The paper presents three Isabelle/HOL embeddings of monadic second‑order logic (MSO): a deep embedding, a maximal‑shallow embedding, and a minimal‑shallow embedding that collapses formulas to bool. It introduces a two‑sorted substitution system that ensures capture‑avoiding substitution and proves the faithfulness of all embeddings. A fully mechanised two‑sorted downward Löwenheim‑Skolem theorem is established, showing that the minimal embedding recovers deep validity relative to countable assignments and aligns with both the general (Henkin‑style) and standard readings of MSO, while also demonstrating differences in classical MSO properties across the embeddings.
By Christoph Benzmueller, Daniel Kirchner
arXiv:2606. 03655v1 Announce Type: new Abstract: Recent work in defeasible reasoning has seen notions of preferential semantics and entailment in the style of Kraus et al.
By Nicholas Leisegang, Thomas Meyer, Ivan Varzniczak
arXiv:2609.36279v1 Announce Type: cross
Abstract: The shallow embedding of higher-order modal logic in classical higher-order logic, used in Benzm\"uller and Scott's Notes on G\"odel's and Scott's va...
By Christoph Benzm\"uller
Recent work in defeasible reasoning has seen notions of preferential semantics and entailment in the style of Kraus et al. applied to modal logics.
This paper reports a full, structure‑preserving port of the Isabelle/HOL dataset on G"odel’s and Scott’s modal ontological arguments to Lean 4. The port consists of 30 Lean 4 modules that mirror the original theories in section structure, declaration order, and naming, and a comparison tool confirms that all 548 statements are identical. The Lean 4 development reproduces every proof from the Isabelle/HOL version—including the inconsistency of G"odel’s 1970 axioms, the repaired variants, Scott’s variant, modal collapse, monotheism, and the ultrafilter property—while also proving five previously unproven statements and documenting the remaining 45 as unresolved.
"whyItMatters":"The work demonstrates that Lean 4 can faithfully replicate a complex formal development from Isabelle/HOL, providing a new, lean‑based foundation for further exploration of modal ontological arguments."
By Christoph Benzm\"uller
arXiv:2606. 08503v1 Announce Type: new Abstract: In this paper, we integrate the defeasible logic of Kraus, Lehmann and Magidor (KLM) with the standpoint logic framework of G\'omez \'Alvarez and Rudolph.
By Nicholas Leisegang, Thomas Meyer, Sebastian Rudolph