Mining Logic under Uncertainty: Probabilistic Soft Logic with Energy-Based Inference for Chain-of-Thought Verification

Chain-of-Thought reasoning improves the explainability of large language models, yet verifying such reasoning remains difficult when intermediate steps express epistemic uncertainty rather than deterministic entailment. Neuro-symbolic verifiers typically translate reasoning steps into rigid first-order or SMT-style constraints. This forces each proposition into a binary truth regime and creates a Rigidity--Ambiguity Mismatch: uncertainty-marked claims such as may, likely, or suggests are either over-hardened into deterministic assertions or rejected as unverifiable. We propose ProbVeri, a probabilistic soft-logic framework for formal verification under uncertainty. The key idea is to represent uncertainty-marked reasoning steps as weighted soft logical constraints over continuous truth assignments, so that partial evidential support remains expressible. Building on this soft-logic representation, we formulate verification as a contrastive energy minimization problem: a step is considered supported when its natural interpretation has lower verification energy than its negated counterpart under the same evidence, soft rules, and assumption regularization. Probabilistic Soft Logic defines the logical energy through weighted hinge-loss constraints. We introduce a support-grounded assumption penalty that discourages unsupported latent bridge assumptions between context and claim. The resulting energy gap provides a confidence-scored verification signal that separates unsupported steps from weakly but consistently supported inferences. Experiments on ProofWriter, BioASQ, and LegalBench-SARA show that ProbVeri recovers more verified-correct reasoning traces than rigid verifiers while reducing over-acceptance compared with soft-only PSL. These results suggest that uncertainty-aware verification benefits from soft logical formalization and contrastive comparison between natural and negated interpretations.