 
Summary: A variant of the doublenegation translation
Jeremy Avigad
August 21, 2006
Abstract
An efficient variant of the doublenegation translation explains the
relationship between Shoenfield's and G¨odel's versions of the Dialectica
interpretation.
Fix a classical firstorder 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 to the formula
= ¬, so is supposed to represent an intuitionistic version of the negation
of . The map from to is defined recursively, as follows:
= ¬, when is atomic
(¬) = ¬
( ) =
( ) =
(x ) = x
Note that we can eliminate either or and retain a complete set of connectives.
If is the set of classical formulas {1, . . . , k}, let
