This paper shows that the interpolation theorem fails in the intuitionistic logic of constant domains. This result refutes two previously published claims that the interpolation property holds.

No abstract

Notions of k-asimulation and asimulation are introduced as asymmetric counterparts to k-bisimulation and bisimulation, respectively. It is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula over the class of intuitionistic Kripke models iff it is invariant with respect to asimulations between intuitionistic models. §1. Introduction.Van Benthem's well-known modal characterization theorem (Theorem 4.9 below) states that a first-order formula is equivalent to a standard translation of a modal propositional formula iff it is invariant with respect to bisimulations. There is also a weaker 'parametrized' version of this result stating that a first-order formula is equivalent to a standard translation of a modal propositional formula iff this formula is invariant with respect to k-bisimulations for some k. Although both results yield a convenient model-theoretical technique distinguishing 'modal' first-order formulas from 'nonmodal' ones, van Benthem's characterization theorem, unlike its parametrized version, also isolates a single property defining expressive powers of modal propositional logic and thus gives us an important insight into its nature when this logic is viewed as a fragment of first-order logic.Given that the view of intuitionistic logic as a fragment of modal propositional logic has a long and established tradition dating back to Tarski-Gödel translation of this logic into S4, it was natural to ask whether one can add to the bisimulation invariance some extra condition to get a criterion distinguishing 'intuitionistic propositional' first-order formulas from nonintuitionistic ones. As was shown by Visser et al. (1995), bisimulation invariance plus a form of monotonicity with respect to the accessibility relation 'almost' gives such a criterion: i. e., it tells intuitionistic propositional first-order formulas from nonintuitionistic ones over the set of intuitionistic models. It was not known, however, how to remove this restriction and provide a criterion distinguishing intuitionistic propositional first-order formulas from nonintuitionistic ones in the general case.In this paper, we take a different strategy and amend the notion of bisimulation itself rather than adding new conditions to it. In this way, we get a notion of asimulation and we show that asimulation invariance allows one to tell intuitionistic propositional first-order formulas from nonintuitionistic ones in the general case, thus providing a criterion for the

Abstract. Notions of asimulation and k-asimulation introduced in [Olkhovikov 2011] are extended onto the level of predicate logic. We then prove that a firstorder formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula over a class of intuitionistic models (intuitionistic models with constant domain) iff it is invariant with respect to asimulations between intuitionistic models (intuitionistic models with constant domain).Van Benthem's well-known modal characterization theorem shows that expressive power of modal propositional logic as a fragment of first-order logic can be described via the notion of bisimulation invariance. Moreover, it is known that modal predicate logic, initially considered as an extension of first-order logic, can also be viewed as its fragment, although somewhat bigger than the fragment induced by propositional modal logic. Expressive power of modal predicate logic, from this vantage point, is described by the notion of world-object bisimulation which appears to be a rather direct combination of bisimulation and partial isomorphism (see, e. g. [Van Benthem 2010, p. 124, Theorem 21]).Although intuitionistic logic has been treated as a fragment of modal logic for quite a long while, results analogous to propositional and predicate version of Van Benthem's modal characterization theorem were not obtained for it until recently. In [Olkhovikov 2011] we filled this gap for intuitionistic propositional logic. In this paper we introduced the notion of asimulation and its parametrized version, k-asimulation, and showed that they can be used to characterize expressive power of intuitionistic propositional logic in much the same way bisimulation and k-bisimulation are used to characterize modal propositional logic. In this paper we do the same job for intuitionistic predicate logic without identity.The layout of the paper is as follows. Starting from some notational conventions and 1

It is shown that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain topological property reminiscent of compactness, the Tarski union property and preservation under asimulations.

We formulate a Hilbert-style axiomatic system for STIT logic of imagination recently proposed by H. Wansing in [2] and prove its completeness by the method of canonical models.

The paper treats 4 different fragments of first-order logic induced by their respective versions of Kripke style semantics for modal intuitionistic logic. In order to capture these fragments, the notion of asimulation, defined in [7], is modified and extended to yield Van Benthem type of semantic characterization of their respective expressive powers. It is shown further, that this characterization can be easily carried over to arbitrary first-order definable subclasses of classical first-order models.

In the style of Lindström’s theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of the previous work done by the authors for intuitionistic logic (Badia and Olkhovikov, 2020, Notre Dame Journal of Formal Logic, 61, 11–30).