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Linear Reasoning vs. Proof by Cases: Obstacles for Large Language Models in FOL Problem Solving

Yuliang Ji, Fuchen Shen, Jian Wu, Qiujie Xie, Yue Zhang · Feb 24, 2026 · Citations: 0

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Data freshness

Extraction: Fresh

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Metadata refreshed

Feb 24, 2026, 2:53 PM

Stale

Extraction refreshed

Apr 13, 2026, 6:41 AM

Fresh

Extraction source

Persisted extraction

Confidence 0.15

Abstract

To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets. However, most existing datasets primarily focus on linear reasoning, neglecting other parts such as proof by contradiction and proof by cases, which are crucial for investigating LLMs' reasoning abilities. To address this limitation, we first introduce a novel first-order logic (FOL) dataset named PC-FOL, annotated by professional mathematicians, focusing on case-based reasoning problems. All instances in this dataset are equipped with a manually written natural language proof, clearly distinguishing it from conventional linear reasoning datasets. Our experimental results over leading LLMs demonstrate a substantial performance gap between linear reasoning and case-based reasoning problems. To further investigate this phenomenon, we provide a theoretical analysis grounded in graphical model, which provides an explanation for the observed disparity between the two types of reasoning problems. We hope this work can reveal the core challenges in the field of automated natural language mathematical proof generation, paving the way for future research.

Low-signal caution for protocol decisions

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  • Extraction flags indicate low-signal or possible false-positive protocol mapping.
  • Extraction confidence is 0.15 (below strong-reference threshold).
  • No explicit evaluation mode was extracted from available metadata.
  • No benchmark/dataset or metric anchors were extracted.
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HFEPX Relevance Assessment

This paper is adjacent to HFEPX scope and is best used for background context, not as a primary protocol reference.

Best use

Background context only

Use if you need

Background context only.

Main weakness

Extraction flags indicate low-signal or possible false-positive protocol mapping.

Trust level

Low

Eval-Fit Score

0/100 • Low

Treat as adjacent context, not a core eval-method reference.

Human Feedback Signal

Not explicit in abstract metadata

Evaluation Signal

Weak / implicit signal

HFEPX Fit

Adjacent candidate

Extraction confidence: Low

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Field Provenance & Confidence

Each key protocol field shows extraction state, confidence band, and data source so you can decide whether to trust it directly or validate from full text.

Human Feedback Types

missing

None explicit

Confidence: Low Source: Persisted extraction missing

No explicit feedback protocol extracted.

Evidence snippet: To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.

Evaluation Modes

missing

None explicit

Confidence: Low Source: Persisted extraction missing

Validate eval design from full paper text.

Evidence snippet: To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.

Quality Controls

missing

Not reported

Confidence: Low Source: Persisted extraction missing

No explicit QC controls found.

Evidence snippet: To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.

Benchmarks / Datasets

missing

Not extracted

Confidence: Low Source: Persisted extraction missing

No benchmark anchors detected.

Evidence snippet: To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.

Reported Metrics

missing

Not extracted

Confidence: Low Source: Persisted extraction missing

No metric anchors detected.

Evidence snippet: To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.

Rater Population

missing

Unknown

Confidence: Low Source: Persisted extraction missing

Rater source not explicitly reported.

Evidence snippet: To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.

Human Data Lens

  • Uses human feedback: No
  • Feedback types: None
  • Rater population: Unknown
  • Unit of annotation: Unknown
  • Expertise required: Math
  • Extraction source: Persisted extraction

Evaluation Lens

  • Evaluation modes:
  • Agentic eval: None
  • Quality controls: Not reported
  • Confidence: 0.15
  • Flags: low_signal, possible_false_positive

Protocol And Measurement Signals

Benchmarks / Datasets

No benchmark or dataset names were extracted from the available abstract.

Reported Metrics

No metric terms were extracted from the available abstract.

Research Brief

Deterministic synthesis

To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets. HFEPX protocol signal is limited in abstract-level metadata, so treat it as adjacent context. Updated from current HFEPX corpus.

Generated Apr 13, 2026, 6:41 AM · Grounded in abstract + metadata only

Key Takeaways

  • To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.
  • However, most existing datasets primarily focus on linear reasoning, neglecting other parts such as proof by contradiction and proof by cases, which are crucial for investigating…
  • Abstract shows limited direct human-feedback or evaluation-protocol detail; use as adjacent methodological context.

Researcher Actions

  • Treat this as method context, then pivot to protocol-specific HFEPX hubs.
  • Identify benchmark choices from full text before operationalizing conclusions.
  • Verify metric definitions before comparing against your eval pipeline.

Caveats

  • Generated from title, abstract, and extracted metadata only; full-paper implementation details are not parsed.
  • Low-signal flag detected: protocol relevance may be indirect.

Research Summary

Contribution Summary

  • To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets.
  • However, most existing datasets primarily focus on linear reasoning, neglecting other parts such as proof by contradiction and proof by cases, which are crucial for investigating LLMs' reasoning abilities.
  • To address this limitation, we first introduce a novel first-order logic (FOL) dataset named PC-FOL, annotated by professional mathematicians, focusing on case-based reasoning problems.

Why It Matters For Eval

  • Abstract shows limited direct human-feedback or evaluation-protocol detail; use as adjacent methodological context.

Researcher Checklist

  • Gap: Human feedback protocol is explicit

    No explicit human feedback protocol detected.

  • Gap: Evaluation mode is explicit

    No clear evaluation mode extracted.

  • Gap: Quality control reporting appears

    No calibration/adjudication/IAA control explicitly detected.

  • Gap: Benchmark or dataset anchors are present

    No benchmark/dataset anchor extracted from abstract.

  • Gap: Metric reporting is present

    No metric terms extracted.

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