Large Language Models as an Indirect Reasoner: Contrapositive and
  Contradiction for Automated Reasoning
Yanfang Zhang Yiliu Sun Yibing Zhan Dapeng Tao Dacheng Tao Chen Gong
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.