Double-negation translation

In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic. Typically it is done by translating formulas to formulas that are classically equivalent but intuitionistically inequivalent. Particular instances of double-negation translations include Glivenko's translation for propositional logic, and the Gödel–Gentzen translation and Kuroda's translation for first-order logic.

Propositional logic
The easiest double-negation translation to describe comes from Glivenko's theorem, proved by Valery Glivenko in 1929. It maps each classical formula φ to its double negation ¬¬φ.

Results
Glivenko's theorem states:
 * If φ is a propositional formula, then φ is a classical tautology if and only if ¬¬φ is an intuitionistic tautology.

Glivenko's theorem implies the more general statement:
 * If T is a set of propositional formulas and φ a propositional formula, then T ⊢ φ in classical logic if and only if T ⊢ ¬¬φ in intuitionistic logic.

In particular, a set of propositional formulas is intuitionistically consistent if and only if it is classically satisfiable.

First-order logic
The Gödel–Gentzen translation (named after Kurt Gödel and Gerhard Gentzen) associates with each formula φ in a first-order language another formula φN, which is defined inductively:

as above, but furthermore and otherwise
 * If φ is atomic, then φN is ¬¬φ
 * (φ ∨ θ)N is ¬(¬φN ∧ ¬θN)
 * (∃x φ)N is ¬(∀x ¬φN)
 * (φ ∧ θ)N is φN ∧ θN
 * (φ → θ)N is φN → θN
 * (¬φ)N is ¬φN
 * (∀x φ)N is ∀x φN

This translation has the property that φN is classically equivalent to φ. Troelstra and Van Dalen (1988, Ch. 2, Sec. 3) give a description, due to Leivant, of formulas that do imply their Gödel–Gentzen translation in intuitionistic first-order logic also. There, this is not the case for all formulas. (This is related to the fact that propositions with additional double-negations can be stronger than their simpler variant. E.g., ¬¬φ → θ always implies φ → θ, but the schema in the other direction would imply double-negation elimination.)

Equivalent variants
Due to constructive equivalences, there are several alternative definitions of the translation. For example, a valid De Morgan's law allows one to rewrite a negated disjunction. One possibility can thus succinctly be described as follows: Prefix "¬¬" before every atomic formula, but also to every disjunction and existential quantifier, Another procedure, known as Kuroda's translation, is to construct a translated φ by putting "¬¬" before the whole formula and after every universal quantifier. This procedure exactly reduces to the propositional translation whenever φ is propositional.
 * (φ ∨ θ)N is ¬¬(φN ∨ θN)
 * (∃x φ)N is ¬¬∃x φN

Thirdly, one may instead prefix "¬¬" before every subformula of φ, as done by Kolmogorov. Such a translation is the logical counterpart to the call-by-name continuation-passing style translation of functional programming languages along the lines of the Curry–Howard correspondence between proofs and programs.

The Gödel-Gentzen- and Kuroda-translated formulas of each φ are provenly equivalent to one another, and this result holds already in minimal propositional logic. And further, in intuitionistic propositional logic, the Kuroda- and Kolmogorov-translated formulas are equivalent also.

The mere propositional mapping of φ to ¬¬φ does not extend to a sound translation of first-order logic, as the so called double negation shift
 * ∀x ¬¬φ(x) → ¬¬∀x φ(x)

is not a theorem of intuitionistic predicate logic. So the negations in φN have to be placed in a more particular way.

Results
Let TN consist of the double-negation translations of the formulas in T.

The fundamental soundness theorem (Avigad and Feferman 1998, p. 342; Buss 1998 p. 66) states:
 * If T is a set of axioms and φ is a formula, then T proves φ using classical logic if and only if TN proves φN using intuitionistic logic.

Arithmetic
The double-negation translation was used by Gödel (1933) to study the relationship between classical and intuitionistic theories of the natural numbers ("arithmetic"). He obtains the following result:
 * If a formula φ is provable from the axioms of Peano arithmetic then φN is provable from the axioms of Heyting arithmetic.

This result shows that if Heyting arithmetic is consistent then so is Peano arithmetic. This is because a contradictory formula θ &and; ¬θ is interpreted as θN &and; ¬θN, which is still contradictory. Moreover, the proof of the relationship is entirely constructive, giving a way to transform a proof of θ &and; ¬θ in Peano arithmetic into a proof of θN &and; ¬θN in Heyting arithmetic.

By combining the double-negation translation with the Friedman translation, it is in fact possible to prove that Peano arithmetic is Π02 -conservative over Heyting arithmetic.