In this critical survey, we analyze typical claims on the relationship between explainable AI (XAI) and fairness to disentangle the multidimensional relationship between these two concepts. Based on a systematic literature review and a subsequent qualitative content analysis, we identify seven archetypal claims from 175 papers on the alleged fairness benefits of XAI. We present crucial caveats with respect to these claims and provide an entry point for future discussions around the potentials and limitations of XAI for specific fairness desiderata. While the literature often suggests XAI to be an enabler for several fairness desiderata, we notice a divide between these desiderata and the capabilities of XAI. We encourage to conceive XAI as one of many tools to approach the multidimensional, sociotechnical challenge of algorithmic fairness and to be more specific about how exactly what kind of XAI method enables whom to address which fairness desideratum.
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