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.

The advancement of Large Language Models (LLMs) has made ensuring their trustworthiness increasingly critical, especially in terms of fairness across diverse human groups. While modern LLMs are aligned with user preferences through Reinforcement Learning from Human Feedback (RLHF), the reward models used for alignment are trained on preference data that may both reflect societal biases and suffer from demographic skewness, as labeler populations are often uneven due to systemic accessibility or participation gaps. In this work, we reveal that reward models can exhibit significant discrepancies across different demographic groups, posing a fundamental challenge to fair and robust alignment. Using real-world datasets, we conduct the most comprehensive study to date, auditing various state-of-the-art reward models across nine sensitive attributes, including age, gender, ethnicity, etc. Our evaluation spans both (1) the agreement level between reward models and specific user groups, and (2) the reward model’s preference toward responses associated with different groups. Based on these findings, we propose the first method to mitigate group disparities in reward modeling. Code is available at https://github.com/Violet24K/FaRM.