Completeness theorems for some intermediate predicate calculi (Q1061734)

Completeness theorems, with respect to Kripke's semantics, are given for two kinds of negation-free intermediate predicate calculi and for one with negation. PD is the usual positive predicate calculus with the additional axiom \(\forall x(\alpha (x)\vee \beta)\to \forall x\alpha (x)\vee \beta\). The calculi treated in this paper are: (1) \(PD+(\alpha \to \beta)\vee (\beta \to \alpha)\), (2) \(PD+\alpha_ 0\vee (\alpha_ 0\to \alpha_ 1)\vee...\vee (\alpha_ n\to \alpha_{n+1})\), (3) \((1)+intuitionistic\) \(negation+\neg \forall x\alpha \to \exists x\neg \alpha\).