Everything about Double Negative Elimination totally explained
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 isn't raining.
and
» It is raining.
Formally, the rule double negative elimination is:
¬¬A
∴ A
Formally, the rule double negative introduction is:
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):
» ,
» .
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
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 wouldn't 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
.
In
set theory also we've 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.
Further Information
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