A good metric is one in which close proximity correlates well with the likelihood of being in the same class.
when applying clustering to some task, people typically decide on the clustering quality measure q S (h) they want to optimize, and then chose a specific clustering algorithm A and a distance metric d to generate a 'good' partition function h. it is clear that without any supervision, the resulting function is not guaranteed to agree with the target function p (or one's original intention).