On the Computational Interpretation of Negation

We investigate the possibility of giving a computational interpretation of an involutive negation in classical natural deduction. We first show why this cannot be simply achieved by adding ¬¬A = A to typed λ-calculus: the main obstacle is that an involutive negation cannot be a particular case of implication at the computational level. It means that one has to go out typed λ-calculus in order to have a safe computational interpretation of an involutive negation. We then show how to equip λμ-calculus in a natural way with an involutive negation: the abstraction and application associated to negation are simply the operators μ and [ ] from λμ-calculus. The resulting system is called symmetric λμ-calculus. Finally we give a translation of symmetric λ-calculus in symmetric λμcalculus, which doesn’t make use of the rule of μ-reduction of λμ-calculus (which is precisely the rule which makes the difference between classical and intuitionistic proofs in the context of λμ-calculus). This seems to indicate that an involutive negation generates an original way of computing. Because symmetric λμ-calculus contains both ways, it should be a good framework for further investigations.