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

Recently, increasing attention has been focused on improving the ability of Large Language Models (LLMs) to perform complex reasoning. Advanced methods, such as Chain-of-Thought (CoT) and its variants, are found to enhance their reasoning skills by designing suitable prompts or breaking down complex problems into more manageable sub-problems. However, little concentration has been put on exploring the reasoning process, textiti.e., we discovered that most methods resort to Direct Reasoning (DR) and disregard Indirect Reasoning (IR). This can make LLMs difficult to solve IR tasks, which are often encountered in the real world. To address this issue, we propose a Direct-Indirect Reasoning (DIR) method, which considers DR and IR as multiple parallel reasoning paths that are merged to derive the final answer. We stimulate LLMs to implement IR by crafting prompt templates incorporating the principles of contrapositive and contradiction. These templates trigger LLMs to assume the negation of the conclusion as true, combine it with the premises to deduce a conclusion, and utilize the logical equivalence of the contrapositive to enhance their comprehension of the rules used in the reasoning process. Our DIR method is simple yet effective and can be straightforwardly integrated with existing variants of CoT methods. Experimental results on four datasets related to logical reasoning and mathematic proof demonstrate that our DIR method, when combined with various baseline methods, significantly outperforms all the original methods.