Double negation


In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not, or by the formula A ≡ ~ where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.
Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica'' as:

Elimination and introduction

'Double negation elimination and double negation introduction are two valid rules of replacement. They are the inferences that if A is true, then not not-A is true and its converse, that, if not not-A is true, then A is true. The rule allows one to introduce or eliminate a negation from a formal proof. The rule is based on the equivalence of, for example, It is false that it is not raining. and It is raining.
The double negation introduction rule is:
and the double negation elimination rule is:
Where "" is a metalogical symbol representing "can be replaced in a proof with."
In logics that have both rules, negation is an involution.

Formal notation

The double negation introduction rule may be written in sequent notation:
The double negation elimination rule may be written as:
In rule form:
and
or as a tautology :
and
These 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 well-formed formula.
Double negative elimination is a theorem of classical logic, but not of weaker logics such as intuitionistic logic and minimal logic. Double negation introduction is a theorem of both intuitionistic logic and minimal logic, as is.
Because of their constructive character, 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.

Proofs

In classical propositional calculus system

In Hilbert-style deductive systems for propositional logic, double negation is not always taken as an axiom, and is rather a theorem. We describe a proof of this theorem in the system of three axioms proposed by Jan Łukasiewicz:
We use the lemma proved here, which we refer to as, and use the following additional lemma, proved here:
We first prove. For shortness, we denote by φ0. We also use repeatedly the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps.
We now prove.
And the proof is complete.