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
arXiv:2607. 21199v1 Announce Type: cross Abstract: Grounding, the translation of high-level theories into equivalent quantifier-free formulas, is a crucial step in declarative solving, yet it has so far escaped the proof-logging revolution.
By Daimy Van Caudenberg, Alexander Ek, Carlos Cantero, Bart Bogaerts
arXiv:2609.26806v2 Announce Type: replace-cross
Abstract: This paper presents a complete, structure-preserving port to Lean 4 of the Isabelle/HOL dataset accompanying Benzm\"uller and Scott's study o...
By Christoph Benzm\"uller
Grounding, the translation of high-level theories into equivalent quantifier-free formulas, is a crucial step in declarative solving, yet it has so far escaped the proof-logging revolution. When this grounding step is not certifying, there is no way of knowing that the obtained solutions actually correspond to the original problem specification, resulting in a trust gap.
arXiv:2608. 14004v1 Announce Type: new Abstract: In-context learning is commonly formalized as inference from examples of a function.
By Faizanuddin Ansari, Debanjan Dutta, Swagatam Das
The paper introduces GUARD, a neuro‑symbolic system that autoformalizes argumentative material by completing missing premises (guards) before formal verification. It uses large language models to generate candidate guards, Isabelle/HOL to verify them, and a contrastive test to ensure the proof depends on the original premises and does not over‑generalize. Experiments on Debatepedia and ARCT show that GUARD improves verified‑faithful scores by over 30 points and reduces leakage by about 20 points compared to prior LLM‑driven theorem proving methods.
By Xin Quan, Reto Gubelmann, Andr\'e Freitas
arXiv:2602.02940v2 Announce Type: replace-cross
Abstract: Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model...
By Daniel Quigley