Technical Report -- A Context-Sensitive Multi-Level Similarity Framework for First-Order Logic Arguments: An Axiomatic Study
Authors: Victor David, Jérôme Delobelle, Jean-Guy Mailly
Organizations: University Cˆote d’Azur, Inria, CNRS, I3S, Sophia-Antipolis, France · Universit´e Paris Cit´e, LIPADE, Paris, France · Universit´e Toulouse Capitole, IRIT, Toulouse, France
Similarity in formal argumentation has recently gained attention due to its significance in problems such as argument aggregation in semantics and enthymeme decoding. While existing approaches focus on propositional logic, we address the richer setting of First-Order Logic (FOL), where similarity must account for structured content. We introduce a comprehensive framework for FOL argument similarity, built upon: (1) an extended axiomatic foundation; (2) a four-level parametric model covering predicates, literals, clauses, and formulae similarity; (3) two model families, one syntax-sensitive via language models, both integrating contextual weights for nuanced and explainable similarity; and (4) formal constraints enforcing desirable properties.
Figures & tables
Figure 1: Histogram explaining the formulae similarity for [ CNF_T1 , CNF_T2 ] (green line), based on weighted clause similarities (blue) and their proportion of importance (red).
Figure 2: Similarity on the premises of A1 vs A2 where 1=α1 , 2=α2 , 3=α3 , 4=α4 , 5=α5 , 6=α6 , 7=α7 , 8=α8 and 9=α9 on the x-axis.
Figure 3: Similarity on the claims of A1 vs A2 where 1=β1 and 2=β2 on the x-axis.
Figure 4: Influence of η on argument similarity: higher values prioritize the support, lower values the claim.
Figure 5: Similarity on the premises of A2 vs A3 where 1=α5 , 2=α6 , 3=α7 , 4=α8 , 5=α9 , 6=α10 , 7=α11 , 8=α12 , 9=α13 and 10=α14 on the x-axis.
Figure 6: Similarity on the claims of A2 vs A3 where 1=β2 and 2=β3 on the x-axis.
We propose a new framework for machine-learning-oriented argument analysis tasks. Our proposal involves replacing traditional contextualized word embeddings used in most NLP tasks with logical embeddings, an alternative encoding that directly exploits argumentation structures. In essence, logical embeddings encapsulate the logical semantics of an argument, allowing for a better representation of its meaning. Supporting these embeddings is a mathematical logic-based similarity measure that offers a transparent notion of proximity and is guaranteed to satisfy several desirable theoretical properties that current cosine similarity-based contextualized word embeddings cannot assure. This similarity measure induces a positive semi-definite kernel on the set of arguments, enabling us to uniquely define logical embeddings using the theory of Reproducing Kernel Hilbert Spaces (RKHS). Moreover, we prove that this encoding is optimal, in the sense that no logical information is lost in the process. As with other RKHS applications, logical embeddings can be used in numerous supervised and unsupervised tasks. We provide an implementation of the method and aim to test it against literature benchmarks. Additionally, we demonstrate that logical embeddings outperform most standard embedding methods on a classification task.
Leander Heldring, Santiago Torres
Kellogg School of Management Northwestern University 2211 Campus Drive, Evanston, IL 60208 · Department of Economics MIT 50 Memorial Drive, Cambridge, MA 02142
Large Language Models (LLMs) achieve strong performance in analyzing and generating text, yet they struggle with explicit, transparent, and verifiable reasoning over complex texts such as those containing debates. In particular, they lack structured representations that capture how arguments support or attack each other and how their relative strengths determine overall acceptability. We encompass these limitations by proposing a framework that integrates learning-based argument mining with quantitative reasoning and ontology-based querying. Starting from a raw debate text, the framework extracts a fuzzy argumentative knowledge base, where arguments are explicitly represented as entities, linked by attack and support relations, and annotated with initial fuzzy strengths reflecting plausibility w.r.t. the debate's context. Quantitative argumentation semantics are then applied to compute final argument strengths by propagating the effects of supports and attacks. These results are then embedded into a fuzzy description logic setting, enabling expressive query answering through efficient rewriting techniques. The proposed approach provides a transparent, explainable, and formally grounded method for analyzing debates, overcoming purely statistical LLM-based analyses.
Gianvincenzo Alfano, Sergio Greco, Lucio La Cava +2
Department of Informatics, Modeling, Electronics and System Engineering, University of Calabria, Italy
Large Language Models (LLMs) have renewed interest in autoformalization. Yet, when First-Order Logic (FOL) is considered as the target formalism, the field still lacks a unified task formulation and a systematic survey. This paper addresses this gap: we first provide a principled definition for the FOL-autoformalization task by distinguishing Ontology Extraction from Logical Translation, showing how their conflation obscures (cross-study) evaluation; we review existing datasets, evaluation metrics, and LLM-based methods, including fine-tuning, prompting, and verification-based refinement; we identify open challenges in benchmarking, semantic evaluation, ontology-aware methods, and end-to-end applications.