View source for Law of the excluded middle

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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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. van Dalen (ed.) , ''Brouwer's Cambridge lectures on intuitionism'' , Cambridge Univ. Press  (1981)</TD></TR></table>