Can indirect reasoning methods solve problems direct chain-of-thought cannot?
Most language model reasoning follows direct forward steps from premises. But some logical problems may require contrapositive or proof-by-contradiction approaches. Can these indirect methods overcome limitations of direct reasoning alone?
Chain-of-Thought and self-consistency are direct reasoning (DR) frameworks — they march forward from premises to conclusion. But many real tasks (some factual reasoning, mathematical proofs) are hard to solve by direct derivation. This paper adds an indirect reasoning (IR) capability built from two classical logical moves. First, it uses the logical equivalence of the contrapositive to augment the data and rules, making the problem more tractable for the model. Second, it supplies prompt templates that trigger proof by contradiction — logically equivalent to the original direct process but reachable when forward derivation stalls. IR is simple, improves factual reasoning and math-proof tasks, and composes with existing DR methods to boost them further.
The keeper is that the logical form of the reasoning request is a lever independent of the model: giving the model the contrapositive or a contradiction frame changes what it can derive, without changing the model or adding training. Reasoning prompting has a logic-structure dimension, not just a step-count dimension.
This adds a logical-form axis to the vault's CoT-methods cluster. Where most CoT work varies how many steps or how they're selected, IR varies the logical strategy — and it rhymes with Can backward reasoning during training improve forward reasoning?: both exploit that reasoning backward or by contradiction accesses competence forward reasoning alone misses.
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Can backward reasoning during training improve forward reasoning?
Does training models to reason backward—generating inverse questions and solutions—build internal consistency checking that transfers to forward-only inference? This explores whether backward capacity internalized during training without test-time deployment can enhance reasoning quality.
both exploit non-forward logical strategies (backward / contrapositive / contradiction) to access otherwise-missed competence
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Why do chain-of-thought examples fail across different conditions?
Chain-of-thought exemplars show surprising sensitivity to order, complexity level, diversity, and annotator style. Understanding these brittleness dimensions could reveal what makes reasoning prompts robust or fragile.
IR adds a logical-form lever orthogonal to the exemplar-selection levers
Related papers in this collection 8
Papers most semantically related to this note, ranked by cosine similarity in the embedding space.
- Large Language Models as an Indirect Reasoner: Contrapositive and Contradiction for Automated Reasoning
- On the Reasoning Capacity of AI Models and How to Quantify It
- Reasoning Beyond Chain-of-Thought: A Latent Computational Mode in Large Language Models
- The Illusion of Thinking: Understanding the Strengths and Limitations of Reasoning Models via the Lens of Problem Complexity
- Causal Sufficiency and Necessity Improves Chain-of-Thought Reasoning
- Cumulative Reasoning with Large Language Models
- Reasoning Strategies in Large Language Models: Can They Follow, Prefer, and Optimize?
- Is Chain-of-Thought Reasoning of LLMs a Mirage? A Data Distribution Lens
Original note title
indirect reasoning via contrapositive and proof by contradiction solves problems direct chain-of-thought cannot