The rule of double negation elimination is a valid argument in certain types of logic dealing with negation $\neg$.
This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not intuitionistic propositional logic.
- If we can conclude $\neg \neg \phi$, then we may infer $\phi$.
- $\neg \neg p \vdash p$
Double Negation from Intuitionistic Perspective
The intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom.
This in turn invalidates the Law of Double Negation Elimination from the system of intuitionistic propositional logic.
Hence a difference is perceived between Double Negation Elimination and Double Negation Introduction, whereby it can be seen from the Principle of Non-Contradiction that if a statement is true, then it is not the case that it is false.
However, if all we know is that a statement is not false, we can not be certain that it is actually true without accepting that there are only two possible truth values.
Such distinctions may be important when considering, for example, multi-value logic.
However, when analysing logic from a purely classical standpoint, it is common and acceptable to make the simplification of taking just one Double Negation rule:
- $p \dashv \vdash \neg \neg p$