We show that Pitts' modeling of propositional quantification in intuitionistic logic (as the appropriate interpolants) does not coincide with the topological interpretation. This contrasts with the case of the monadic language and the interpretation over sufficiently regular topological spaces. We also point to the difference between the topological interpretation over sufficiently regular spaces and the interpretation of propositional quantifiers in Kripke models.

A one-premiss rule is said to be archetypal for a consequence relation when not only is the conclusion of any application of the rule a consequence of the premiss, but whenever one formula has another as a consequence, these formulas are respectively equivalent to a premiss and a conclusion of some application of the rule. We are concerned here with the consequence relation of classical propositional logic and with the task of extending the above notion of archetypality to rules with (...) more than one premiss, and providing an informative characterization of the set of rules falling under the more general notion. (shrink)

We introduce the notion of bisimulation for first-order Kripke models. It is defined as a relation that satisfies certain zig-zag conditions involving back-and-forth moves between nodes of Kripke models and, simultaneously, between the domains of their underlying structures. As one of our main results, we prove that if two Kripke models bisimulate to a certain degree, then they are logically equivalent with respect to the class of formulae of the appropriate complexity. Two applications of the notion introduced in the paper (...) are given. First, we consider bisimulations in the context of Kripke models of Heyting Arithmetic. Next, we discuss the notion of elementary submodel of a Kripke model and provide a suitable example. (shrink)

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. (shrink)

We study the monadic fragment of second order intuitionistic propositional logic in the language containing the standard propositional connectives and propositional quantifiers. It is proved that under the topological interpretation over any dense-in-itself metric space, the considered fragment collapses to Heyting calculus. Moreover, we prove that the topological interpretation over any dense-in-itself metric space of fragment in question coincides with the so-called Pitts' interpretation. We also prove that all the nonstandard propositional operators of the form q $\mapsto \exists$p ), where (...) F is an arbitrary monadic formula of the variable p, are definable in the language of Heyting calculus under the topological interpretation of intuitionistic logic over sufficiently regular spaces. (shrink)