Intuitionistic K is a Bisimulation-Invariant Fragment of Intuitionistic First-Order Logic

2026-06-30

AI Summary

This study investigates the expressive power of intuitionistic modal logic IK within intuitionistic first-order logic. By introducing a notion of IK-bisimulation and combining relational semantics with model-theoretic techniques—including an intuitionistic version of Łoś’s theorem, elementary embeddings, and countable saturation—the work provides the first intrinsic characterization of IK: it precisely captures the fragment of intuitionistic first-order logic invariant under IK-bisimulation. This result establishes the exact model-theoretic status of IK and yields a Hennessy–Milner theorem in the intuitionistic setting, thereby furnishing a crucial tool for the development of intuitionistic first-order model theory.

Abstract

We define the notion of IK-bisimulation between the relational semantics for the intuitionistic modal logic IK, and prove that IK arises as the IK-bisimulation-invariant fragment of intuitionistic first-order logic. En route, we provide an intrinsic characterisation result of this logic by way of a Hennessy-Milner-style theorem and develop some intuitionistic first-order model theory, including intuitionistic analogues of Los's Theorem, elementary embeddings and countable saturation.

Problem

• intuitionistic logic
• bisimulation invariance
• first-order logic
• modal logic
• model theory

Innovation

• intuitionistic modal logic
• bisimulation-invariance
• first-order model theory
• Hennessy-Milner theorem
• Łoś's Theorem