Structured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants
For researchers building reliable LLM reasoning systems, this provides a formal framework to prevent logical inconsistency accumulation, though it is an incremental extension of existing weakest link logic to LLM contexts.
LLMs conflate hypothesis generation with verification and allow weak reasoning steps to propagate unchecked. The authors introduce a symbolic reasoning scaffold enforcing five algebraic invariants (Gamma Quintet), with the Weakest Link bound ensuring no conclusion exceeds the reliability of its least-supported premise, verified via 100 properties and 16 fuzz tests over 10^5+ cases.
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.