Algebraic Invariants Enforce Peircean Reasoning to Counter LLM Logical Drift
Paper introduces Gamma Quintet algebraic invariants to structure abductive-deductive-inductive LLM reasoning; framework addresses core reliability gaps via Weakest Link bound, verified through extensive property testing.
Large language models conflate hypothesis generation with verification and permit weak steps to propagate, according to primary source arXiv:2604.15727.
Gilda (2026) operationalizes Peirce's abduction-deduction-induction triad through a symbolic scaffold held by five algebraic invariants (Gamma Quintet), the strongest being the Weakest Link bound that caps conclusion reliability at the least-supported premise; this bound is independently grounded in possibilistic logic (Dubois & Prade, 1988, https://doi.org/10.1016/0888-613X(88)90001-7) and empirically shown in chain-of-thought settings (Wei et al., arXiv:2201.11903). The work verifies all invariants with a property-based suite of 100 properties plus 16 fuzz tests over 10^5 generated cases, supplying a reference implementation absent from prior heuristic prompting literature.
Original coverage overlooked explicit ties to documented CoT unfaithfulness where models generate plausible yet invalid intermediate claims (Turpin et al., arXiv:2305.04388); the algebraic approach also connects to self-consistency methods (Wang et al., arXiv:2203.11171) that sample multiple paths yet lack formal guards against error accumulation. Synthesis of these sources indicates the invariants supply the missing deductive closure and inductive validation layers that current frontier models bypass, converting probabilistic guesswork into bounded inference chains.
AXIOM: The Gamma Quintet invariants convert ad-hoc CoT into formally bounded inference; enforcing the Weakest Link bound could cut unchecked error propagation in multi-step LLM reasoning and supply the verifiable scaffold frontier models currently lack.
Sources (3)
- [1]Structured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants(https://arxiv.org/abs/2604.15727)
- [2]Chain-of-Thought Prompting Elicits Reasoning in Large Language Models(https://arxiv.org/abs/2201.11903)
- [3]Language Models Don't Always Say What They Think(https://arxiv.org/abs/2305.04388)