Cover Semantics for Intuitionistic Modalities

2026-07-13

AI Summary

Traditional Kripke semantics relies on classical reasoning, making it difficult to formalize within constructive type theory. While Goldblatt’s covering semantics offers a constructive alternative, it is constrained by a “modal locality” condition that complicates model construction. This work proposes a conservative extension of relational covering semantics that eliminates this restriction, enabling simpler and more standard model constructions and accommodating various intuitionistic modal logics featuring independent □ and ◇ operators. Building on this framework, we provide a constructive completeness proof that avoids intricate order-theoretic arguments and fully formalize both the semantics and multiple logical systems in Agda. This significantly enhances the applicability and formalizability of modal semantics within constructive settings.

Abstract

Intuitionistic modal logic (IML) has inspired several developments in programming languages including modal type systems for staging, computational effects and language-based security. IMLs are typically studied using Kripke-style relational semantics, which simplifies proofs of meta-theoretic properties, such as completeness and consistency, by making it easy to construct models. Kripke-style relational semantics, however, relies upon classical reasoning principles, which makes it unappealing from a computational perspective and unsuitable for formalization in a constructive type theory. Goldblatt provides an alternative semantics for IMLs by extending Beth-Kripke-Joyal-style "cover" semantics for intuitionistic propositional logic with relations to support modalities. Goldblatt's "relational cover" semantics overcomes classical reasoning but introduces a new limitation: it relies upon a "modal localization" condition that restricts the class of models and complicates model construction. Goldblatt bypasses this restriction by using intricate order-theoretic completion arguments to prove completeness. In this article, we present a conservative extension of relational cover semantics that alleviates this restriction and is amenable to simpler and standard model construction techniques. We formalize our semantics in Agda and prove completeness constructively in the style of Normalization by Evaluation for a variety of IMLs featuring independent box and diamond modalities.

Problem

• Intuitionistic modal logic
• Cover semantics
• Modal localization
• Constructive semantics
• Model construction

Innovation

• relational cover semantics
• intuitionistic modal logic
• constructive completeness
• modal localization
• Normalization by Evaluation