Law of excluded middle
In logic, the law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P ∨ ¬P).
The symbol, '¬', reads 'not', ∨ reads 'or', and ∧ reads 'and'
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 ∧ ¬P) is true. The law of excluded middle only says that the total (P ∨ ¬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 (e.g.; true, false, and indeterminate; true, false, neither, both), but accept the law of excluded middle.
In such logics,(P ∨ ¬P) may be true while P and ¬P are not assigned opposite truth-values like true and false, respectively.
The law of excluded middle can be misapplied, leading to the logical fallacy of the excluded middle, also known as a false dilemma.cs:Zákon o vyloučení třetího de:Satz vom ausgeschlossenen Dritten fr:Principe du tiers exclu ko:배중률 he:כלל השלישי מן הנמנע no:Loven om den ekskluderte tredje pl:Prawo wyłączonego środka ru:Закон исключённого третьего