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

arXiv: https://arxiv.org/abs/2504.04110 · HTML: https://mlsanity.com/paper/2504.04110/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 7 of its 56 references that are in the map.
- First posted on arXiv 5 Apr 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 7 of 7)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Language models show human-like content effects on reasoning tasks](https://mlsanity.com/paper/2207.07051/) (2022)
- [Learning Reasoning Strategies in End-to-End Differentiable Proving](https://mlsanity.com/paper/2007.06477/) (2020)
- [Empowering LLMs with Logical Reasoning: A Comprehensive Survey](https://mlsanity.com/paper/2502.15652/) (2025)
- [Inductive Learning of Logical Theories with LLMs: An Expressivity-Graded Analysis](https://mlsanity.com/paper/2408.16779/) (2024)

## Built on by (most cited first, 7 of 7)

- [Mitigating Content Effects on Reasoning in Language Models through Fine-Grained Activation Steering](https://mlsanity.com/paper/2505.12189/) (2025)
- [Feature Extraction and Steering for Enhanced Chain-of-Thought Reasoning in Language Models](https://mlsanity.com/paper/2505.15634/) (2025)
- [PatchWorld: Gradient-Free Optimization of Executable World Models for Agent Environments](https://mlsanity.com/paper/2605.30880/) (2026)
- [A DbC Inspired Neurosymbolic Layer for Trustworthy Agent Design](https://mlsanity.com/paper/2508.03665/) (2025)
- [Inferring Latent Intentions: Attributional Natural Language Inference in LLM Agents](https://mlsanity.com/paper/2601.08742/) (2026)
- [Decompose-and-Formalise: Recursively Verifiable Natural Language Inference](https://mlsanity.com/paper/2601.19605/) (2026)
- [ANCHOR-RE: An Agentic Neuro-Symbolic Framework for Grounded Biomedical Relation Extraction](https://mlsanity.com/paper/2608.03154/) (2026)

## Abstract (arXiv)

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.