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.
arXiv:2607. 10880v1 Announce Type: new 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.
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.
Recent work in defeasible reasoning has seen notions of preferential semantics and entailment in the style of Kraus et al. applied to modal logics.
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.
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."
The paper introduces the Evidential-Based Higher-Order Set Argumentation Framework (EHSAF), a unified formalism that extends Dung’s abstract argumentation by incorporating evidential support, higher-order relations, and collective interactions. Two complete semantics are defined: an adjacent complete labelling semantics allowing multiple truth values for arguments in support cycles, and an extension-based complete semantics that accepts only well‑founded support chains. The authors provide a propositional encoding in three‑valued Łukasiewicz logic and extend it to continuous fuzzy logics, proving key properties and showing equivalence under support‑acyclicity.
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.
arXiv:2606. 20227v1 Announce Type: new Abstract: Large Language Models (LLMs) have made significant progress in reasoning, particularly in deductive reasoning, which is crucial for high-stakes decision-making.
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.
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.
arXiv:2607. 09729v1 Announce Type: new Abstract: In his 1996 doctoral thesis, Maurice Pagnucco created the first AGM-like abductive expansion operation.