Explainability of algorithmic decision-making systems is both a regulatory objective and an area of intense research. The article argues that a crucial condition for the acceptability of algorithmic decision-making systems is that decisions must be justified in the eyes of their recipients. We make a clear distinction between explanation and justification. Explanations describe how a decision was made, while justifications give reasons that aim to make the decision acceptable. We propose a conceptual framework of explanations and justifications, based on Habermas's theory of communicative action and Perelman's New Rhetoric theory of law. This framework helps to analyze how different forms of explanation can support or fail to support justification. We illustrate our approach with a case study on university admissions in France.

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