The law of the excluded middle is the principle of classical logic that, for every proposition 
, the disjunction of 
and its negation is valid:
 |
It is also called the law of the excluded third, the principle of excluded middle, or simply excluded middle. The law is distinct from the semantic claim that every
proposition is either true
or false. In some nonclassical interpretations, 
remains valid even when 
has neither classical truth value (Aloni 2024). The law is
not accepted as a general principle in intuitionistic
logic, where a proof of 
must construct either a proof of 
or a proof of 
. It can also fail under the usual interpretations of
three-valued logic and fuzzy
logic.
The law should not be confused with the law of noncontradiction, which states that 
and 
cannot both be true. The two laws are distinct even though both hold in classical
logic.
