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

TL;DR

This paper introduces a symbolic reasoning framework for large language models that enforces logical consistency using algebraic invariants, improving structured reasoning and preventing error propagation.

Contribution

It operationalizes Peirce's inference types with algebraic invariants, especially the Weakest Link bound, to ensure reliable multi-step reasoning in LLMs.

Findings

The Weakest Link bound prevents logical errors from propagating in reasoning chains.

A property-based testing suite verifies all invariants over 10^5+ generated cases.

Provides a verified implementation as a foundation for future reasoning benchmarks.

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.