Gödel–Gentzen negative translation

In proof theory, the Gödel–Gentzen negative translation is a method for embedding classical first-order logic into intuitionistic first-order logic. It is one of a number of double-negation translations that are of importance to the metatheory of intuitionistic logic. It is named after logicians Kurt Gödel and Gerhard Gentzen.

Propositional logic

The easiest double-negation translation to describe comes from Glivenko's theorem. It maps each classical formula φ to its double negation ¬¬φ.

Glivenko's theorem states:

If T is a set of propositional formulas and φ a propositional formula, then T proves φ using classical logic if and only if T proves ¬¬φ using intuitionistic logic.

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

First-order logic

The translation associates with each formula φ in a first-order language another formula φ^N, which is defined inductively:

• If φ is atomic, then φ^N is ¬¬φ
• (φ ∧ θ)^N is φ^N ∧ θ^N
• (φ ∨ θ)^N is ¬(¬φ^N ∧ ¬θ^N)
• (φ → θ)^N is φ^N → θ^N
• (¬φ)^N is ¬φ^N
• (∀x φ)^N is ∀x φ^N
• (∃x φ)^N is ¬∀x ¬φ^N

Notice that φ^N is classically equivalent to φ.

The fundamental soundness theorem states:^[1]

If T is a set of axioms and φ a formula, then T proves φ using classical logic if and only if T^N proves φ^N using intuitionistic logic.

Here T^N consists of the double-negation translations of the formulas in T.

Note that φ need not imply its negative translation φ^N in intuitionistic first-order logic. Troelsta and Van Dalen^[2] give a description (due to Leivant) of formulas which do imply their Gödel–Gentzen translation.

Variants

There are several alternative definitions of the negative translation. They are all provably equivalent in intuitionistic logic, but may be easier to apply in particular contexts.

One possibility is to change the clauses for disjunction and existential quantifier to

• (φ ∨ θ)^N is ¬¬(φ^N ∨ θ^N)
• (∃x φ)^N is ¬¬∃x φ^N

Then the translation can be succinctly described as: prefix ¬¬ to every atomic formula, disjunction, and existential quantifier.

Another possibility (described by Kuroda) is to construct φ^N from φ by putting ¬¬ before the whole formula and after every universal quantifier. Notice that this reduces to the simple ¬¬φ translation if φ is propositional.

It is also possible to define φ^N by prefixing ¬¬ 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.

Results

The double-negation translation was used by Gödel (1933) to study the relationship between classical and intutionistic theories of the natural numbers ("arithmetic"). He obtains the following result:

If a formula φ is provable from the axioms of Peano arithmetic then φ^N is provable from the axioms of intuitionistic Heyting arithmetic.

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

The propositional mapping of φ to ¬¬φ does not extend to a sound translation of first-order logic, because ∀x ¬¬φ(x) → ¬¬∀x φ(x) is not a theorem of intuitionistic predicate logic. This explains why φ^N has to be defined in a more complicated way in the first-order case.

See also

• Dialectica interpretation
• Glivenko's theorem

Notes

1. ^ Avigad and Feferman 1998, p. 342; Buss 1998 p. 66
2. ^ Troelsta, van Dalen 1988, Ch. 2, Sec. 3)

References

• J. Avigad and S. Feferman (1998), "Gödel's Functional ("Dialectica") Interpretation", Handbook of Proof Theory, S. Buss, ed., Elsevier. ISBN 0-444-89840-9
• S. Buss (1998), "Introduction to Proof Theory", Handbook of Proof Theory, S. Buss, ed., Elsevier. ISBN 0-444-89840-9
• G. Gentzen (1936), "Die Widerspruchfreiheit der reinen Zahlentheorie", Mathematische Annalen, v. 112, pp. 493–565 (German). Reprinted in English translation as "The consistency of arithmetic" in The collected papers of Gerhard Gentzen, M. E. Szabo, ed.
• K. Gödel (1933), "Zur intuitionistischen Arithmetik und Zahlentheorie", Ergebnisse eines mathematischen Kolloquiums, v. 4, pp. 34–38 (German). Reprinted in English translation as "On intuitionistic arithmetic and number theory" in The Undecidable, M. Davis, ed., pp. 75–81.
• A. N. Kolmogorov (1925), "O principe tertium non datur" (Russian). Reprinted in English translation as "On the principle of the excluded middle" in From Frege to Gödel, van Heijenoort, ed., pp. 414–447.
• A. S. Troelsta (1977), "Aspects of Constructive Mathematics", Handbook of Mathematical Logic", J. Barwise, ed., North-Holland. ISBN 0-7204-2285-X
• A.S. Troelsta and D. van Dalen (1988), Constructivism in Mathematics. An Introduction, volumes 121, 123 of Studies in Logic and the Foundations of Mathematics, North–Holland.

External links

• "Intuitionistic logic", Stanford Encyclopedia of Philosophy.