Double negation

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:

$\mathbf{*4\cdot13}. \ \ \vdash.\ p \ \equiv \ \thicksim(\thicksim p)$^[4]

"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."

The principium contradictiones of modern logicians (particularly Leibniz and Kant) in the formula A is not not-A, differs entirely in meaning and application from the Aristotelian proposition [ i.e. Law of Contradiction: not (A and not-A) i.e. ~(A & ~A), or not (( B is A) and (B is not-A))]. This latter refers to the relation between an affirmative and a negative judgment.

According to Aristotle, one judgment [B is judged to be an A] contradicts another [B is judged to be a not-A]. The later proposition [ A is not not-A ] refers to the relation between subject and predicate in a single judgment; the predicate contradicts the subject. Aristotle states that one judgment is false when another is true; the later writers [Leibniz and Kant] state that a judgment is in itself and absolutely false, because the predicate contradicts the subject. What the later writers desire is a principle from which it can be known whether certain propositions are in themselves true. From the Aristotelian proposition we cannot immediately infer the truth or falsehood of any particular proposition, but only the impossibility of believing both affirmation and negation at the same time.^[5]

Double negative elimination

Double negative elimination (also called double negation elimination, double negative introduction, double negation introduction, double negation, or negation elimination^[6][7][8]) 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 logical 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 $\Rightarrow$ ¬¬P

and the double negation elimination rule is:

¬¬P $\Rightarrow$ P

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

Formal notation

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

$P \vdash \neg \neg P$

The double negation elimination rule may be written as:

$\neg \neg P \vdash P$

In rule form:

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

and

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

or as a tautology (plain propositional calculus sentence):

$P \to \neg \neg P$

and

$\neg \neg P \to P$

These can be combined together into a single biconditional formula:

$\neg \neg P \leftrightarrow P$.

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. 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.) Double negation introduction is a theorem of both intuitionistic logic and minimal logic, as is $\neg \neg \neg A \vdash \neg A$.

In set theory also we have the negation operation of the complement which obeys this property: a set A and a set (A^C)^C (where A^C represents the complement of A) are the same.

See also

• Gödel–Gentzen negative translation

References

1. 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".)
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 *49^o 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.
5. Sigwart 1895:142-143
6. Copi and Cohen
7. Moore and Parker
8. Hurley

Bibliography

• William Hamilton, 1860, Lectures on Metaphysics and Logic, Vol. II. Logic; Edited by Henry Mansel and John Veitch, Boston, Gould and Lincoln. Available online from googlebooks.
• Christoph Sigwart, 1895, Logic: The Judgment, Concept, and Inference; Second Edition, Translated by Helen Dendy, Macmillan & Co. New York. Available online from googlebooks.
• 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 (pbk.)
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, 2nd edition 1927, reprint 1962, Cambridge at the University Press, London UK, no ISBN or LCCCN.