The Gentzen-Kripke construction of the intermediate logic LQ

The Gentzen-Kripke construction of the semantics for intermediate logic LQ, which is obtainable from the intuitionistic propositional logic H by adding the weak law of excluded middle -*A v —• —1>4, is presented. Our construction spans the Gentzen system and the Kripke semantics for LQ by providing the way from the cut-elimination theorem to model-theoretic results. The completeness and decidability theorems are shown in this method. / Introduction An intermediate logic LQ is obtainable from the intuitionistic propositional logic H by adding the weak law of excluded middle -\A v -»-vl. The logic can also be axiomatized by the axiom schema (-\A -> -ii?) v (-ιB -> -ιA). The motivation of LQ is purely technical rather than philosophical in that LQ is one possible extension of H. Our interest thus lies in its characterization in relation to the formalization of H. The semantics for LQ can be given by the class of directed Kripke frames; see Gabbay [4]. Several authors also proposed the Gentzen systems for LQ, none of which is successful. For example, the cut-elimination theorem cannot be proved in the sequent calculus of Boricic [2] as Hosoi pointed out. Recently, Hosoi [6] gave a Gentzen-type formulation GQ for LQ and proved the cut-elimination theorem. The purpose of this paper is to develop the Gentzen-Kripke construction of the semantics for LQ using the subformula models, as a modification of the one developed in Akama [1] for intuitionistic predicate logic. The proposed construction spans the Gentzen system and the Kripke semantics for LQ by providing the way from the cut-elimination theorem to model-theoretic results. The completeness and decidability theorems are shown in this method. 2 Intermediate logic LQ and its Gentzen-type formulation GQ The intermediate logic LQ is one of the extensions of the intuitionistic propositional logic H with the weak law of excluded middle, i.e. -*A v -ι-υ4. The proof and model Received July 10, 1989; revised May 7, 1990 CONSTRUCTION OF INTERMEDIATE LOGIC 149 theories for LQ are modified so that the weak law of excluded middle can be accommodated therein. As for model theory, Gabbay [4] presented a directed Kripke model for LQ. For the detailed exposition of a Kripke semantics for intuitionistic logic, the reader is referred to Fitting [3]. In the following, we review a Gentzen-type formulation GQ of LQ as developed by Hosoi [6]. GQ is a variant of LJ due to Gentzen [5] for intuitionistic logic. It requires two additional axioms for beginning sequents: