Maximality of bi-intuitionistic propositional logic

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). © 2021 The Author(s). Published by Oxford University Press. All rights reserved

On expressive power of basic modal intuitionistic logic as a fragment of classical FOL

The modal characterization theorem by J. van Benthem characterizes classical modal logic as the bisimulation invariant fragment of first-order logic. In this paper, we prove a similar characterization theorem for intuitionistic modal logic. For this purpose we introduce the notion of modal asimulation as an analogue of bisimulations. The paper treats four different fragments of first-order logic induced by their respective versions of Kripke-style semantics for modal intuitionistic logic. It is shown further that this characterization can be easily carried over to arbitrary first-order definable subclasses of classical first-order models. © 2016 Elsevier B.V

Towards a Generic Model Theory: Automatic Bisimulations for Atomic, Molecular and First-order Logics

After observing that the truth conditions of connectives of non-classical logics are generally defined in terms of formulas of first-order logic (FOL), we introduce protologics, a class of logics whose connectives are defined by arbitrary first-order formulas. Then, we identify two subclasses of protologics which are particularly well-behaved. We call them atomic and molecular logics. Notions of invariance for atomic and molecular logics can be automatically defined from the truth conditions of their connectives, bisimulations do not need to be defined by hand on a case by case basis for each logic. Moreover, molecular logics behave as 'paradigmatic logics': every first-order logic and every protologic is as expressive as a molecular logic. Then, we prove a series of model-theoretical results for molecular logics which characterize them as fragments of FOL and which provide criteria for axiomatizability and definability of a class of models in these logics. In particular, we rediscover van Benthem's theorem for modal logic as a specific instance of our generic theorems and other results for modal intuitionistic logic and temporal logic. We also discover a wide range of novel results, such as for the Lambek calculus. Then, we apply our method and generic results to FOL and find out novel invariance notions for FOL, that we call predicate bisimulation and first-order bisimulation. They refine the usual notions of isomorphism and partial isomorphism. We prove generalizations as well as new versions of the Keisler theorems for countable languages in which isomorphisms are replaced by predicate bisimulations and first-order bisimulations