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Concept
Double negation
Résumé
In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.
Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."
Double negation elimination and double negation introduction are two valid rules of replacement. They are the inferences that, if not not-A is true, then A is true, and its converse, that, if A is true, then not not-A is true, respectively. The rule allows one to introduce or eliminate a negation from a formal proof. The rule is based on the equivalence of, for example, It is false that it is not raining. and It is raining.
The double negation introduction rule is:
P P
and the double negation elimination rule is:
P P
Where "" is a metalogical symbol representing "can be replaced in a proof with."
In logics that have both rules, negation is an involution.
The double negation introduction rule may be written in sequent notation:
The double negation elimination rule may be written as:
In rule form:
and
or as a tautology (plain propositional calculus sentence):
and
These can be combined into a single biconditional formula:
Since biconditionality is an equivalence relation, any instance of ¬¬A in a well-formed formula can be replaced by A, leaving unchanged the truth-value of the well-formed formula.
Double negative elimination is a theorem of classical logic, but not of weaker logics such as intuitionistic logic and minimal logic. Double negation introduction is a theorem of both intuitionistic logic and minimal logic, as is .
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