Negation
Logical operation that inverts the truth value of a proposition.
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Negation, also called the logical not or logical complement, is an operation in logic that takes a proposition P to another proposition "not P", written as ¬P, ∼P, P′, or P̅. It is interpreted intuitively as being true when P is false, and false when P is true. Negation is a unary logical connective applicable to propositions, notions, truth values, or semantic values more generally.
- field
- Logic
- known_for
- Logical operation that inverts truth values; classical negation corresponds to complementation in Boolean algebra; intuitionistic negation to pseudocomplementation in Heyting algebra
- variants
- Classical negation, intuitionistic negation
Lore & Background
Negation is defined in classical logic as an operation on one logical value that produces true when its operand is false, and false when its operand is true. For example, if P is "The dog runs", then "not P" is "The dog does not run". An operand of a negation is called a negand or negatum. Negation can be defined in terms of other logical operations, such as ¬P ≡ P → ⊥ (where → is logical consequence and ⊥ is absolute falsehood), or ⊥ can be defined as Q ∧ ¬Q for any proposition Q. These definitions work in classical and intuitionistic logic but not in paraconsistent logic, where contradictions are not necessarily false.
Reader's Guide
Negation is a fundamental operation in logic, serving as the basis for reasoning about truth and falsity. In classical logic, double negation (¬¬P) is equivalent to P, making negation an involution of period two. In intuitionistic logic, a proposition implies its double negation but not conversely, though the weaker equivalence ¬¬¬P ≡ ¬P holds. De Morgan's laws allow distribution of negation over disjunction and conjunction: ¬(P ∨ Q) ≡ (¬P ∧ ¬Q) and ¬(P ∧ Q) ≡ (¬P ∨ ¬Q). Negation also has defined precedence rules, with ¬ having higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. Algebraically, classical negation corresponds to complementation in a Boolean algebra, while intuitionistic negation corresponds to pseudocomplementation in a Heyting algebra. Glivenko's theorem states that in the propositional case, a sentence is classically provable if its double negation is intuitionistically provable.
Did You Know?
- Negation is also called the logical not or logical complement.
- An operand of a negation is called a negand or negatum.
- In intuitionistic logic, the negation of a proposition P is the proposition whose proofs are the refutations of P.
- Negation can be defined in terms of NAND and also in terms of NOR.
Frequently Asked Questions
Who is Negation?
Negation is a unary logical connective that takes a proposition P and produces its complement, commonly written as ¬P, ∼P, P′, or P̅. It is the fundamental operation in logic that flips a truth value to its opposite.
What are Negation's powers and role?
Negation inverts the truth value of whatever it is applied to, turning a true statement into a false one and vice versa. It can be applied not only to simple propositions but also to notions, truth values, and semantic values more broadly.
How does Negation's story resolve in different logical systems?
In classical logic, Negation achieves full complementation, meaning every proposition has exactly one complement and the law of excluded middle holds. In intuitionistic logic, its role is more constrained, corresponding to pseudocomplementation in a Heyting algebra rather than full Boolean complementation.
Why is Negation important to the broader field of logic?
Negation underpins core reasoning tools such as proof by contradiction, the law of non-contradiction, and the law of excluded middle. Without it, one could not formally express that a statement is false or derive conclusions from the impossibility of an assumption.
What are Negation's main variants?
The two principal forms are classical negation, which aligns with complementation in Boolean algebra, and intuitionistic negation, which corresponds to pseudocomplementation in Heyting algebras. They diverge most sharply on undecidable propositions, where classical negation still yields a definite truth value but intuitionistic negation may not.
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