Law of excluded middle

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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

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 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.

Some logics do not accept the law of excluded middle, most notably intuitionistic logic. The article 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.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