# Data-driven Machine Learning Cannot Reach Symbolic-level Logical Reasoning -- The Limit of the Scaling Law

arXiv: https://arxiv.org/abs/2606.26454 · HTML: https://mlsanity.com/paper/2606.26454/

## 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 26 of its 53 references that are in the map.
- First posted on arXiv 24 Jun 2026.

## Placed in

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

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [Gemma: Open Models Based on Gemini Research and Technology](https://mlsanity.com/paper/2403.08295/) (2024)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Selection-Inference: Exploiting Large Language Models for Interpretable Logical Reasoning](https://mlsanity.com/paper/2205.09712/) (2022)
- [BioMistral: A Collection of Open-Source Pretrained Large Language Models for Medical Domains](https://mlsanity.com/paper/2402.10373/) (2024)
- [Inductive Biases for Deep Learning of Higher-Level Cognition](https://mlsanity.com/paper/2011.15091/) (2020)
- [Chain of Thought Empowers Transformers to Solve Inherently Serial Problems](https://mlsanity.com/paper/2402.12875/) (2024)
- [Goedel-Prover: A Frontier Model for Open-Source Automated Theorem Proving](https://mlsanity.com/paper/2502.07640/) (2025)
- [What Formal Languages Can Transformers Express? A Survey](https://mlsanity.com/paper/2311.00208/) (2023)
- [STP: Self-play LLM Theorem Provers with Iterative Conjecturing and Proving](https://mlsanity.com/paper/2502.00212/) (2025)
- [Siamese Masked Autoencoders](https://mlsanity.com/paper/2305.14344/) (2023)
- [Abstract Diagrammatic Reasoning with Multiplex Graph Networks](https://mlsanity.com/paper/2006.11197/) (2020)
- [A Systematic Comparison of Syllogistic Reasoning in Humans and Language Models](https://mlsanity.com/paper/2311.00445/) (2023)
- [Exploring Reasoning Biases in Large Language Models Through Syllogism: Insights from the NeuBAROCO Dataset](https://mlsanity.com/paper/2408.04403/) (2024)
- [A Systematic Analysis of Large Language Models as Soft Reasoners: The Case of Syllogistic Inferences](https://mlsanity.com/paper/2406.11341/) (2024)
- [SylloBio-NLI: Evaluating Large Language Models on Biomedical Syllogistic Reasoning](https://mlsanity.com/paper/2410.14399/) (2024)
- [Sphere Neural-Networks for Rational Reasoning](https://mlsanity.com/paper/2403.15297/) (2024)
- [On the Representational Capacity of Recurrent Neural Language Models](https://mlsanity.com/paper/2310.12942/) (2023)
- [An AI Monkey Gets Grapes for Sure -- Sphere Neural Networks for Reliable Decision-Making](https://mlsanity.com/paper/2601.00142/) (2026)

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

- [Logical Judgments Under Pressure: Diagnosing Syllogistic Stability with Learned Soft Prefixes](https://mlsanity.com/paper/2607.18228/) (2026)

## Abstract (arXiv)

By promoting vectors to spheres and enabling explicit model construction, neural networks can perform symbolic-level syllogistic reasoning without training data. We identify two fundamental limitations that prevent conventional data-driven machine learning systems from achieving this capability: training data generated by the combination table cannot distinguish all 24 valid syllogism types, and end-to-end premise-to-conclusion mapping creates contradictory targets within neural components. Experiments with two representative conventional systems, GPT-5 using linguistic inputs and Euler Net using visual inputs, support this analysis. ChatGPT GPT-5 may reach 100% accuracy in syllogistic reasoning, but with hallucinations. Because the learning process terminates upon reaching 100% accuracy, the system cannot progress beyond empirical accuracy to symbolic level reasoning. Random test data reduced Euler Net's accuracy to 56%. Repeatedly expanding the training set increased its accuracy to 97%, with perfect performance on 8 syllogism types. However, because unintended inputs cannot be exhaustively covered, even 100% test accuracy does not imply symbolic-level reasoning. Since syllogistic reasoning underpins logical reasoning and human rationality, these results suggest that increasing data and training time alone cannot ensure symbolic level logical reasoning.