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While large pre-trained language models are powerful, their predictions often lack logical consistency across test inputs. For example, a state-of-the-art Macaw question-answering (QA) model answers 'Yes' to 'Is a sparrow a bird?' and 'Does a bird have feet?' but answers 'No' to 'Does a sparrow have feet?'. To address this failure mode, we propose a framework, Consistency Correction through Relation Detection, or ConCoRD, for boosting the consistency and accuracy of pre-trained NLP models using pre-trained natural language inference (NLI) models without fine-tuning or re-training. Given a batch of test inputs, ConCoRD samples several candidate outputs for each input and instantiates a factor graph that accounts for both the model's belief about the likelihood of each answer choice in isolation and the NLI model's beliefs about pair-wise answer choice compatibility. We show that a weighted MaxSAT solver can efficiently compute high-quality answer choices under this factor graph, improving over the raw model's predictions. Our experiments demonstrate that ConCoRD consistently boosts accuracy and consistency of off-the-shelf closed-book QA and VQA models using off-the-shelf NLI models, notably increasing accuracy of LXMERT on ConVQA by 5% absolute. See https://ericmitchell.ai/emnlp-2022-concord/ for code and data., Comment: 16 pages. EMNLP 2022 Camera Ready. See https://ericmitchell.ai/emnlp-2022-concord/ for code and data