---
tag: CX032ACTMZ
kind: statement
role: proposition
name: Law of the weak excluded middle
aliases: [weak-excluded-middle, weak-law-of-the-excluded-middle]
msc: [03B05]
sources:
  - wikipedia: en:17637@r1370685960
    locator: Law of excluded middle, section Analogous laws > Law of the weak excluded middle
    note: adopted
    mention: wikipedia://en/17637#AS5S
    sections: [Statement]
inventory: [1BQ7ACFGDN]
---

## Statement

A particularly well-studied intermediate logic is given by De Morgan logic, which adds the axiom $\neg P \lor \neg\neg P$ to intuitionistic logic, which is sometimes called the law of the weak excluded middle.

This is equivalent to a few other statements:

- Satisfying all of [[De Morgan's laws|09PKJKA6J0]] including $\neg(P\land Q) \,\leftrightarrow\, \neg P \lor \neg Q$
- $$(P \to Q) \lor (\neg P \to \neg Q)$$
- $$(P \to (Q \lor \neg R)) \to ((P \to Q) \lor (P \to \neg R))$$