Abstract
Three-way logical question answering (QA) assigns True/False/Unknown to a hypothesis H given a premise set S. While modern large language models (LLMs) can be accurate on isolated examples, we identify two recurring failure modes in 3-way logic QA: (i) negation inconsistency, where answers to H and
eg H violate the deterministic label mapping, and (ii) epistemic Unknown, where the model predicts Unknown due to uncertainty or instability even when S entails one side. We present CGD-PD, a lightweight test-time layer that (a) queries a single 3-way classifier on both H and a mechanically negated form of H, (b) projects the pair onto a negation-consistent decision when possible, and (c) invokes a proof-driven disambiguation step that uses targeted binary entailment probes to selectively resolve Unknown outcomes, requiring only an average of 4-5 model calls. On the FOLIO benchmark's first-order-logic fields, CGD-PD yields consistent gains across frontier LLMs, with relative improvements in accuracy of up to 16% over the base model, while also reducing Unknown predictions.
