Double negation is a rule of inference and rule of replacement for classical propositional logic. It states that the negation of a negation of a proposition is logically equivalent to the affirmation of that proposition. [1][2] It is often expressed symbolically as so:

$ \lnot \lnot P \Leftrightarrow P $

or even

$ \lnot \lnot P = P $.

Where $ \Leftrightarrow $ means "replaceable".

In standard rule form, this rule may be represented as two subrules:

$ \frac {\lnot \lnot P} {P} , \frac{P}{\lnot \lnot P} $

Please note it is not universally accepted. Some formulations of propositional logic omit this rule of inference (such as intuitionistic propositional logic).


  1. Double negation rule. Retrevied January 27, 2017 from
  2. Double Negation. Retrieved January 27 2017 from