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 $A\lor\neg A$, 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).



[a1] D. van Dalen (ed.) , Brouwer's Cambridge lectures on intuitionism , Cambridge Univ. Press (1981)
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Law of the excluded middle. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article