We propose a novel method for explaining the predictions of any classifier. In our approach, local explanations are expected to explain both the outcome of a prediction and how that prediction would change if ‘things had been different’. Furthermore, we argue that satisfactory explanations cannot be dissociated from a notion and measure of fidelity, as advocated in the early days of neural networks’ knowledge extraction. We introduce a definition of fidelity to the underlying classifier for local explanation models which is based on distances to a target decision boundary. A system called CLEAR: Counterfactual Local Explanations via Regression, is introduced and evaluated. CLEAR generates b-counterfactual explanations that state minimum changes necessary to flip a prediction’s classification. CLEAR then builds local regression models, using the b-counterfactuals to measure and improve the fidelity of its regressions. By contrast, the popular LIME method [17], which also uses regression to generate local explanations, neither measures its own fidelity nor generates counterfactuals. CLEAR’s regressions are found to have significantly higher fidelity than LIME’s, averaging over 40% higher in this paper’s five case studies.