law of double negation

logic

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June 25, 2026

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The law of double negation is a principle in logic that states that negating a statement twice is equivalent to the original statement. In simpler terms, saying "it is not true that X is not happening" is the same as saying "X is happening." This concept is related to the law of contradiction, which posits that a thing cannot both be and not be at the same time. The law of double negation is a fundamental aspect of classical logic, though some alternative logical systems may treat it differently.12

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formal logic
Logical manipulations in LPC
Similarly, because the law of double negation permits the deletion of a pair of consecutive negation signs, ∼(∃x) may be replaced by (∀x)∼, and ∼(∀x) by (∃x)∼.
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Similarly, because the law of double negation permits the deletion of a pair of consecutive negation signs, ∼(∃x) may be replaced by (∀x)∼, and ∼(∀x) by (∃x)∼.
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formal logic
Nonstandard versions of PC
…accept the validity of the law of double negation in the form ∼∼p ⊃ p. They do, however, regard a demonstration that p is true as showing that the negation of p is false and hence accept p ⊃ ∼∼p as valid. For somewhat similar reasons, these mathematicians also refuse…
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…accept the validity of the law of double negation in the form ∼∼p ⊃ p. They do, however, regard a demonstration that p is true as showing that the negation of p is false and hence accept p ⊃ ∼∼p as valid. For somewhat similar reasons, these mathematicians also refuse…
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