A short proof of Glivenko theorems for intermediate predicate logics

We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic. We say that a set L of first order formulas is an intermediate predicate logic between intuitionistic and classical predicate logic if: 1. every formula provable in intuitionistic predicate logic IL belongs to L, and every formula in L is provable in the classical predicate logic CL. 2. L is closed under modus ponens, substitution and generalization. (we refer the reader to [1] for a complete definition of the notions in 2). Let L and M be two intermediate predicate logics, where the signature has been previously fixed. We say that Glivenko’s theorem for L overM holds if and only if, whenever `L φ, then we also have `M ¬¬φ. The case where L (resp. M) are precisely the classical (resp. intuitionistic) propositional logic had been first considered by Glivenko in [4]. As observed by Kleene in [5], this situation cannot be generalized to the predicate case; instead, one gets a Glivenko theorem setting M to be the logic obtained from the intuitionistic logic by adding the so called double negation shift (DNS), which is the axiom scheme ∀x¬¬φ(x) → ¬¬∀xφ(x). Several proofs of this fact are known, both syntactic and semantic, involving generally some kind of induction at the meta-level (see [6], [7], [9]) or descriptions of Kripke models which are enough to characterize DNS ([3]). Other generalizations to substructural logics are also known ([2]). The main purpose of this note is to prove that Glivenko’s theorem, in the propositional and the predicate version, are immediate consequences of the corresponding deduction theorems. Consider the restricted excluded middle scheme REM consisting of those instances of the principle of excluded middle that involve only closed formulas. In what follows, we write L+M for the logic axiomatized by formulas in both L andM. We use the Hilbert style axiomatization of intuitionistic logic, as exposed, for example, in [8]. Results obtained by using Gentzen systems, as those of [9] and [10], can be transferred to this context in view of the equivalence of such systems, which is proved in [8]. Our main result is: ∗This research was financed by the project “Constructive and category-theoretic foundations of mathematics (dnr 2008-5076)” from the Swedish Research Council (VR).