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:

^{[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 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.

### Formal notation

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 .

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.

## Proofs

### In classical propositional calculus system

In Hilbert-style deductive systems for propositional logic, double negation is not always taken as an axiom (see list of Hilbert systems), and is rather a theorem. We describe a proof of this theorem in the system of three axioms proposed by Jan Łukasiewicz:

- A1.
- A2.
- A3.

We use the lemma proved here, which we refer to as (L1), and use the following additional lemma, proved here:

- (L2)

We first prove . For shortness, we denote by φ_{0}. We also use repeatedly the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps.

- (1) (instance of (A1))
- (2) (instance of (A3))
- (3) (instance of (A3))
- (4) (from (2) and (3) by the hypothetical syllogism metatheorem)
- (5) (instance of (A1))
- (6) (from (4) and (5) by the hypothetical syllogism metatheorem)
- (7) (instance of (L2))
- (8) (from (1) and (7) by modus ponens)
- (9) (from (6) and (8) by the hypothetical syllogism metatheorem)

We now prove .

- (1) (instance of the first part of the theorem we have just proven)
- (2) (instance of (A3))
- (3) (from (1) and (2) by modus ponens)

And the proof is complete.

## See also

## References

**^**Or alternate symbolism such as A ↔ ¬(¬A) or Kleene's *49^{o}: A ∾ ¬¬A (Kleene 1952:119; in the original Kleene uses an elongated tilde ∾ for logical equivalence, approximated here with a "lazy S".)**^**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)**^**The^{o}of Kleene's formula *49^{o}indicates "the demonstration is not valid for both systems [classical system and intuitionistic system]", Kleene 1952:101.**^**PM 1952 reprint of 2nd edition 1927 pp. 101–02, 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.