Philosophy Index

Double negation rule

The double negation rule is a rule in logic that holds that a proposition is equivalent to its double negation (double negative).

The double negation takes the form of "A is equivalent to not not A", or "A implies and is implied by not not A".

The rule may also take the form of:

α is equivalent to ¬¬α

Where α is any proposition or formula. The symbol, is equivalent to, is used to signify two-way inference, or equivalence. The symbol ⇔ is sometimes used for the same purpose.

Proof of the double negation rule by contradiction

Although it is very simple, the double negation rule is not a basic rule of inference — that is, it may be described by more fundamental rules. To justify the double negation rule using basic rules, we may construct two proofs, one of each direction of inference, as follows.

To show that A infers ¬¬A:

1 A Pr.
RTP ¬¬A
2 ¬A AIP
RTP Contradiction
3 A R, 1
4 ¬¬A ¬I, 2–3

And to show that ¬¬A infers A:

1 ¬¬A Pr.
RTP A
2 ¬A AIP
RTP Contradiction
3 ¬¬A R, 1
4 A ¬E, 2–3

The basic rules used to demonstrate this derived rule are the introduction and elimination rules for negation (¬). These are the two versions of the proof by contradiction (reductio ad absurdum method).