Double negative elimination
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{{Unreferenced|date=December 2009}}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.
and
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):
Applying the
deduction theorem to each of these two
inference rules produces the pair of valid conditional formulas
which can be combined together into a single biconditional formula
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 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
≠g ≠g ≠g A vdash ≠g A
.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.
See also
{{Logic-stub}}
Eliminatie van dubbele negatie二重否定の除去Dupla negaçãoЗакон двойного отрицания双重否定除去
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