Structured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants

Researchers propose a symbolic reasoning framework that implements Peirce's abductive-deductive-inductive reasoning model to address systematic weaknesses in large language model logical reasoning. The system enforces logical consistency through five algebraic invariants, with the Weakest Link bound preventing unreliable premises from corrupting multi-step inference chains.

This research addresses a fundamental limitation in current LLM architectures: their inability to maintain logical rigor across complex reasoning chains. Large language models frequently conflate hypothesis generation with evidence validation, allowing weak assumptions to propagate through inference without degradation signals. The proposed framework operationalizes classical Peirce reasoning by separating abduction (conjecture generation), deduction (logical derivation), and induction (empirical validation) into discrete, auditable steps.

The algebraic invariants, termed the Gamma Quintet, provide formal guarantees about inference quality. The Weakest Link bound proves particularly significant as it mathematically constrains conclusion reliability to match the weakest supporting premise—preventing the accumulation of uncertainty errors typical in chain-of-thought reasoning. This principle aligns with possibilistic logic theory and empirical observations about how errors compound in sequential reasoning tasks.

The verification methodology demonstrates rigor: property-based testing across 100 formal properties and 16 fuzz tests on 10^5+ generated cases provides strong evidence the framework functions as specified. This creates a verified reference implementation usable for standardizing future reasoning benchmarks.

For AI development, this represents progress toward more trustworthy and interpretable LLM reasoning, particularly valuable for applications requiring auditable decision-making. However, the framework's practical integration into production systems remains unclear. The structured protocol may impose computational overhead and could limit the flexible, emergent reasoning that makes LLMs valuable for creative tasks. The research establishes theoretical foundations for symbolic-neural hybrid systems but requires empirical validation on realistic reasoning problems.

• →Researchers formalize three classical reasoning modes into an explicit LLM protocol enforcing logical consistency across inference chains.
• →The Weakest Link bound mathematically prevents unreliable premises from corrupting multi-step conclusions.
• →Comprehensive property-based testing (10^5+ cases) validates the framework's invariants and provides a verified reference implementation.
• →The approach addresses LLMs' tendency to conflate hypothesis generation with evidence validation in structured reasoning tasks.
• →Framework applicability to production systems and computational overhead trade-offs remain unresolved.