Generation, Validation, and Selection: A Proof-by-Contradiction Reasoning Chain Data Synthesis Method for Enhancing Indirect Reasoning in Large Language Models

Reasoning ability has attracted extensive attention due to its importance in large language models (LLMs). Most existing approaches for enhancing reasoning focus on direct reasoning while overlooking indirect reasoning, which can be more effective for certain complex problems. A recent study introduced proof by contradiction (PbC) into prompts to guide LLMs in performing indirect reasoning. To fundamentally strengthen this ability, we adopt fine-tuning driven by high-quality big data to explicitly teach LLMs how to conduct indirect reasoning. For data construction, we design a PbC reasoning chain data synthesis method consisting of three stages: generation, validation, and selection, which are responsible for producing data in the required format, verifying correctness, and selecting data suitable for PbC. Using this method, we synthesized a dataset of 3439 examples with input-output pairs and corresponding PbC reasoning chains. Experiments on both reasoning models and non-reasoning models show that fine-tuning with our dataset significantly improves LLMs' indirect reasoning capabilities, while also yielding gains in direct reasoning.