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 tilde symbol, '¬', reads 'not'.)

Related Topics:
Logic - Not

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For example, if P is

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: Joe is bald

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then the inclusive disjunction

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: Joe is bald, or Joe is not bald

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is true.

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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.

Related Topics:
Principle of bivalence - Law of noncontradiction - Many-valued logic

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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.

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In such logics,(P ∨ ¬P) may be true while P and ¬P are not assigned opposite truth-values like true and false, respectively.

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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.

Related Topics:
Intuitionistic logic - Bivalence and related laws

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

Related Topics:
Logical fallacy - False dilemma

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