Logical negation exchanges zero and one: ¬0=1 and ¬1=0. For a numerical bit x, this is 1−x; applying logical negation twice restores x. It acts on the entire propositional formula in its scope. Parentheses therefore matter: ¬(P∨Q) says that neither condition holds, while (¬P)∨Q also accepts every case where Q holds. At (P,Q)=(1,1) their values are zero and one.
On a set, logical negation becomes taking a complement in a fixed universal set. In Boolean operations on events, “not even” selects {1,3,5}. “Not both even and at least four” selects {1,2,3,5}; it does not require both qualifications to fail. De Morgan's laws make the scope of these compound negations precise.
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Incoming links (99)
- Balanced parentheses in propositional syntax
- Capture-safe equality replacement
- Complete one-type
- Endpoint-false disjunctive normal form
- Fixed point of boxed negated provability
- Formation rule
- Functional completeness
- Maximal consistency in propositional logic
- Maximal consistency of a first-order theory
- Maximal consistent set of first-order sentences
- Negation elimination rule
- Negation introduction rule
- Negation normal form
- Negation with one binary connective
- N-variable Lindenbaum algebra
- One-consistency
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- Prefix-freeness of fully parenthesized formulas
- Propositional consistency
- Pull-down network
- Rosser provability predicate
- Separation of number sets in an arithmetic theory
- Sigma-0 soundness
- Syntax tree of a formula
- Three-sample run detector
- Truth-matrix countermodel for a Hilbert axiom set
- Unique readability theorem
- Universal theory and substructures of elementarily equivalent models
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Negation is a linguistic and logical concept that involves the denial or contradiction of a proposition or statement. In simpler terms, it refers to the process of stating that something is not the case or that a particular assertion is false. In language, negation is often expressed using words such as "not," "no," "never," and "nothing." For example: - The statement "It is raining" can be negated to "It is not raining.