Kripke submodels and universal sentences

Abstract

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including and ⊥) using ∧, ∨, ∀, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model-consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory ∆ if and only if every rooted Kripke model of ∆ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence.

Preservation theorems for Kripke models

“…We use the following notational conventions adapted from [5]. If C is an arbitrary set of constants, then L(C) is the language L extended with all constants in C. The set At ⊆ L is the set of atomic formulas in L. Analogously, At(C) ⊆ L(C) is the set of atomic formulas in L(C).…”

“…We follow the method used in [5], where theories closed under (their notion of) submodels are characterized. The difference is that we fix the frame of the models in the extensions.…”

“…The following definition that plays an essential role in our proof of the completeness theorem is a modified version of the one given in [5,Definition 3.2]. Definition 3.4 Let C and D be sets of constants with C ⊆ D. Let Γ be a consistent theory over L(C), and let Δ be a consistent theory over L(D).…”

“…In [5], the authors choose the third definition and prove that an intuitionistic theory is preserved under Kripke submodels if and only if it is axiomatized by semipositive universal sentences, as they naturally define (they call these formulas universal and denote them by U).…”

“…We also characterize theories that are preserved under extensions. For these we use a version of the Henkin method applied in [5] that itself is based on the one in [7]. We also consider the notion of elementary submodel and give some results in this direction.…”