [[Negation|negation]]. From the intuitionistic (constructive) point of view, establishing the truth of a statement $A\lor\neg A$ means establishing the truth of either $A$ or $\neg A$. Since there is no general method for establishing in a finite number of steps the truth of an arbitrary statement, or of that of its negation, the law of the excluded middle was subjected to criticism by representatives of the intuitionistic and constructive directions in the foundations of mathematics (cf. [[Intuitionism|Intuitionism]]; [[Constructive mathematics|Constructive mathematics]]).
  
  

Latest revision as of 12:49, 17 March 2014

The law in classical logic stating that one of the two statements "A" or "not A" is true. The law of the excluded middle is expressed in mathematical logic by the formula $A\lor\neg A$, where $\lor$ denotes disjunction and $\neg$ denotes negation. From the intuitionistic (constructive) point of view, establishing the truth of a statement $A\lor\neg A$ means establishing the truth of either $A$ or $\neg A$. Since there is no general method for establishing in a finite number of steps the truth of an arbitrary statement, or of that of its negation, the law of the excluded middle was subjected to criticism by representatives of the intuitionistic and constructive directions in the foundations of mathematics (cf. Intuitionism; Constructive mathematics).


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References

[a1] D. van Dalen (ed.) , Brouwer's Cambridge lectures on intuitionism , Cambridge Univ. Press (1981)
How to Cite This Entry:
Law of the excluded middle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Law_of_the_excluded_middle&oldid=17830
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article