Retrieval-Augmented Generation (RAG) is widely used to ground large language models (LLMs) in external knowledge and improve factual accuracy. Prior work has explored iterative and self-reflective mechanisms to refine reasoning, but these approaches rely on internal model judgment and lack formally grounded, verifiable feedback. As a result, RAG systems may still produce logically inconsistent or contradictory answers in multi-step reasoning. In this paper, we propose LCR-RAG, a framework that integrates neuro-symbolic verification with reinforcement learning to explicitly optimize logical consistency. The core of our approach is a Logic-Consistency-driven Reward (LCR), which converts discrete logical signals—such as contradictions or incomplete inference chains—into a structured reward signal. This reward guides a PPO-based agent to iteratively rewrite queries and correct reasoning errors. Experiments on HotpotQA, ASQA, and TriviaQA show that LCR-RAG consistently outperforms strong RAG baselines, with ablation results indicating that the LCR mechanism is the primary source of improvement, even under noisy or conflicting retrieval conditions.