# Discovering a Shared Logical Subspace: Steering LLM Logical Reasoning via Alignment of Natural-Language and Symbolic Views

arXiv: https://arxiv.org/abs/2604.19716 · HTML: https://mlsanity.com/paper/2604.19716/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 20 of its 24 references that are in the map.
- First posted on arXiv 21 Apr 2026.

## Placed in

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

## Builds on (most cited first, 20 of 20)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Logic-LM: Empowering Large Language Models with Symbolic Solvers for Faithful Logical Reasoning](https://mlsanity.com/paper/2305.12295/) (2023)
- [Language Models Are Greedy Reasoners: A Systematic Formal Analysis of Chain-of-Thought](https://mlsanity.com/paper/2210.01240/) (2022)
- [ProofWriter: Generating Implications, Proofs, and Abductive Statements over Natural Language](https://mlsanity.com/paper/2012.13048/) (2020)
- [Large Language Model Guided Tree-of-Thought](https://mlsanity.com/paper/2305.08291/) (2023)
- [ReClor: A Reading Comprehension Dataset Requiring Logical Reasoning](https://mlsanity.com/paper/2002.04326/) (2020)
- [Evaluating the Logical Reasoning Ability of ChatGPT and GPT-4](https://mlsanity.com/paper/2304.03439/) (2023)
- [LINC: A Neurosymbolic Approach for Logical Reasoning by Combining Language Models with First-Order Logic Provers](https://mlsanity.com/paper/2310.15164/) (2023)
- [Are Large Language Models Really Good Logical Reasoners? A Comprehensive Evaluation and Beyond](https://mlsanity.com/paper/2306.09841/) (2023)
- [Empowering LLMs with Logical Reasoning: A Comprehensive Survey](https://mlsanity.com/paper/2502.15652/) (2025)
- [Dissecting Logical Reasoning in LLMs: A Fine-Grained Evaluation and Supervision Study](https://mlsanity.com/paper/2506.04810/) (2025)
- [Boosting Logical Fallacy Reasoning in LLMs via Logical Structure Tree](https://mlsanity.com/paper/2410.12048/) (2024)
- [Enhancing Logical Reasoning in Language Models via Symbolically-Guided Monte Carlo Process Supervision](https://mlsanity.com/paper/2505.20415/) (2025)

## Abstract (arXiv)

Large Language Models (LLMs) still struggle with multi-step logical reasoning. Existing approaches either purely refine the reasoning chain in natural language form or attach a symbolic solver as an external module. In this work, we instead ask whether LLMs contain a shared internal logical subspace that simultaneously aligns natural-language and symbolic-language views of the reasoning process. Our hypothesis is that this logical subspace captures logical reasoning capabilities in LLMs that are shared across views while remaining independent of surface forms. To verify this, we employ Canonical Correlation Analysis on the paired residual activations from natural-language and symbolic-language reasoning chains, learning a low-dimensional subspace with maximum cross-view correlation. Furthermore, we design a training-free approach that steers LLMs reasoning chain along this logical subspace, thereby leveraging the complementary reasoning signals from both views. Experiments on four logical reasoning benchmarks demonstrate the effectiveness of our approach, improving accuracy by up to 11 percentage points and generalizing well on out-of-domain problems.