nLab weak excluded middle

Weak excluded middle

Weak excluded middle


In logic, the principle of weak excluded middle says that for any proposition PP we have ¬P¬¬P\neg P \vee \neg\neg P.

This follows from the full principle of excluded middle, which says that for any QQ we have Q¬QQ \vee \neg Q (take Q=¬PQ = \neg P). Thus, in classical mathematics weak excluded middle is just true. But in constructive mathematics (i.e. intuitionistic logic), it is a weaker assumption than full excluded middle.

Equivalent forms

De Morgan’s Law

In intuitionistic logic, de Morgan’s law often refers to the one of de Morgan's four laws that is not an intuitionistic tautology, namely ¬(PQ)(¬P¬Q)\neg (P \wedge Q) \to (\neg P \vee \neg Q) for any P,QP,Q.


De Morgan’s law is equivalent to weak excluded middle.


If de Morgan’s law holds, then since ¬(P¬P)\neg (P \wedge \neg P), we have ¬P¬¬P\neg P \vee \neg\neg P, as desired. Conversely, if weak excluded middle holds and we have ¬(PQ)\neg (P\wedge Q), then from weak excluded middle we get ¬P¬¬P\neg P \vee \neg\neg P and ¬Q¬¬Q\neg Q \vee \neg\neg Q which give four cases. In three of those cases ¬P¬Q\neg P \vee \neg Q holds, while in the fourth we have ¬¬P\neg\neg P and ¬¬Q\neg\neg Q, which together imply ¬¬(PQ)\neg\neg(P\wedge Q) (see the first lemma here and its proof), contradicting the assumption of ¬(PQ)\neg (P\wedge Q); so the fourth case is impossible.

 See also

Last revised on November 16, 2022 at 20:13:05. See the history of this page for a list of all contributions to it.