Large Language Models as an Indirect Reasoner: Contrapositive and Contradiction for Automated Reasoning

Recently, increasing attention has been focused drawn on to improve the ability of Large Language Models (LLMs) to perform complex reasoning. However, previous methods, such as Chain-of-Thought and Self-Consistency, mainly follow Direct Reasoning (DR) frameworks, so they will meet difficulty in solving numerous real-world tasks which can hardly be solved via DR. Therefore, to strengthen the reasoning power of LLMs, this paper proposes a novel Indirect Reasoning (IR) method that employs the logic of contrapositives and contradictions to tackle IR tasks such as factual reasoning and mathematic proof. Specifically, our methodology comprises two steps. Firstly, we leverage the logical equivalence of contrapositive to augment the data and rules to enhance the comprehensibility of LLMs. Secondly, we design a set of prompt templates to trigger LLMs to conduct IR based on proof by contradiction that is logically equivalent to the original DR process. Our IR method is simple yet effective and can be straightforwardly integrated with existing DR methods to further boost the reasoning abilities of LLMs. The experimental results on popular LLMs, such as GPT-3.5-turbo and Gemini-pro, show that our IR method enhances the overall accuracy of factual reasoning by 27.33% and mathematical proof by 31.43%, when compared with traditional DR methods. Moreover, the methods combining IR and DR significantly outperform the methods solely using IR or DR, further demonstrating the effectiveness of our strategy.