In-depth logical reasoning in domains such as legal analysis remains challenging for large language models (LLMs), exhibiting brittle and inconsistent performance on tasks requiring multi-step deductive inference. This limitation is expected, as LLMs primarily rely on learned distributional associations rather than performing explicit step-by-step logical inference through formal rule application. To address this gap, we propose a novel framework, called PercepLog, that incorporates a classical deductive reasoning algorithm, backward chaining, within an outer control loop, while utilizing LLMs for primitive operations such as substitution and unification. By following the backward chaining algorithm, the framework benefits from the interpretability and explainability of symbolic reasoning procedures while retaining the flexibility of LLM perception for processing natural-language legal text. We evaluate the proposed method on statutory reasoning tasks across a range of legal domains as well general logical reasoning. The results demonstrate that our approach enables more robust, explainable, and principled reasoning over unstructured legal text, where exhaustive formalization is often brittle and prone to hallucination. Our method yields average relative performance improvements of 58% over direct LLM reasoning and 16% over formal LLM-based translation and Prolog inference pipelines.