Syllogistic Reasoning

Inspired by the analogy between the all-or-none activation of biological neurons and the true-false of logic circuits, McCulloch and Pitts proposed a mathematical model of the biological neuron in 1943 with the aim of simulating logical reasoning. Although Norbert Wiener considered the model overly simplistic at the time, it later became a building block of modern neural networks, including large language models. Despite their remarkable empirical success, however, conventional neural networks remain largely black-box systems and struggle with reliable symbolic reasoning. Viewing vectors as spheres of zero radius, we relax the zero-radius constraint and represent concepts as spheres with non-zero radii. We argue that sphere-based representations can capture additional features of biological cognition, including representing spatial concepts as regions and reasoning through model construction and inspection. Building on sphere embeddings, we introduce Sphere Neural Networks (SphNNs), which use a transition map to achieve symbolic-level Aristotelian syllogistic reasoning without training data, thereby taking an initial step toward the original goal of McCulloch and Pitts. We suggest that the zero-radius constraint may reflect an aspect of the oversimplification identified by Wiener and may limit conventional neural networks' capacity for symbolic-level logical reasoning.

Diagrams are widely used to support logical reasoning, and prior studies suggest that representations such as Euler diagrams can improve human reasoning performance. Recent work has also explored their effects on large language models (LLMs). In this paper, we compare four representational conditions for syllogistic reasoning: natural language, logical notation, linear diagrams, and Euler diagrams. Using 285 problems from Ando et al. (2024), we evaluate two contemporary LLMs, Claude 3.5~Sonnet and GPT-4o-mini. Our results show that diagrammatic representations do not consistently improve performance. Although the models perform well on entailment and contradiction problems, they struggle with neutral problems and often make systematic conversion errors. Overall, the results suggest that the tested models gain limited benefit from diagrams in logical reasoning tasks.

Language models (LMs) struggle with logical tasks like reasoning on syllogisms. It has been shown that Knowledge Representation (KR) plays a crucial role in expressing input information to help models solve tasks. This observation motivates our study of the impact of different formal KR notations on syllogistic reasoning by extending the FOLIO and P-FOLIO datasets. Our experiments on Small Language Models (SLMs) in Supervised Fine-Tuning (SFT) and Zero-Shot (ZS) settings show that the choice of input notation can yield performances competitive with natural language while enabling faster inference. We also propose a syllogistic categorization method (SEF) and use it to enrich ZS prompts with logical definitions, which boost reasoning in small models. We open-source our framework, Common Logic Grammar Construction (CLGC), as the first Python library for automatically generating syllogisms in KR notations and defining their SEF categories.

To test how correct logical judgments respond to learned context, we prepend a soft prefix to an exactly labeled syllogistic reasoning benchmark while keeping the model fixed. Soft prefixes are opaque continuous vectors, so we characterize them through the behavior they induce across controlled variations in logical form and interface. By studying which prefixes succeed and how their effects generalize, we characterize how learned contextual pressure can override correct judgments and expose limits in a model's logical stability. Across Qwen3.6-35B-A3B MoE, Qwen3-8B, and Gemma 4 31B, learned prefixes redirect many correct answers and remain effective across unseen forms and interface changes. In repeated tests with Qwen3.6 MoE and Gemma, they outperform paired random controls in all 16 model--direction--split comparisons by 37 to 99 percentage points. Qwen3.6 MoE flip rates remain between 72% and 90% across wording and prompt changes, while Gemma validity prefixes retain 54% to 56% flip compared with less than 1% for matched random prefixes. Diagnostic tests show that the dominant effect is a broad preference for one answer meaning rather than fixed-symbol forcing or a logical operation that transfers reliably between tasks. The form of this bias differs across models. In both Qwen models, simple score models often predict which judgments will flip but not how far their margins will move, whereas Gemma's overall response is more closely approximated by the same models. These results show that the dominant behavioral effect of successful soft prefixes is a broad answer preference, while the remaining response reveals substantial model-specific differences in logical stability.

Sphere neural networks have achieved symbolic level syllogistic reasoning without training data, raising the question of where the limit of the scaling law for logical reasoning lies, i.e., whether data-driven machine learning systems can achieve the same level by increasing training data and training time. We show two methodological limitations that prevent supervised deep learning from reaching the symbolic-level syllogistic reasoning: (1) training data can not distinguish all 24 types of valid syllogistic reasoning; (2) end-to-end mapping from premises to conclusion introduces contradictory training targets between neural components for pattern recognition and logical reasoning. Beside theoretical analysis, we experimentally illustrate that Euler Net cannot achieve rigorous syllogistic reasoning. We further challenge the most recent ChatGPTs (GPT-5-nano and GPT-5) to determine the satisfiability of syllogistic statements in four surface forms (patterns): words, double words, simple symbols, and long random symbols, showing that surface forms affect the reasoning performance and that ChatGPT GPT-5 may reach 100% accuracy but still provide incorrect explanations. As empirical training processes are stopped after achieving 100% accuracy, we conclude that supervised machine learning systems will not attain the rigour of symbolic logical reasoning.

This paper revisits our pipeline called Syllogistic Evaluation Framework-Common Logic Grammar Construction (SEF-CLGC). We combine formal logical notations with Small Language Models (SLMs) to evaluate reasoning performance on the SemEval-2026 Task 11 Subtask 1: Disentangling Content and Formal Reasoning in Large Language Models. Our experiments show that by relying solely on SLMs, trained on a combination of natural and symbolic languages, our best model achieves a content score of 27.80% on the task while significantly lowering the content bias in reasoning.

This paper describes our system submitted to SemEval-2026 Task 11: Disentangling Content and Formal Reasoning in Large Language Models. We present an efficient modular neuro-symbolic approach, combining a symbolic prover with small reasoning LLMs (4B parameters). The system consists of an LLM-based parser that translates natural language syllogisms to a first-order logic (FOL) representation, an automated theorem prover, and two optional modules: machine translation for multilingual inputs and a symbolic retrieval component for the identification of relevant premises. The system achieves competitive accuracy and relatively low content effect on most subtasks. Our ablations show that this approach outperforms LLM-based zero-shot baselines in this parameter size range, but also reveal limited multilingual capabilities of small LLMs. Finally, we include a discussion of the task's main ranking metric and analyze its limitations.

We present FregeLogic, a hybrid neuro-symbolic system for SemEval-2026 Task 11 (Subtask 1), which addresses syllogistic validity prediction while reducing content effects on predictions. Our approach combines an ensemble of five LLM classifiers, spanning three open-weights models (Llama 4 Maverick, Llama 4 Scout, and Qwen3-32B) paired with varied prompting strategies, with a Z3 SMT solver that serves as a formal logic tiebreaker. The central hypothesis is that LLM disagreement within the ensemble signals likely content-biased errors, where real-world believability interferes with logical judgment. By deferring to Z3's structurally-grounded formal verification on these disputed cases, our system achieves 94.3% accuracy with a content effect of 2.85 and a combined score of 41.88 in nested 5-fold cross-validation on the dataset (N=960). This represents a 2.76-point improvement in combined score over the pure ensemble (39.12), with a 0.9% accuracy gain, driven by a 16% reduction in content effect (3.39 to 2.85). Adopting structured-output API calls for Z3 extraction reduced failure rates from ~22% to near zero, and an Aristotelian encoding with existence axioms was validated against task annotations. Our results suggest that targeted neuro-symbolic integration, applying formal methods precisely where ensemble consensus is lowest, can improve the combined accuracy-plus-content-effect metric used by this task.

This paper asks whether a bounded neural architecture can exhibit a meaningful division of labor between intuition and deliberation on a classic 64-item syllogistic reasoning benchmark. More broadly, the benchmark is relevant to ongoing debates about world models and multi-stage reasoning in AI. It provides a controlled setting for testing whether a learned system can develop structured internal computation rather than only one-shot associative prediction. Experiment 1 evaluates a direct neural baseline for predicting full 9-way human response distributions under 5-fold cross-validation. Experiment 2 introduces a bounded dual-path architecture with separate intuition and deliberation pathways, motivated by computational mental-model theory (Khemlani & Johnson-Laird, 2022). Under cross-validation, bounded intuition reaches an aggregate correlation of r = 0.7272, whereas bounded deliberation reaches r = 0.8152, and the deliberation advantage is significant across folds (p = 0.0101). The largest held-out gains occur for NVC, Eca, and Oca, suggesting improved handling of rejection responses and c-a conclusions. A canonical 80:20 interpretability run and a five-seed stability sweep further indicate that the deliberation pathway develops sparse, differentiated internal structure, including an Oac-leaning state, a dominant workhorse state, and several weakly used or unused states whose exact indices vary across runs. These findings are consistent with reasoning-like internal organization under bounded conditions, while stopping short of any claim that the model reproduces full sequential processes of model construction, counterexample search, and conclusion revision.

Research in AI using Large-Language Models (LLMs) is rapidly evolving, and the comparison of their performance with human reasoning has become a key concern. Prior studies have indicated that LLMs and humans share similar biases, such as dismissing logically valid inferences that contradict common beliefs. However, criticizing LLMs for these biases might be unfair, considering our reasoning not only involves formal deduction but also abduction, which draws tentative conclusions from limited information. Abduction can be regarded as the inverse form of syllogism in its basic structure, that is, a process of drawing a minor premise from a major premise and conclusion. This paper explores the accuracy of LLMs in abductive reasoning by converting a syllogistic dataset into one suitable for abduction. It aims to investigate whether the state-of-the-art LLMs exhibit biases in abduction and to identify potential areas for improvement, emphasizing the importance of contextualized reasoning beyond formal deduction. This investigation is vital for advancing the understanding and application of LLMs in complex reasoning tasks, offering insights into bridging the gap between machine and human cognition.