PEIRCE: Unifying Material and Formal Reasoning via LLM-Driven Neuro-Symbolic Refinement

A persistent challenge in AI is the effective integration of material and formal inference - the former concerning the plausibility and contextual relevance of arguments, while the latter focusing on their logical and structural validity. Large Language Models (LLMs), by virtue of their extensive pre-training on large textual corpora, exhibit strong capabilities in material inference. However, their reasoning often lacks formal rigour and verifiability. At the same time, LLMs' linguistic competence positions them as a promising bridge between natural and formal languages, opening up new opportunities for combining these two modes of reasoning. In this paper, we introduce PEIRCE, a neuro-symbolic framework designed to unify material and formal inference through an iterative conjecture-criticism process. Within this framework, LLMs play the central role of generating candidate solutions in natural and formal languages, which are then evaluated and refined via interaction with external critique models. These critiques include symbolic provers, which assess formal validity, as well as soft evaluators that measure the quality of the generated arguments along linguistic and epistemic dimensions such as plausibility, coherence, and parsimony. While PEIRCE is a general-purpose framework, we demonstrate its capabilities in the domain of natural language explanation generation - a setting that inherently demands both material adequacy and formal correctness.

Improving Chain-of-Thought Reasoning via Quasi-Symbolic Abstractions

Chain-of-Though (CoT) represents a common strategy for reasoning in Large Language Models (LLMs) by decomposing complex tasks into intermediate inference steps. However, explanations generated via CoT are susceptible to content biases that negatively affect their robustness and faithfulness. To mitigate existing limitations, recent work has proposed using logical formalisms coupled with external symbolic solvers. However, fully symbolic approaches possess the bottleneck of requiring a complete translation from natural language to formal languages, a process that affects efficiency and flexibility. To achieve a trade-off, this paper investigates methods to disentangle content from logical reasoning without a complete formalisation. In particular, we present QuaSAR (for Quasi-Symbolic Abstract Reasoning), a variation of CoT that guides LLMs to operate at a higher level of abstraction via quasi-symbolic explanations. Our framework leverages the capability of LLMs to formalise only relevant variables and predicates, enabling the coexistence of symbolic elements with natural language. We show the impact of QuaSAR for in-context learning and for constructing demonstrations to improve the reasoning capabilities of smaller models. Our experiments show that quasi-symbolic abstractions can improve CoT-based methods by up to 8% accuracy, enhancing robustness and consistency on challenging adversarial variations on both natural language (i.e. MMLU-Redux) and symbolic reasoning tasks (i.e., GSM-Symbolic).

Formalizing Complex Mathematical Statements with LLMs: A Study on Mathematical Definitions

Thanks to their linguistic capabilities, LLMs offer an opportunity to bridge the gap between informal mathematics and formal languages through autoformalization. However, it is still unclear how well LLMs generalize to sophisticated and naturally occurring mathematical statements. To address this gap, we investigate the task of autoformalizing real-world mathematical definitions -- a critical component of mathematical discourse. Specifically, we introduce two novel resources for autoformalisation, collecting definitions from Wikipedia (Def_Wiki) and arXiv papers (Def_ArXiv). We then systematically evaluate a range of LLMs, analyzing their ability to formalize definitions into Isabelle/HOL. Furthermore, we investigate strategies to enhance LLMs' performance including refinement through external feedback from Proof Assistants, and formal definition grounding, where we guide LLMs through relevant contextual elements from formal mathematical libraries. Our findings reveal that definitions present a greater challenge compared to existing benchmarks, such as miniF2F. In particular, we found that LLMs still struggle with self-correction, and aligning with relevant mathematical libraries. At the same time, structured refinement methods and definition grounding strategies yield notable improvements of up to 16% on self-correction capabilities and 43% on the reduction of undefined errors, highlighting promising directions for enhancing LLM-based autoformalization in real-world scenarios.

Integrating Expert Knowledge into Logical Programs via LLMs

This paper introduces ExKLoP, a novel framework designed to evaluate how effectively Large Language Models (LLMs) integrate expert knowledge into logical reasoning systems. This capability is especially valuable in engineering, where expert knowledge-such as manufacturer-recommended operational ranges-can be directly embedded into automated monitoring systems. By mirroring expert verification steps, tasks like range checking and constraint validation help ensure system safety and reliability. Our approach systematically evaluates LLM-generated logical rules, assessing both syntactic fluency and logical correctness in these critical validation tasks. We also explore the models capacity for self-correction via an iterative feedback loop based on code execution outcomes. ExKLoP presents an extensible dataset comprising 130 engineering premises, 950 prompts, and corresponding validation points. It enables comprehensive benchmarking while allowing control over task complexity and scalability of experiments. We leverage the synthetic data creation methodology to conduct extensive empirical evaluation on a diverse set of LLMs including Llama3, Gemma, Mixtral, Mistral, and Qwen. Results reveal that while models generate nearly perfect syntactically correct code, they frequently exhibit logical errors in translating expert knowledge. Furthermore, iterative self-correction yields only marginal improvements (up to 3%). Overall, ExKLoP serves as a robust evaluation platform that streamlines the selection of effective models for self-correcting systems while clearly delineating the types of errors encountered. The complete implementation, along with all relevant data, is available at GitHub.

Eliciting Critical Reasoning in Retrieval-Augmented Language Models via Contrastive Explanations

Retrieval-augmented generation (RAG) has emerged as a critical mechanism in contemporary NLP to support Large Language Models(LLMs) in systematically accessing richer factual context. However, the integration of RAG mechanisms brings its inherent challenges, as LLMs need to deal with potentially noisy contexts. Recent studies have shown that LLMs still struggle to critically analyse RAG-based in-context information, a limitation that may lead to incorrect inferences and hallucinations. In this paper, we investigate how to elicit critical reasoning in RAG via contrastive explanations. In particular, we propose Contrastive-RAG (C-RAG), a framework that (i) retrieves relevant documents given a query, (ii) selects and exemplifies relevant passages, and (iii) generates explanations that explicitly contrast the relevance of the passages to (iv) support the final answer. We show the impact of C-RAG building contrastive reasoning demonstrations from LLMs to instruct smaller models for retrieval-augmented tasks. Extensive experiments demonstrate that C-RAG improves state-of-the-art RAG models while (a) requiring significantly fewer prompts and demonstrations and (b) being robust to perturbations in the retrieved documents.

SylloBio-NLI: Evaluating Large Language Models on Biomedical Syllogistic Reasoning

Syllogistic reasoning is crucial for Natural Language Inference (NLI). This capability is particularly significant in specialized domains such as biomedicine, where it can support automatic evidence interpretation and scientific discovery. This paper presents SylloBio-NLI, a novel framework that leverages external ontologies to systematically instantiate diverse syllogistic arguments for biomedical NLI. We employ SylloBio-NLI to evaluate Large Language Models (LLMs) on identifying valid conclusions and extracting supporting evidence across 28 syllogistic schemes instantiated with human genome pathways. Extensive experiments reveal that biomedical syllogistic reasoning is particularly challenging for zero-shot LLMs, which achieve an average accuracy between 70% on generalized modus ponens and 23% on disjunctive syllogism. At the same time, we found that few-shot prompting can boost the performance of different LLMs, including Gemma (+14%) and LLama-3 (+43%). However, a deeper analysis shows that both techniques exhibit high sensitivity to superficial lexical variations, highlighting a dependency between reliability, models' architecture, and pre-training regime. Overall, our results indicate that, while in-context examples have the potential to elicit syllogistic reasoning in LLMs, existing models are still far from achieving the robustness and consistency required for safe biomedical NLI applications.

Reasoning with Natural Language Explanations

Explanation constitutes an archetypal feature of human rationality, underpinning learning and generalisation, and representing one of the media supporting scientific discovery and communication. Due to the importance of explanations in human reasoning, an increasing amount of research in Natural Language Inference (NLI) has started reconsidering the role that explanations play in learning and inference, attempting to build explanation-based NLI models that can effectively encode and use natural language explanations on downstream tasks. Research in explanation-based NLI, however, presents specific challenges and opportunities, as explanatory reasoning reflects aspects of both material and formal inference, making it a particularly rich setting to model and deliver complex reasoning. In this tutorial, we provide a comprehensive introduction to the field of explanation-based NLI, grounding this discussion on the epistemological-linguistic foundations of explanations, systematically describing the main architectural trends and evaluation methodologies that can be used to build systems capable of explanatory reasoning.

A Mechanistic Interpretation of Syllogistic Reasoning in Auto-Regressive Language Models

Recent studies on logical reasoning in auto-regressive Language Models (LMs) have sparked a debate on whether such models can learn systematic reasoning principles during pre-training or merely exploit superficial patterns in the training data. This paper presents a mechanistic interpretation of syllogistic reasoning in LMs to further enhance our understanding of internal dynamics. Specifically, we present a methodology for circuit discovery aimed at disentangling content-independent reasoning mechanisms from world knowledge acquired during pre-training. Through two distinct intervention methods, we uncover a sufficient and necessary circuit involving middle-term suppression that elucidates how LMs transfer information to derive valid conclusions from premises. Furthermore, we investigate how belief biases manifest in syllogistic reasoning, finding evidence of partial contamination from additional attention heads responsible for encoding commonsense and contextualized knowledge. Finally, we explore the generalization of the discovered mechanisms across various syllogistic schemes and model sizes, finding that the identified circuit is sufficient and necessary for all the schemes on which the model achieves high downstream accuracy ($\geq$ 60\%). Overall, our findings suggest that LMs indeed learn transferable content-independent reasoning mechanisms, but that, at the same time, such mechanisms do not involve generalisable and abstract logical primitives, being susceptible to contamination by the same world knowledge acquired during pre-training.

Verification and Refinement of Natural Language Explanations through LLM-Symbolic Theorem Proving

Natural language explanations have become a proxy for evaluating explainable and multi-step Natural Language Inference (NLI) models. However, assessing the validity of explanations for NLI is challenging as it typically involves the crowd-sourcing of apposite datasets, a process that is time-consuming and prone to logical errors. To address existing limitations, this paper investigates the verification and refinement of natural language explanations through the integration of Large Language Models (LLMs) and Theorem Provers (TPs). Specifically, we present a neuro-symbolic framework, named Explanation-Refiner, that augments a TP with LLMs to generate and formalise explanatory sentences and suggest potential inference strategies for NLI. In turn, the TP is employed to provide formal guarantees on the logical validity of the explanations and to generate feedback for subsequent improvements. We demonstrate how Explanation-Refiner can be jointly used to evaluate explanatory reasoning, autoformalisation, and error correction mechanisms of state-of-the-art LLMs as well as to automatically enhance the quality of human-annotated explanations of variable complexity in different domains.

SemEval-2024 Task 2: Safe Biomedical Natural Language Inference for Clinical Trials

Large Language Models (LLMs) are at the forefront of NLP achievements but fall short in dealing with shortcut learning, factual inconsistency, and vulnerability to adversarial inputs.These shortcomings are especially critical in medical contexts, where they can misrepresent actual model capabilities. Addressing this, we present SemEval-2024 Task 2: Safe Biomedical Natural Language Inference for ClinicalTrials. Our contributions include the refined NLI4CT-P dataset (i.e., Natural Language Inference for Clinical Trials - Perturbed), designed to challenge LLMs with interventional and causal reasoning tasks, along with a comprehensive evaluation of methods and results for participant submissions. A total of 106 participants registered for the task contributing to over 1200 individual submissions and 25 system overview papers. This initiative aims to advance the robustness and applicability of NLI models in healthcare, ensuring safer and more dependable AI assistance in clinical decision-making. We anticipate that the dataset, models, and outcomes of this task can support future research in the field of biomedical NLI. The dataset, competition leaderboard, and website are publicly available.