🤖 AI Summary
This work addresses systematic deficiencies in large language models (LLMs) regarding structured logical reasoning—such as conflation of hypotheses with verification and unconstrained propagation of weak inference steps—by proposing a symbolic reasoning framework that integrates Peircean triadic inference (abduction, deduction, and induction). Central to this approach is the Gamma Quintet algebraic invariant system, particularly its “Weakest Link bound,” which mitigates inconsistency accumulation across multi-step reasoning from a possibilistic logic perspective, thereby providing the first formal logical guarantees for LLMs. Leveraging attribute-based testing and fuzz testing, we validate all invariants across over 10⁵ generated cases, releasing an open-source, reproducible implementation that establishes a new benchmark for structured reasoning.
📝 Abstract
Large language models exhibit systematic limitations in structured logical reasoning: they conflate hypothesis generation with verification, cannot distinguish conjecture from validated knowledge, and allow weak reasoning steps to propagate unchecked through inference chains. We present a symbolic reasoning scaffold that operationalizes Peirce's tripartite inference -- abduction, deduction, and induction -- as an explicit protocol for LLM-assisted reasoning. The framework enforces logical consistency through five algebraic invariants (the Gamma Quintet), the strongest of which -- the Weakest Link bound -- ensures that no conclusion in a reasoning chain can exceed the reliability of its least-supported premise. This principle, independently grounded as weakest link resolution in possibilistic logic and empirically validated for chain-of-thought reasoning, prevents logical inconsistencies from accumulating across multi-step inference. We verify all invariants through a property-based testing suite of 100 properties and 16 fuzz tests over 10^5+ generated cases, providing a verified reference implementation of the invariants suitable as a foundation for future reasoning benchmarks.