Double negative elimination

In logic and the propositional calculus, double negative elimination is a rule that states that double negatives can be removed from a proposition without changing its meaning:

It's not the case that it's not raining.

means the same as:

It's raining.

Formally:

  ¬ ¬ A
  ∴ A

The rule of double negative introduction states the converse, that double negatives can be added without changing the meaning of a proposition.

These two rules — double negative elimination and introduction — can be restated as follows (in sequent notation):

<math> \neg \neg A \vdash A <math>,

<math> A \vdash \neg \neg A <math>.

Applying the Deduction Theorem to each of these two inference rules produces the pair of valid conditional formulas

<math> \vdash \neg \neg A \rightarrow A <math>,

<math> \vdash A \rightarrow \neg \neg A <math>,

which can be combined together into a single biconditional formula

<math> \neg \neg A \leftrightarrow A <math>.

Since biconditionality is an equivalence relation, any instance of ¬ ¬ A in a well-formed formula can be replaced by A, leaving unchanged the truth-value of the wff.

Double negative elimination is a theorem of classical logic, but not intuitionistic logic. Because of the constructive flavor of intuitionistic logic, 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.) Double negation introduction is a theorem of intuitionistic logic, as is <math> \neg \neg \neg A \vdash \neg A <math>.

In set theory also we have the negation operation of the complement which obeys this property: a set A and a set (A^C)^C (where A^C represents the complement of A) are the same.