We study how to postpone the application of the reductio ad absurdum rule (RAA) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of RAA, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

We study how to postpone the application of the reductio ad absurdum rule ) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of \, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

A widely debated issue in philosophy of logic concerns the possibility of an inferentialist account of classical logic. Many proposals to show that classical logic satisfies the requirements of inferentialist semantics (such as harmony) demand to modify the ordinary natural deduction rules. In this paper, we try to explain why the ordinary natural deduction rules for classical logic are not harmonious and therefore not directly justifiable within an inferentialist framework. We show however that an indirect justification of classical logic, passing (...) through negative translation, can be acceptable from an inferentialist point of view. (shrink)