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Humans have been shown to give contrastive explanations, which explain why an observed event happened rather than some other counterfactual event (the contrast case). Despite the influential role that contrastivity plays in how humans explain, this property is largely missing from current methods for explaining NLP models. We present Minimal Contrastive Editing (MiCE), a method for producing contrastive explanations of model predictions in the form of edits to inputs that change model outputs to the contrast case. Our experiments across three tasks--binary sentiment classification, topic classification, and multiple-choice question answering--show that MiCE is able to produce edits that are not only contrastive, but also minimal and fluent, consistent with human contrastive edits. We demonstrate how MiCE edits can be used for two use cases in NLP system development--debugging incorrect model outputs and uncovering dataset artifacts--and thereby illustrate that producing contrastive explanations is a promising research direction for model interpretability.