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A variant of the double-negation translation Jeremy Avigad
 

Summary: A variant of the double-negation translation
Jeremy Avigad
August 21, 2006
Abstract
An efficient variant of the double-negation translation explains the
relationship between Shoenfield's and G¨odel'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 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

  

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

 

Collections: Mathematics