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

arXiv: https://arxiv.org/abs/2604.15727 · HTML: https://mlsanity.com/paper/2604.15727/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 10 of its 32 references that are in the map.
- First posted on arXiv 17 Apr 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 10 of 10)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [FOLIO: Natural Language Reasoning with First-Order Logic](https://mlsanity.com/paper/2209.00840/) (2022)
- [Logical Neural Networks](https://mlsanity.com/paper/2006.13155/) (2020)
- [A Chain-of-Thought Is as Strong as Its Weakest Link: A Benchmark for Verifiers of Reasoning Chains](https://mlsanity.com/paper/2402.00559/) (2024)
- [AIRS-Bench: a Suite of Tasks for Frontier AI Research Science Agents](https://mlsanity.com/paper/2602.06855/) (2026)
- [Epistemology gives a Future to Complementarity in Human-AI Interactions](https://mlsanity.com/paper/2601.09871/) (2026)
- [Typed Chain-of-Thought: A Curry-Howard Framework for Verifying LLM Reasoning](https://mlsanity.com/paper/2510.01069/) (2025)
- [AI-Assisted Engineering Should Track the Epistemic Status and Temporal Validity of Architectural Decisions](https://mlsanity.com/paper/2601.21116/) (2026)

## Built on by (most cited first, 1 of 1)

- [Position: Evaluation Scores Are Perishable Knowledge Claims](https://mlsanity.com/paper/2607.26191/) (2026)

## Abstract (arXiv)

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.