Double negative elimination
In propositional logic, the inference rules double negative elimination (also called double negation elimination, double negative introduction, double negation introduction, or simply double negation) allow deriving the double negative equivalent by adding (for double negative introduction) or removing (for double negative elimination) a pair of negation signs. This is based on the equivalence of, for example,
- It is false that it is not raining.
- It is raining.
Formally, the rule of double negative elimination is
The rule of double negative introduction states the converse, that double negatives can be added without changing the meaning of a proposition. Formally, the rule of double negative introduction is
These two rules — double negative elimination and introduction — can be restated as follows (in sequent notation):
which can be combined together into a single biconditional formula
Double negative elimination is a theorem of classical logic, but not of weaker logics such as intuitionistic logic and minimal logic. Because of their constructive flavor, 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 both intuitionistic logic and minimal logic, as is .
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