A semantic approach to conservativity

The aim of this paper is to describe from a semantic perspective the problem of conservativity of classical first-order theories over their intuitionistic counterparts. In particular, we describe a class of formulae for which such conservativity results can be proven in case of any intuitionistic theory T which is complete with respect to a class of T-normal Kripke models. We also prove conservativity results for intuitionistic theories which are closed under the Friedman translation and complete with respect to a class of conversely well-founded Kripke models. The results can be applied to a wide class of intuitionistic theories and can be viewed as generalization of the results obtained by syntactic methods

Bisimulation reducts and submodels of intuitionistic first-order Kripke models

We consider elementary submodels of a given intuitionistic Kripke model K meant as models that share the same theory with K and result in restricting the frame of K and/or replacing some of its worlds with their elementary substructures. We introduce the notion of bisimulation reduct of the Kripke model wich allows us to construct elementary submodels of given Kripke models in the sense of the definition above. As it was observed by A. Visser in [6], the notion of submodel can be desined for intuitionistic sirst-order Kripke models in several different ways. We can either consider models on the same frame, where the worlds of submodels are substructures of the worlds of the original model, or we can define a submodel to be the result of restricting the frame of the given model, or we can combine both of these operations. All of these possibilities were considered in the literature, see [1], [6] and [2] respectively, however it seems that we should accept the third notion as the correct one. The reason for that is, that not only such defined notion of submodel coincides with the classical notion of substructure in the case of the simplest Kripke model, but also because the well-known classical Tarski- łoś preservation theorem concerning substructures becomes a particular case of the result proven in [2]; i.e. the class of the formulas that are preserved under Kripke submodels is the class of an intuitionistic variant of universal formulas

Partially-elementary extension Kripke models and Burr's hierarchy

We investigate Kripke models of subtheories /'k,, of Heyting Arithmetic. The theories Al*,,, defined by W. Burr, can be regarded as the natural intuitionistic counterparts of subtheories Inn of Peano Arithmetic. In the paper we consider n-elementary extension Kripke models which are models whose worlds are ordered by the elementary extension relation with respect to A„ formulae instead of merely the (weak) submodel relation. We prove that every Inn-normal, n-elementary extension model is a model of /'k,,. This suggests a method of constructing non-trivial Kripke models of /'k„. We also show that every (n + 1)- elementary extension model of /'k„ is Inn-normal

Anti-chains, focuses and projective formulas

We characterize projective formulas in intuitionistic propositional logic in terms of properties of subsets of universal Kripke models they define. The characterization allows us to prove some properties of the formulas in question

Quantified intuitionistic propositional logic and Cantor space

We consider propositional quantification in intuitionistic logic. We prove that, under topological interpretation over the Cantor space, it enjoys surprising and interesting properties such as the maximum property and a kind of distribution of existential quantifier over conjunction. Moreover, by pointing to appropriate examples, we show that the set of quantified formulas valid in the Cantor space strictly contains the set of formulas provable in the minimal system of intuitionistic logic with propositional quantification