A Variant of the Double-Negation Translation

An efficient variant of the double-negation translation explains the relationship between Shoenfield’s and Gödel’s versions of the Dialectica interpretation. Fix a classical first-order language, based on the connectives ∨, ∧, ¬, and ∀. We will define a translation to intuitionistic (even minimal) logic, based on the usual connectives. The translation maps each formula φ to the formula φ∗ = ¬φ∗, so φ∗ is supposed to represent an intuitionistic version of the negation of φ. The map from φ to φ∗ is defined recursively, as follows: φ∗ = ¬φ, when φ is atomic (¬φ)∗ = ¬φ∗ (φ ∨ ψ)∗ = φ∗ ∧ ψ∗ (φ ∧ ψ)∗ = φ∗ ∨ ψ∗ (∀x φ)∗ = ∃x φ∗ Note that we can eliminate either ∨ or ∧ and retain a complete set of connectives. If Γ is the set of classical formulas {φ1, . . . , φk}, let Γ∗ denote the set of formulas {φ1, . . . , φk}. The main theorem of this note is the following: Theorem 0.1 1. Classical logic proves φ ↔ φ∗. 2. If φ is provable from Γ in classical logic, then φ∗ is provable from Γ∗ in minimal logic. ∗Carnegie Mellon Technical Report CMU-PHIL 179. This note was written in response to a query from Grigori Mints. After circulating a draft, I learned that Thomas Streicher and Ulrich Kohlenbach had hit upon the same solution [5], and that the version of the doublenegation translation described below is due to Streicher and Reus [6], inspired by a similar translation by Jean-Louis Krivine. These results now appear as exercises in [3]. Since this variant of the double-negation translation and its application to the Dialectica translation are not well known, however, posting this note seemed worthwhile. Similar double-negation translations, for formulas in negation-normal form, can be found in [1, 2]. July 6, 2007: I am grateful to Jaime Gaspar for pointing out an error in the statement of Proposition 0.4, which has now been corrected.