Unveiling the constructive core of classical theories: A contribution to 90 years of Glivenko’s theorem
Glivenko’s well known result of 1929 established that a negated propositional formula provable in classical logic is even provable intuitionistically. Similar later transfers from classical to intuitionistic provability therefore fall under the nomenclature of Glivenko-style results: these are results about classes of formulas for which classical provability yields intuitionistic provability. The interest in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been gathered together under the conceptual umbrella “computational content of classical theories.” They also belong to a more general shift of perspective in foundations: rather than developing constructive mathematics separately, as in Brouwer’s program, one studies which parts of classical mathematics can be directly translated into constructive terms.
We shall survey how Glivenko-style results can be easily obtained by the choice of suitable sequent calculi for classical and intuitionistic logic, by the conversion of axioms into inference rules, and by the procedure of geometrization of first order logic.