# Deductive Logic in Language Models: Horizontal vs Vertical Reasoning

arXiv: https://arxiv.org/abs/2510.09340 · HTML: https://mlsanity.com/paper/2510.09340/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 10 Oct 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 25 of 25)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Towards Reasoning in Large Language Models: A Survey](https://mlsanity.com/paper/2212.10403/) (2022)
- [Selection-Inference: Exploiting Large Language Models for Interpretable Logical Reasoning](https://mlsanity.com/paper/2205.09712/) (2022)
- [ProofWriter: Generating Implications, Proofs, and Abductive Statements over Natural Language](https://mlsanity.com/paper/2012.13048/) (2020)
- [Transformers as Soft Reasoners over Language](https://mlsanity.com/paper/2002.05867/) (2020)
- [Curriculum Learning: A Survey](https://mlsanity.com/paper/2101.10382/) (2021)
- [How to use and interpret activation patching](https://mlsanity.com/paper/2404.15255/) (2024)
- [How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model](https://mlsanity.com/paper/2305.00586/) (2023)
- [Post-hoc Interpretability for Neural NLP: A Survey](https://mlsanity.com/paper/2108.04840/) (2021)
- [A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis](https://mlsanity.com/paper/2305.15054/) (2023)
- [A Primer on the Inner Workings of Transformer-based Language Models](https://mlsanity.com/paper/2405.00208/) (2024)
- [Transformers, parallel computation, and logarithmic depth](https://mlsanity.com/paper/2402.09268/) (2024)
- [LogicBench: Towards Systematic Evaluation of Logical Reasoning Ability of Large Language Models](https://mlsanity.com/paper/2404.15522/) (2024)
- [Logical Reasoning in Large Language Models: A Survey](https://mlsanity.com/paper/2502.09100/) (2025)
- [A Mechanistic Analysis of a Transformer Trained on a Symbolic Multi-Step Reasoning Task](https://mlsanity.com/paper/2402.11917/) (2024)
- [Reasoning Circuits in Language Models: A Mechanistic Interpretation of Syllogistic Inference](https://mlsanity.com/paper/2408.08590/) (2024)
- [JustLogic: A Comprehensive Benchmark for Evaluating Deductive Reasoning in Large Language Models](https://mlsanity.com/paper/2501.14851/) (2025)
- [How Do LLMs Perform Two-Hop Reasoning in Context?](https://mlsanity.com/paper/2502.13913/) (2025)
- [MIRAGE: Evaluating and Explaining Inductive Reasoning Process in Language Models](https://mlsanity.com/paper/2410.09542/) (2024)
- [How do Transformers Learn Implicit Reasoning?](https://mlsanity.com/paper/2505.23653/) (2025)
- [A Implies B: Circuit Analysis in LLMs for Propositional Logical Reasoning](https://mlsanity.com/paper/2411.04105/) (2024)
- [Arithmetic with Language Models: from Memorization to Computation](https://mlsanity.com/paper/2308.01154/) (2023)

## Built on by (most cited first, 1 of 1)

- [Wiring the 'Why': A Unified Taxonomy and Survey of Abductive Reasoning in LLMs](https://mlsanity.com/paper/2604.08016/) (2026)

## Abstract (arXiv)

Recent language models exhibit significant logical reasoning abilities, yet the mechanisms supporting deductive inference remain poorly understood. This paper studies small transformer-based language models trained from scratch on multi-step deductive tasks, focusing on the distinction between horizontal reasoning, where intermediate steps are generated autoregressively, and vertical reasoning, where inference unfolds implicitly across layers before the first output token is produced. We analyze two synthetic tasks: logical consequence over chains of symbolic implications and root-to-leaf navigation in binary trees. Mechanistic interpretability reveals that Chain-of-Thought supervision enables models to learn rule-based inference rather than statistical shortcuts. In the horizontal setting, a shallow attention-only model develops interpretable circuits for rule completion, rule chaining, and final decision making, largely implemented through induction-head-like mechanisms. We further introduce a truncated pseudoinverse method to decode the information carried by queries, keys, and values. For vertical reasoning, Chain-of-Thought appears to act less as explicit step-by-step guidance and more as a form of curriculum learning, helping the model acquire increasingly complex reasoning patterns. Without Chain-of-Thought, models tend to memorize or exploit dataset biases. These results provide a low-level account of how transformers can implement deductive reasoning and suggest how Chain-of-Thought may serve different functions in horizontal and vertical reasoning.