Large Language Models (LLMs) often struggle with rigorous logical reasoning, suffering from hallucinations and inconsistent deduction. Chain-of-Thought (CoT) prompting relies heavily on the frozen model’s inherent capabilities, while neuro-symbolic approaches face a "translation reliability" bottleneck in which semantic parsing errors mislead external solvers. Recent work attempts to infuse symbolic capabilities via fine-tuning, but typically mimics forward reasoning, suffering from search-space explosion and irrelevant intermediate facts that dilute attention. We propose SOLID (Symbolic Oriented Logic Infusion via Dual-verification), a framework that synthesizes high-quality supervision through a Dual-Verification Mechanism: bidirectional consistency checks filter out hallucinated translations. Crucially, instead of forward chaining, we execute validated translations to generate backward reasoning traces—verifying necessary premises from the goal. By fine-tuning on these focused, goal-oriented paths, SOLID infuses the model with the efficient deduction process of a symbolic engine. Experiments on ProntoQA, LogicNLI, and ProofWriter demonstrate state-of-the-art accuracy, significantly outperforming CoT baselines, solver-based methods, and forward-chaining variants. Code and data are available at: https://github.com/ejie22/SOLID.

Geometry Problem Solving (GPS) is a fundamental task in mathematical reasoning, with applications in intelligent education and automated theorem proving. Existing neuro-symbolic solvers often rely on two restrictive assumptions: textual conditions are expressed as atomic geometric statements, and target derivation can be reduced to eager elimination of intermediate variables. These assumptions break down in realistic problems, where coupled geometric descriptions lead to combinatorial growth in parsing patterns, and nonlinear constraints make explicit elimination inefficient or even intractable. We propose RIGOR, a neuro-symbolic GPS framework based on Atomic Semantic Normalization and Declarative Constraint Reasoning. RIGOR uses a Geometric Semantic Normalizer trained on our newly constructed GSN-15K dataset to normalize geometric texts into atomic statements for reliable formalization. It then adopts a Declarative Symbolic Engine that globally accumulates derived geometric constraints and determines target values through constraint entailment, while also reconstructing proof-grounded reasoning traces. Experiments on multiple standard GPS benchmarks show that RIGOR outperforms state-of-the-art methods while producing interpretable derivations. Our project is available at https://github.com/dongyuhang22/RIGOR.