A secondary semantics for Second Order Intuitionistic Propositional Logic

We propose a Kripke-style semantics for second order intuitionistic propositional logic. Our semantics can be viewed as a secondary semantics with nested domains in the sense of Skvortsov [6]. Namely, let F be a Kripke frame, that is a partially ordered set, and let D(F) be the Heyting algebra of the upward-closed subsets of F . In principal semantics, quantifiers ranges over all the elements of D(F) and, as proved in [6], the set of formulas valid in such a semantics is non-recursively axiomatizable (according to [4] such a set is even non-arithmetical). On the other hand, in secondary semantics propositional quantifiers range over proper subsets of D(F), and in [6] some examples of axiomatizable logics with a secondary semantics are given. The logic Ipl generated from our semantics corresponds to H2 of [6] and can be seen as a variant of the ones of Gabbay [2, 3] and Sobolev [7]. Such a semantics has an impredicative character connected with the distinction between pseudomodels and models, the latter being pseudomodels where every closed formula is simulated by an appropriate propositional constant. The domain of every element of a model is a set of propositional constants and propositional quantifiers range over these sets. In this paper we prove that Ipl meets the disjunction property (A∨B ∈ Ipl implies A ∈ Ipl or B ∈ Ipl) and the explicit definability property (∃pA(p) ∈ Ipl implies A(H/p) ∈ Ipl for some formula H). Our proof is semantical and, as far as we know, no semantical proof of constructivity for a secondary semantics has been given in the literature. In the paper we also provide a tableau calculus T -Ipl for Ipl obtained by adding to a standard tableau calculus for intuitionistic propositional logic (see, e. g., [1]) the rules for quantifiers and a special rule. In the last section of the paper we show that T -Ipl is sound and complete.