First-Order Modal Logic in HOL: Deep and Shallow Embeddings with Automated Faithfulness (Extended Preprint)

2026-07-12

AI Summary

This work addresses the challenge of formalizing first-order modal logic within higher-order logic under constant-domain Kripke semantics, where uncountable individual domains complicate the surjectivity of variable assignments. Three embedding approaches—deep embedding, heavyweight maximal shallow embedding, and lightweight minimal shallow embedding—are proposed in Isabelle/HOL and uniformly modeled via parameterized locales. The study extends deep–shallow embedding techniques from propositional to first-order modal logic for the first time, mechanizing the countable downward Löwenheim–Skolem theorem to handle uncountable domains. It also implements capture-avoiding substitution and α-renaming mechanisms to support quantifier reasoning. Finally, an automatic soundness proof is established between the deep and minimal shallow embeddings, guaranteeing their equivalence under constant-domain semantics.

Abstract

We extend, in Isabelle/HOL, the deep-and-shallow embedding methodology of our prior work from propositional to first-order modal logic (FML) with constant-domain Kripke semantics. Three embeddings of FML into classical higher-order logic (HOL) are provided side by side: a deep embedding, a heavyweight maximal-shallow embedding, and a lightweight minimal-shallow embedding. The minimal-shallow embedding is presented as an Isabelle/HOL locale, parametrised by an accessibility relation, a world-indexed interpretation, a universe of worlds, and a variable assignment; the locale form admits a global faithfulness theorem, stating that quantifying over all minimal-shallow interpretations recovers exactly deep validity. A central technical contribution is a mechanisation, for FML under constant-domain Kripke semantics, of the (countable) downward Löwenheim-Skolem theorem, which underpins the automation of our faithfulness proof between the deep and minimal-shallow embeddings. Deploying it inside an extension of the minimal-shallow locale resolves the surjectivity problem that arises against an uncountable domain of individuals -- where the locale's variable assignment, having countable domain V = nat, cannot be surjective onto the domain -- and thereby yields faithfulness over the full domain. Since prior work treats only the propositional fragment, we develop here the substitution machinery (free/bound-variable predicates, the fresh-variable function, capture-avoiding substitution, alphabetic renaming, the substitutability predicate, the substitution lemma, and size-based induction principles) needed for the first-order quantifiers.

Problem

• first-order modal logic
• constant-domain Kripke semantics
• faithfulness
• shallow embedding
• downward Löwenheim-Skolem theorem

Innovation

• first-order modal logic
• shallow embedding
• Löwenheim-Skolem theorem
• Isabelle/HOL
• faithfulness