# Double negation

## Propositional logic theorem / From Wikipedia, the free encyclopedia

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In propositional logic, the **double negation** of a statement states that "it is not the case that the statement is not true". In classical logic, every statement is logically equivalent to its double negation, but this is not true in intuitionistic logic; this can be expressed by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign **~** expresses negation.

**Quick Facts**Type, Field ...

Type | Theorem |
---|---|

Field | |

Statement | If a statement is true, then it is not the case that the statement is not true." |

Symbolic statement | $A\equiv \sim (\sim A)$ |

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,^{[1]} but it is disallowed by intuitionistic logic.^{[2]} The principle was stated as a theorem of propositional logic by Russell and Whitehead in *Principia Mathematica* as:

- $\mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)$
^{[3]} - "This is the principle of double negation,
*i.e.*a proposition is equivalent of the falsehood of its negation."

- $\mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)$