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 (notA), 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 ⋅ 13 . ⊢ . p ≡ ∼ ( ∼ p ) displaystyle mathbf *4cdot 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." Contents 1
Double negative
1.1 Formal notation 2 See also 3 References 4 Bibliography
Double negative
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." Formal notation[edit] The double negation introduction rule may be written in sequent notation: P ⊢ ¬ ¬ P displaystyle Pvdash neg neg P The double negation elimination rule may be written as: ¬ ¬ P ⊢ P displaystyle neg neg Pvdash P In rule form: P ¬ ¬ P displaystyle frac P neg neg P and ¬ ¬ P P displaystyle frac neg neg P P or as a tautology (plain propositional calculus sentence): P → ¬ ¬ P displaystyle Pto neg neg P and ¬ ¬ P → P displaystyle neg neg Pto P These can be combined together into a single biconditional formula: ¬ ¬ P ↔ P displaystyle neg neg Pleftrightarrow P . Since biconditionality is an equivalence relation, any instance of
¬¬A in a wellformed formula can be replaced by A, leaving unchanged
the truthvalue of the wellformed formula.
Double negative
¬ ¬ ¬ A ⊢ ¬ A displaystyle neg neg neg Avdash 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. In set theory also we have the negation operation of the complement which obeys this property: a set A and a set (AC)C (where AC represents the complement of A) are the same. See also[edit] Gödel–Gentzen negative translation References[edit] ^ 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".)
^ Hamilton is discussing
Hegel
Bibliography[edit] 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, NorthHolland Publishing Company,
Amsterdam NY, ISBN 0720421039.
Stephen C. Kleene, 1967, Mathematical Logic, Dover edition 2002, Dover
Publications, Inc, Mineola N.Y. ISBN 0486425339
Alfred North Whitehead
