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Explainable artificial intelligence and interpretable machine learning areresearch domains growing in importance. Yet, the underlying concepts remainsomewhat elusive and lack generally agreed definitions. While recentinspiration from social sciences has refocused the work on needs andexpectations of human recipients, the field still misses a concreteconceptualisation. We take steps towards addressing this challenge by reviewingthe philosophical and social foundations of human explainability, which we thentranslate into the technological realm. In particular, we scrutinise the notionof algorithmic black boxes and the spectrum of understanding determined byexplanatory processes and explainees' background knowledge. This approachallows us to define explainability as (logical) reasoning applied totransparent insights (into, possibly black-box, predictive systems) interpretedunder background knowledge and placed within a specific context -- a processthat engenders understanding in a selected group of explainees. We then employthis conceptualisation to revisit strategies for evaluating explainability aswell as the much disputed trade-off between transparency and predictive power,including its implications for ante-hoc and post-hoc techniques along withfairness and accountability established by explainability. We furthermorediscuss components of the machine learning workflow that may be in need ofinterpretability, building on a range of ideas from human-centredexplainability, with a particular focus on explainees, contrastive statementsand explanatory processes. Our discussion reconciles and complements currentresearch to help better navigate open questions -- rather than attempting toaddress any individual issue -- thus laying a solid foundation for a groundeddiscussion and future progress of explainable artificial intelligence andinterpretable machine learning.

Explainability Is in the Mind of the Beholder: Establishing the Foundations of Explainable Artificial Intelligence