Law of excluded middle
The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). (The tilde symbol, '~', reads 'not'.)
For example, if P is
- Joe is bald
then the inclusive disjunction
- Joe is bald, or Joe is not bald
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 the law of noncontradiction, which states that ~(P and ~P) is true. The law of excluded middle only says that the total (P or ~P) is true, but does not comment on what truth values P itself may take. In any case, the semantics of any bivalent logic will assign opposite truth values to P and ~P (i.e., if P is true, then ~P is false), so the law of excluded middle will be equivalent to the principle of bivalence in a bivalent logic. However, the same cannot be said about non-bivalent logics, or many-valued logics.
Certain systems of logic may reject bivalence by allowing more than two truth values (i.e., true, false, and indeterminate), but accept the law of excluded middle. In such logics, (P or ~P) may be true while P and ~P are not assigned opposite truth-values like true and false, respectively.