Open-world logical entailment requires deciding whether a set of premises entails a conclusion, entails its negation, or leaves it undecided. Many existing methods are deductive in spirit, searching for a proof and reading the label from whether that proof succeeds. When implemented through LLM-based proof search, this makes the prediction depend on a single unstable trajectory, and a failed or partial search is easily mismapped to the wrong label. We propose ABED (Assumption-Based Bidirectional Entailment Diagnosis), a training-free inference-time framework that reframes entailment classification as missing assumption diagnosis. Instead of asking only whether a conclusion can be proved, ABED asks what assumption is missing for it to hold, and classifies that gap as already closed by the premises (Null), contradicted by them (Contra), or left open (Admissible). It applies this diagnosis to both a conclusion and its negation, and determines the label from the resulting dual gap pair. Across FOLIO, ProofWriter, ProntoQA, and LogicNLI, ABED consistently outperforms prompting-based, symbolic, solver-based, and deductive baselines, improving over the strongest baseline on each dataset by 3.4, 5.3, 0.8, and 4.4 points with GPT-4.1.