# Double negation

(Redirected from Double negative elimination)

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.[1]

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,[2] but it is disallowed by intuitionistic logic.[3] The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

${\displaystyle \mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)}$[4]
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."

## Elimination and introduction

'Double negation elimination and double negation introduction are two valid rules of replacement. They are the inferences that if A is true, then not not-A is true and its converse, that, if not not-A is true, then A is true. 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 ${\displaystyle \Rightarrow }$ ${\displaystyle \neg }$${\displaystyle \neg }$P

and the double negation elimination rule is:

${\displaystyle \neg }$${\displaystyle \neg }$P ${\displaystyle \Rightarrow }$ P

Where "${\displaystyle \Rightarrow }$" is a metalogical symbol representing "can be replaced in a proof with."

In logics that have both rules, negation is an involution.

### Formal notation

The double negation introduction rule may be written in sequent notation:

${\displaystyle P\vdash \neg \neg P}$

The double negation elimination rule may be written as:

${\displaystyle \neg \neg P\vdash P}$

In rule form:

${\displaystyle {\frac {P}{\neg \neg P}}}$

and

${\displaystyle {\frac {\neg \neg P}{P}}}$

or as a tautology (plain propositional calculus sentence):

${\displaystyle P\to \neg \neg P}$

and

${\displaystyle \neg \neg P\to P}$

These can be combined together into a single biconditional formula:

${\displaystyle \neg \neg P\leftrightarrow P}$.

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 ${\displaystyle \neg \neg \neg A\vdash \neg A}$.

Because of their constructive character, a statement such as It's not the case that it's not raining is weaker than It's raining. The latter requires a proof of rain, whereas the former merely requires a proof that rain would not be contradictory. This distinction also arises in natural language in the form of litotes.

## References

1. ^ Or alternate symbolism such as A ↔ ¬(¬A) or Kleene's *49o: A ∾ ¬¬A (Kleene 1952:119; in the original Kleene uses an elongated tilde ∾ for logical equivalence, approximated here with a "lazy S".)
2. ^ Hamilton is discussing Hegel in the following: "In the more recent systems of philosophy, the universality and necessity of the axiom of Reason has, with other logical laws, been controverted and rejected by speculators on the absolute.[On principle of Double Negation as another law of Thought, see Fries, Logik, §41, p. 190; Calker, Denkiehre odor Logic und Dialecktik, §165, p. 453; Beneke, Lehrbuch der Logic, §64, p. 41.]" (Hamilton 1860:68)
3. ^ The o of Kleene's formula *49o indicates "the demonstration is not valid for both systems [classical system and intuitionistic system]", Kleene 1952:101.
4. ^ PM 1952 reprint of 2nd edition 1927 pages 101-102, page 117.

## Bibliography

• William Hamilton, 1860, Lectures on Metaphysics and Logic, Vol. II. Logic; Edited by Henry Mansel and John Veitch, Boston, Gould and Lincoln.
• Christoph Sigwart, 1895, Logic: The Judgment, Concept, and Inference; Second Edition, Translated by Helen Dendy, Macmillan & Co. New York.
• Stephen C. Kleene, 1952, Introduction to Metamathematics, 6th reprinting with corrections 1971, North-Holland Publishing Company, Amsterdam NY, ISBN 0-7204-2103-9.
• Stephen C. Kleene, 1967, Mathematical Logic, Dover edition 2002, Dover Publications, Inc, Mineola N.Y. ISBN 0-486-42533-9
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, 2nd edition 1927, reprint 1962, Cambridge at the University Press.