Abstract
AbstractThe celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a ‘continuous’ binary relation. The proof of our theorem generalizes Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem.1
Highlights
Kripke models are relational structures that provide standard models for modal logic
Bisimulations are relations on Kripke models that are indistinguishable by modal logic in the sense that the truth of modal formulae is preserved under these relations
This is often formulated more succinctly by saying that modal logic is the bisimulation-invariant fragment of first-order logic
Summary
Kripke models are relational structures that provide standard models for modal logic. The theorem has inspired many generalizations and alternative versions, including a similar characterization for intuitionistic logic [24, 25], neighbourhood models [16] and numerous coalgebraic generalizations [29] Another notable result is the Janin–Walukiewicz theorem [19], showing that the modal μ-calculus is the bisimulation-invariant fragment of monadic second-order logic. A result that is of particular importance to this paper was given by Rosen in [27], showing that over finite models, too, modal logic is the bisimulation-invariant fragment of first-order logic This is remarkable because the compactness theorem of first-order logic features prominently in the proof of the original van Benthem characterization theorem, while the class of finite models crucially lacks the compactness property. The final section contains a brief summary of the paper and points to a number of possible new research directions
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