Law of excluded middle

The law of excluded middle states that for any proposition P, it is true that (P or not-P).

For example, if P is

Joe is bald

then the inclusive disjunction

Joe is bald, or Joe is not bald

is true.

This is not quite the same as the principle of bivalence, which states that P must be either true or false. It also differs from Law of non-contradiction, which states (P and not-P) is false. The law of excluded middle only says that the total (P or not-P) is true, but does not comment on what truth values P itself may take.

This leaves open the possibility that certain systems of logic may reject bivalence (by allowing more than 2 truth values) but accept the law of excluded middle, by accepting that (P or not-P) is always true, even when P itself is neither true nor false.

The distinction is far less important in traditional logic, however, where bivalence is accepted. Some nontraditional logics, most notably intuitionistic logic, are not bivalent, and in these logics the law of excluded middle does not hold.

The page bivalence and related laws discusses this issue in greater detail.

The law of excluded middle can be misapplied, leading to the logical fallacy of the excluded middle, also known as a False dilemma.