In logic, the principle of weak excluded middle says that for any proposition we have .
This follows from the full principle of excluded middle, which says that for any we have (take ). 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.
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 for any .
De Morgan’s law is equivalent to weak excluded middle.
If de Morgan’s law holds, then since , we have , as desired. Conversely, if weak excluded middle holds and we have , then from weak excluded middle we get and which give four cases. In three of those cases holds, while in the fourth we have and , which together imply (see the first lemma here and its proof), contradicting the assumption of ; so the fourth case is impossible.
Last revised on November 16, 2022 at 20:13:05. See the history of this page for a list of all contributions to it.