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 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. The law of excluded middle only says that (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.

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

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The law of excluded middle holds for any bivalent truth system. If we remove the law of excluded middle from a formal logical system, the result will be a system called intuitionistic logic, which is the logic of mathematical intuitionism.