Learning to Prove, Not Just to Answer: Reinforcement Learning from Formal Verification for Natural-Language Logical Reasoning

Large language models (LLMs) are increasingly deployed for natural-language logical reasoning, where the final answer is easy to check but the proof behind it is not. In natural-language logical reasoning, intermediate conclusion should follow from its premises, and the resulting derivation should support the final answer. Existing methods lack machine-checkable verification of intermediate conclusions and answer-supporting proof dependencies, so they may assign credit to invalid or answer-irrelevant steps. We propose Proof-R1, a RL framework from formal verification that trains LLMs to construct verifiable proofs for natural-language logical reasoning. Proof-R1 admits a generated conclusion into the verified proof state only when the corresponding reasoning action satisfie proof obligations through UNSAT-based machine-checkable formal verification. Proof-R1 also recovers the answer-supporting dependency closure to trace the proof structure of the final answer and align outcome credit with the proof dependencies. Experiments demonstrate that Proof-R1 improves answer accuracy across three logical reasoning benchmarks and four backbone models, and outperforms training-free agents and training-based methods in terms of reasoning-process verifiability.