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Sample sentences
AND operator
IF/THEN operator
NOT operator
OR operator
XOR operator


Chain Rule
Conjunctive Addition
DeMorgan's Law
Disjunctive Addition
Disjunctive Inference
Disjunctive Infer. (XOR)
Double Negation
Modus Ponens
Modus Tollens
Mutual Exclusion


5-step or more
Bad Argument

Double Negation

Double negation is a rule of inference pertaining to the NOT operator.

Double negation states that when two NOT operators are applied to a statement, the statement's truth value remains the same.

Informally, we say that two NOT operators cancel each other out, leaving the statement they have been applied to unchanged.

Imagine we are given a statement: "Today is a cold day." If we apply two NOT operators to it, we obtain "It is not true that today is not a cold day." This is a very complicated way of saying "Today is a cold day." The meaning of the sentence did not change.

Formally, we would write:

~~p: "It is not true that today is not a cold day."
p: "Today is a cold day."

The statement ~~p is above the line of dashes, and the conclusion p obtained by applying the Double Negation rule is below the line.


Other examples of Double Negation

~~A: "It is not true that the play was not fascinating."
A: "The play was fascinating."

~~X: "It is not true that Christmas is not our favorite holiday."
X: "Christmas is our favorite holiday."


Links to Relevant Problems

These are links to validity proof problems whose solutions contain Double Negation.

5-step problem