<p>Unknown unknowns are things we currently do not know we don’t know; as such, these things cannot be described beforehand, let alone be assigned probabilities of occurrence. Examples include breakthrough scientific discoveries, disruptive innovations, germs mutations, and new art movements or fashions. Numerous accounts show in retrospect that significant gains or losses might have been achieved or avoided had such contingencies been timely prefigured. But getting some grasp at the unknown unknowns before they show up remains elusive, both conceptually and in practice. Standard risk analysis is of little help here, because it deals with known unknowns, i.e. situations in which all relevant possible outcomes (even extremely rare ones like ‘black swans’) can be listed explicitly. This paper uses formal concept analysis (FCA)—an approach which is increasingly deployed for mining and organizing data—to show that: (i) if the unknown unknowns’ eventual description and the known items’ current representation shall fit a common ‘context’ (as the notion is defined in FCA), then there should be tangible clues about the unknown unknowns within the actual data; (ii) one can unveil these clues by ‘searching outside then inside the box’ in a well-defined systematic way. Potential uses in creativity support systems, moral decision support, and technology governance are briefly sketched.</p>